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The Projects Physics Course 

Text and Handbook 




Project Physics Course 

Text and Handbook 


UNIT I Concepts of Motion 

HfHaiiBi Published by 

A Component of the UhBH HOLT, RINEHART and WINSTON, Inc. 
Project Physics Course ISQl New York, Toronto 

The following is a partial list of the 
contributors whose creative assistance 
fashioned the basis for the Project 
Physics Course material (the affilia- 
tions indicated are those just prior to 
or during their association with the 

Directors of Harvard Project Physics 

F. James Rutherford, Capuchino High School, San 

Bruno, Calif. 
Gerald Holton, Dept. of Physics, Harvard 

Fletcher G. Watson, Harvard Graduate School of 


Special Consultant 
to Project Physics 

Andrew Ahlgren, Harvard Graduate School of 

Advisory Committee 

E. G. Begle, Stanford University, Calif. 

Paul F. Brandwein, Harcourt, Brace & World. 

Inc., San Francisco, Calif. 
Robert Brode, University of California, Berkeley 
Erwin Hiebert, University of Wisconsin, Madison 
Harry Kelly, North Carolina State College, Raleigh 
William C. Kelly, National Research Council, 

Washington, D.C. 
PhUippe LeCorbeiller, New School for Social 

Research, New York, N.Y. 
Thomas Miner, Garden City High School, New 

York, N.Y. 
Philip Morrison, Massachusetts Institute of 

Technology, Cambridge 
Ernest Nagel, Columbia University, New York, 

Leonard K. Nash, Harvard University 
I. I. Rabi, Columbia University, New York. N.Y. 

This Text-Handbook is one of the many instruc- 
tional materials developed for the Project Physics 
Course, including texts, laboratory experiments, 
films, and teacher guides. Development of the 
course has profited from the help of many col- 
leagues listed at the front of the text units. 

Copyright ® 1970, Project Physics 
01234 58 987654321 

Cover Photograph, 

Dr. Harold E. Edgerton, M.I.T. 

Staff and Consultants 

L. K. Akers, Oak Ridge Associated Universities, 

Roger A. Albrecht. Osage Community Schools, 

David Anderson, Oberlin College, Ohio 
Gary Anderson, Harvard University 
Donald Armstrong, American Science Film 

Association, Washington, D.C. 
Arnold Arons. University of Washington 
Sam Ascher, Henry Ford High School. Detroit, 

Ralph Atherton, Talawanda High School, Oxford, 

Albert V. Baez, UNESCO, Paris 
William G. Banick. Fulton High School, Atlanta. 

Arthur Bardige, Nova High School. Fort 

Lauderdale, Fla. 
Rolland B. Bartholomew, Henry M. Gunn High 

School, Palo Alto. Calif. 
O. Theodor Benfey. Earlham College. Richmond. 

Richard Berendzen. Harvard College Observatory 
Alfred M. Bork, Reed College, Portland, Ore. 
F. David Boulanger, Mercer Island High School. 

Alfred Brenner. Harvard University 
Robert Bridgham. Harvard University 
Richard Brinckerhoff. Phillips Exeter Academy, 

Exeter. N.H. 

Donald Brittain, National Film Board of Canada, 

Joan Bromberg, Harvard University 
Vinson Bronson, Newton South High School, 

Newton Centre, Mass. 
Stephen G. Brush, Lawrence Radiation Laboratory. 

University of California, Livermore 
Michael Butler, CIASA Films Mundiales, S. A., 

Leon Callihan, St. Mark's School of Texas, Dallas 
Douglas Campbell, Harvard University 
Dean R. Casperson, Harvard University 
Bobby Chambers, Oak Ridge Associated 

Universities, Tenn. 
Robert Chesley, Thacher School, Ojai, Calif. 
John Christensen, Oak Ridge Associated 

Universities, Tenn. 
Dora Clark, W. G. Enloe High School, Raleigh, 

David Clarke, Browne and Nichols School, 

Cambridge, Mass. 
Robert S. Cohen, Boston University, Mass. 

Brother Columban Francis, F.S.C., Mater Christi 

Diocesan High School, Long Island City, N.Y. 
Arthur Compton, Phillips Exeter Academy, 

Exeter, N.H. 
David L. Cove, Los Altos High School, Calif. 
William Cooley, University of Pittsburgh, Pa. 
Ann Couch, Harvard University 
Paul Cowan, Hardin-Simmons University, 

Abilene, Tex. 
Charles Davis, Fairfax County School Board, 

Fairfax, Va. 
Michael Dentamaro, Senn High School, Chicago, 

Raymond Dittman, Newton High School, Mass. 
Elsa Dorfman, Educational Services Inc., 

Watertown, Mass. 
Vadim Drozin, Bucknell University, Lewisburg, 

Neil F. Dunn, Burlington High School, Mass. 
R. T. Ellickson, University of Oregon, Eugene 
Thomas Embry, Nova High School, Fort 

Lauderdale, Fla. 
Walter Eppenstein, Rensselaer Polytechnic 

Institute, Troy, N.Y. 
Herman Epstein, Brandeis University, Waltham, 

Thomas F. B. Ferguson, National Film Board of 

Canada, Montreal 
Thomas von Foerster, Harvard University 
Kenneth Ford, University of California, Irvine 
Robert Gardner, Harvard University 
Fred Geis, Jr., Harvard University 
Nicholas J. Georgis, Staples High School, 

Westport, Conn. 
H. Richard Gerfin, Somers Middle School, 

Somers, N.Y. 
Owen Gingerich, Smithsonian Astrophysical 

Observatory, Cambridge, Mass. 

Stanley Goldberg, Antioch College, Yellow Springs, 

Leon Goutevenier, Paul D. Schreiber High School, 

Port Washington, N.Y. 
Albert Gregory, Harvard University 
Julie A. Goetze, Weeks Jr. High School, Newton, 

Robert D. Haas, Clairemont High School, San 

Diego, Calif. 
Walter G. Hagenbuch, Plymouth-Whitemarsh 

Senior High School, Plymouth Meeting, Pa. 
John Harris, National Physical Laboratory of 

Israel, Jerusalem 
Jay Hauben, Harvard University 
Peter Heller, Brandeis University, Waltham, Mass. 
Robert K. Henrich, Kennewick High School, 

Ervin H. HofFart, Raytheon Education Co., Boston 
Banesh Hoffmann, Queens College, Flushing, N.Y. 
Elisha R. Huggins, Dartmouth College, Hanover, 

Lloyd Ingraham, Grant High School, Portland, 

John Jared, John Rennie High School, Pointe 

Claire, Quebec 
Harald Jensen, Lake Forest College, 111. 
John C. Johnson, Worcester Polytechnic Institute, 

Kenneth J. Jones, Harvard University 
LeRoy Kallemeyn, Benson High School, Omaha, 

Irving Kaplan, Massachusetts Institute of 

Technology, Cambridge 
Benjamin Karp, South Philadelphia High School, 

Robert Katz, Kansas State University, Manhattan, 

Harry H. Kemp, Logan High School, Utah 
Ashok Khosla, Harvard University 
John Kemeny, National Film Board of Canada, 

Merritt E. Kimball, Capuchino High School, San 

Bruno, Calif. 
Walter D. Knight, University of California, 

Donald Kreuter, Brooklyn Technical High School, 

Karol A. Kunysz, Laguna Beach High School, 

Douglas M. Lapp, Harvard University 
Leo Lavatelli, University of Illinois, Urbana 
Joan Laws, American Academy of Arts and 

Sciences, Boston 
Alfred Leitner, Michigan State University, East 

Robert B. LUlich, Solon High School, Ohio 
James Lindblad, Lowell High School, Whittier, 

Noel C. Little, Bowdoin College, Brunswick, Me. 
Arthur L. Loeb, Ledgemont Laboratory, Lexington, 


Richard T. Mara, Gettysburg College, Pa. 
Robert H. Maybury, UNESCO, Paris 
John McClain, University of Beirut, Lebanon 
E. Wesley McNair, W. Charlotte High School, 

Charlotte, N.C. 
William K. Mehlbach, Wheat Ridge High School, 

Priya N. Mehta, Harvard University 
Glen Mervyn, West Vancouver Secondary School, 

B.C., Canada 
Franklin Miller, Jr., Kenyon College, Gambler, 

Jack C. Miller, Pomona College, Claremont, Calif. 
Kent D. Miller, Claremont High School, Calif. 
James A. Minstrell, Mercer Island High School, 

James F. Moore, Canton High School, Mass. 
Robert H. Mosteller, Princeton High School, 

Cincinnati, Ohio 
William Naison, Jamaica High School, N.Y. 
Henry Nelson, Berkeley High School, Calif. 
Joseph D. Novak, Purdue University, Lafayette, 

Thorir Olafsson, Menntaskolinn Ad, Laugarvatni, 

Jay Orear, Cornell University, Ithaca, N.Y. 
Paul O'Toole, Dorchester High School, Mass. 
Costas Papaliolios, Harvard University 
Jacques Parent, National Film Board of Canada, 

Eugene A. Platten, San Diego High School, Calif. 
L. Eugene Poorman, University High School, 

Bloomington, Ind. 
Gloria Poulos, Harvard University 
Herbert Priestley, Knox College, Galesburg, 111. 
Edward M. Purcell, Harvard University 
Gerald M. Rees, Ann Arbor High School, Mich. 
James M. Reid, J. W. Sexton High School, 

Lansing, Mich. 
Robert Resnick, Rensselaer Polytechnic Institute, 

Troy, N.Y. 
Paul I. Richards, Technical Operations, Inc., 

Burlington, Mass. 
John Rigden, Eastern Nazarene College, Quincy, 

Thomas J. Ritzinger, Rice Lake High School, Wise. 
Nickerson Rogers, The Loomis School, Windsor, 

Sidney Rosen, University of Illinois, Urbana 
John J. Rosenbaum, Livermore High School, 

William Rosenfeld, Smith College, Northampton, 

Arthur Rothman, State University of New York, 

Daniel Rufolo, Clairemont High School, San 

Diego, Calif. 

Bemhard A. Sachs, Brooklyn Technical High 

School. N.Y. 
Morton L. Schagrin, Denison University, Granville, 

Rudolph Schiller, Valley High School. Las Vegas, 

Myron O. Schneiderwent, Interlochen Arts 

Academy, Mich. 
Guenter Schwarz, Florida State University, 

Sherman D. Sheppard, Oak Ridge High School. 

William E. Shortall, Lansdowne High School. 

Baltimore, Md. 
Devon Showley, Cypress Junior College. Calif. 
William Shurcliff, Cambridge Electron 

Accelerator, Mass. 
George I. Squibb, Harvard University 
Sister M. Suzanne Kelley, O.S.B., Monte Casino 

High School, Tulsa. Okla. 
Sister Mary Christine Martens, Convent of the 

Visitation, St. Paul, Minn. 
Sister M. Helen St. Paul, O.S.F., The Catholic 

High School of Baltimore. Md. 
M. Daniel Smith, Earlham College. Richmond. 

Sam Standring, Santa Fe High School. Santa Fe 

Springs. Calif. 
Albert B. Stewart, Antioch College, Yellow 

Springs, Ohio 
Robert T. Sullivan, Burnt Hills-Ballston Lake 

Central School, N.Y. 
Loyd S. Swenson. University of Houston. Texas 
Thomas E. Thorpe, West High School. Phoenix. 

June Goodfield Toulmin, Nuffield Foundation, 

London. England 
Stephen E. Toulmin. Nuffield Foundation, London, 

Emily H. Van Zee, Harvard University 
Ann Venable, Arthur D. Little, Inc., Cambridge, 

W. O. Viens, Nova High School, Fort Lauderdale, 

Herbert J. Walberg, Harvard University 
Eleanor Webster, Wellesley College. Mass. 
Wayne W. Welch, University of Wisconsin, 

Richard Weller. Harvard University 
Arthur Western, Melbourne High School, Fla. 
Haven Whiteside, University of Maryland, College 

R. Brady Williamson, Massachusetts Institute of 

Technology, Cambridge 
Stephen S. Winter, State University of New York. 


Science is an adventure of the whole human race to learn to live in and perhaps to 
love the universe in which they are. To be a part of it is to understand, to understand 
oneself, to begin to feel that there is a capacity within man far beyond what he felt 
he had, of an infinite extension of human possibilities 

I propose that science be taught at whatever level, from the lowest to the highest, 
in the humanistic way. It should be taught with a certain historical understanding, 
with a certain philosophical understanding, with a social understanding and a 
human understanding in the sense of the biography, the nature of the people who 
made this construction, the triumphs, the trials, the tribulations. 


Nobel Laureate in Physics 


Background The Project Physics Course is based on the ideas and 
research of a national curriculum development project that worked 
in three phases. First, the authors — a high school physics teacher, 
a university physicist, and a professor of science education — 
collaborated to lay out the main goals and topics of a new 
introductory physics course. They worked together from 1962 to 
1964 with financial support from the Carnegie Corporation of New 
York, and the first version of the text was tried out in two schools 
with encouraging results. 

These preliminary results led to the second phase of the 
Project when a series of major grants were obtained from the U.S. 
Office of Education and the National Science Foundation, starting 
in 1964. Invaluable additional financial support was also provided 
by the Ford Foundation, the Alfred P. Sloan Foundation, the 
Carnegie Corporation, and Harvard University. A large number of 
collaborators were brought together from all parts of the nation, 
and the group worked together for over four years under the title 
Harvard Project Physics. At the Project's center, located at 
Harvard University, Cambridge, Massachusetts, the staff and 
consultants included college and high school physics teachers, 
astronomers, chemists, historians and philosophers of science, 
science educators, psychologists, evaluation specialists, engineers, 
film makers, artists and graphic designers. The teachers serving as 
field consultants and the students in the trial classes were also of 
vital importance to the success of Harvard Project Physics. As each 
successive experimental version of the course was developed it was 
tried out in schools throughout the United States and Canada. The 
teachers and students in those schools reported their criticisms and 
suggestions to the staflTin Cambridge. These reports became the 
basis for the next year's revision. The number of participating 

teachers during this period grew from 2 in 1962-63 to over 100 in 
1967-68. In that year over five thousand students participated in a 
large-scale formal research program to evaluate the results 
achieved with the course materials. 

During 1968, the last of the experimental course materials was 
completed. With the culmination of course development and data 
gathering activities, the final phase of Harvard Project Physics got 
under way. During 1968-69 and 1969-70 the work of the Project 
concentrated on developing and conducting special training 
programs for teachers, disseminating information about the course 
to physics teachers, science department heads, school administrators 
and other interested persons, analyzing the large pool of final 
evaluation data and writing a complete report on the results, and 
trying to find out how the course might be reshaped to fit special 

We wish it were possible to list in detail the contributions of 
each person who participated in some part of Harvard Project 
Physics. Unhappily it is not feasible, since more staff members 
worked on a variety of materials and had multiple responsibilities. 
Furthermore, every text chapter, experiment, piece of apparatus, 
film or other item in the experimental program benefitted from the 
contributions of a great many people. On the preceding pages is a 
partial list of contributors to Harvard Project Physics. There were, 
in fact, many other contributors too numerous to mention. These 
include school administrators in participating schools, directors 
and staff members of training institutes for teachers, teachers who 
tried the course after the evaluation year, and most of all the 
thousands of students who not only agreed to take the experimental 
version of the course, but who were also willing to appraise it 
critically and contribute their opinions and suggestions. 

Aims. From the beginning Harvard Project Physics had three 
major goals in mind. These were to design a humanistically oriented 
physics course, to attract more students to the study of introductory 
physics, and to find out more about the factors that influence the 
learning of science in schools. The last of these involved extensive 
educational research, and has now been reported to the teaching 
profession in books and journals. 

About ten years ago it became clear that a new physics course, 
having far wider appeal than the existing ones, was needed. 
Students who plan to go to college to study the humanities or 
social sciences, those already intent on scientific careers, and those 
who may not wish to go to college at all, can all benefit from a good 
introductory physics course. The challenge facing Harvard Project 
Physics was to design a humanistic course that would be useful 
and interesting to students with widely differing skills, backgrounds, 
and career plans. In practice, this meant designing a course that 
would have the following effect: 

1. To help students increase their knowledge of the physical 
world by concentrating on ideas that characterize physics as a 
science at its best, rather than concentrating on isolated bits of 

2. To help students see physics as the wonderfully many-sided 
human activity that it really is. This meant presenting the subject 
in historical and cultural perspective, and showing that the ideas 
of physics have a tradition as well as ways of evolutionary 
adaptation and change. 

3. To increase the opportunity for each student to have 
immediately rewarding experiences in science even while gaining 
the knowledge and skill that will be useful in the long run. 

4. To make it possible for teachers to adapt the course to the 
wide range of interests and abilities of their students. 

5. To take into account the importance of the teacher in the 
educational process, and the vast spectrum of teaching situations 
that prevail. 

How well did Harvard Project Physics meet the challenge? In a 
sense each student who takes this course must answer that 
question himself. It is a pleasure to report, however, that the large- 
scale study of student achievement and student opinion in the 
participating schools throughout the United States and Canada 
showed gratifying results -ranging from the excellent scores on the 
College Entrance Examination Board achievement test in physics 
to the personal satisfaction of individual students. It is clear that 
the diverse array of individual students in the experimental groups 
responded well to the physics content, the humanistic emphasis of 
the course, and to its flexible multimedia course materials. 

The Project Physics Course Today. Using the last of the 
experimental versions of the course developed by Harvard Project 
Physics in 1964-68 as a starting point, and taking into account the 
evaluation results from the try-outs, the three original collaborators 
set out to develop the version suitable for large-scale publication. 
We take particular pleasure in acknowledging the assistance of 
Dr. Andrew Ahlgren of Harvard University. Dr. Ahlgren was 
invaluable because of his skill as a physics teacher, his editorial 
talent, his versatility and energy, and above all, his commitment 
to the goals of Harvard Project Physics. 

We would also especially like to thank Miss Joan Laws whose 
administrative skills, dependability, and thoughtfulness contributed 
so much to our work. The publisher. Holt, Rinehart and Winston, 
Inc. of New York, provided the coordination, editorial support, and 
general backing necessary to the large undertaking of preparing the 
final version of all components of the Project Physics Course, 
including texts, laboratory apparatus, films, etc. Damon, located in 
Needham, Massachusetts, worked closely with us to improve the 
engineering design of the laboratory apparatus and to see that it 
was properly integrated into the program. 

Since their last use in experimental form, all of the instruc- 
tional materials have been more closely integrated and rewritten in 
final form. The course now consists of a large variety of coordinated 
learning materials of which this textbook is only one; in addition 
there are readers, handbooks, programmed instruction booklets, 
film loops, documentary films, transparencies, apparatus and various 
materials for teachers. With the aid of these materials and the 
guidance of your teacher, with your own interest and effort, you can 
look forward to a successful and worthwhile experience. 

In the years ahead, the learning materials of the Project Physics 
Course will be revised as often as is necessary to remove remaining 
ambiguities, clarify instructions, and to continue to make the 
materials more interesting and relevant to students. We therefore 
urge all students and teachers who use this course to send to us 
(in care of Holt, Rinehart and Winston, Inc., 383 Madison Avenue, 
New York, New York 10017) any criticisms or suggestions they may 
have. And now — welcome to the study of physics! 



Prologue 1 

Chapter 1 : The Language of Motion 

The motion of things 9 

A motion experiment that does not quite work 1 1 

A better experiment 1 3 

Leshe's "50" and the meaning of average speed 15 

Graphing motion and finding the slope 18 

Time out for a warning 21 

Instantaneous speed 23 

Acceleration— by comparison 28 

Chapter 2: Free Fall— Galileo Describes Motion 

The Aristotelian theory of motion 37 

Galileo and his time 43 

Galileo's Two New Sciences 43 

Why study the motion of freely falling bodies? 47 

GalUeo chooses a definition of uniform acceleration 47 

Galileo cannot test his hypothesis directly 49 

Looking for logical consequences of Galileo's hypothesis 50 

Galileo turns to an indirect test 53 

Doubts about Galileo's procedure 56 

Consequences of Galileo's work on motion 57 

Chapter 3: The Birth of Dynamics— Newton Explains Motion 

Explanation and the laws of motion 67 
The Aristotelian explanation of motion 69 
Forces in equilibrium 70 
About vectors 73 
Newton's first law of motion 75 
The significance of the first law 78 
Newton's second law of motion 79 
Mass, weight, and free fall 83 
Newton's third law of motion 86 
Using Newton's laws of motion 88 
Nature's basic forces 90 
Chapter 4: Understanding Motion 

A trip to the moon 99 

Projectile motion 101 

What is the path of a projectile? 103 

Moving frames of reference 105 

Circular motion 107 

Centripetal acceleration and centripetal force 109 

The motion of earth satellites 113 

What about other motions? 116 

Epilogue 118 

Contents Handbook Section 127 

Index 193 

Answers to End of Section Questions 197 

Brief Answers to Study Guide 199 

i t ^« -{ • If: 'M'Ji 


— --.- * T 1 





y <;^W1 ^^S5*^wi^^^^^V'— " linne" 


T~ ^""^ i'^' 


Physicist Enrico Fermi (1901-1954) 
at different stages of his career in 
Italy and America. Mrs. Laura Fermi 
is shown in the photograph at the top 
left of the page. 



Concepts of Motion 


1 The Language of Motion 

2 Free Fall -Galileo Describes Motion 

3 The Birth of Dynamics -Newton Explains Motion 

4 Understanding Motion 

PROLOGUE It is January 1934, a dreary month in the city of Paris. A 
husband and wife, working in a university laboratory, are exposing a 
piece of ordinary aluminunn to a stream of tiny charged bits of matter 
called alpha particles. Stated so simply, this certainly does not sound like 
a momentous event. But let us look more closely, for it is momentous 

Never mind the technical details. Don't let them get in the way of 
the story. It all began as something of a family affair. The husband and 
wife are the French physicists Frederic Joliot and Irene Curie. The alpha 
particles they are using in their experiment are shooting out of a piece 
of naturally radioactive metal, polonium, discovered 36 years before 
by Irene's parents, Pierre and Marie Curie, the famous discoverers of 
radium. What Frederic and Irene have found is that when the aluminum 
is bombarded by alpha particles, the commonplace bit of material 
becomes radioactive for a while. 

This is a surprise. Until this moment, nothing like this-a familiar, 
everyday substance becoming artificially radioactive — has ever been 
observed. But physicists in the laboratory cannot force new phenomena 
on nature, they can only show more clearly what nature is like. We 
know now that this sort of thing is a frequent occurrence. It happens, 
for example, in stars and in our atmosphere when it is bombarded by 
cosmic rays. 

The news was exciting to scientists and traveled rapidly, though it 
made few, if any, newspaper headlines. Enrico Fermi, a young physicist 
on the staff of the University of Rome, became intrigued by the 
possibility of repeating the experiment of Frederic and Irene- repeating 
it with one significant alteration. The story is told in the book Atoms in 
the Family by Enrico Fermi's wife, Laura. She writes: 

... he decided he would try to produce artificial radioactivity 
with neutrons [instead of alpha particles]. Having no electric 

The Language of Motion 

charge, neutrons are neither attracted by electrons nor 
repelled by nuclei; their path inside matter is much longer 
than that of alpha particles; their speed and energy remain 
higher; their chances of hitting a nucleus with full impact are 
much greater. 

All quotations in the Prologue are 
from Laura Fermi, Atoms in the 
Family: IVIy Life Witt) Enrico Fermi, 
University of Chicago Press, Chicago, 
1954 (available as a paperback 
book in the Phoenix Books series). 
Fermi was one of the major 
physicists of the twentieth century. 

Usually a physicist has some theory to guide him in setting up an 
experiment. This time, no good theory had yet been developed. Only 
through actual experiment could one tell whether or not neutrons would 
be good projectiles for triggering artificial radioactivity in the target 
nuclei. Therefore, Fermi, already an outstanding theoretical physicist 
at the age of 33, decided to design some experiments that could settle 
the issue. His first task was to obtain instruments suitable for detecting 
the particles emitted by radioactive materials. The best such laboratory 
instruments by far were Geiger counters, but in 1934 Geiger counters 
were still relatively new and not readily available. Therefore, Fermi built 
his own. 

The counters were soon in operation detecting the radiation from 
radioactive materials. But Fermi also needed a source of neutrons. This 
he made by enclosing beryllium powder and the radioactive gas radon 
in a glass tube. Alpha particles from the radon, striking the beryllium, 
caused it to emit neutrons, which passed freely through the glass tube. 

Now Enrico was ready for the first experiments. Being a 
man of method, he did not start by bombarding substances 
at random, but proceeded in order, starting from the lightest 
element, hydrogen, and following the periodic table of 
elements. Hydrogen gave no results; when he bombarded 
water with neutrons, nothing happened. He tried lithium next, 
but again without luck. He went on to beryllium, then to 
boron, to carbon, to nitrogen. None were activated. Enrico 
wavered, discouraged, and was on the point of giving up his 
researches, but his stubbornness made him refuse to yield. He 
would try one more element. That oxygen would not become 
radioactive he knew already, for his first bombardment had 
been on water. So he irradiated fluorine. Hurrah! He was 
rewarded. Fluorine was strongly activated, and so were other 
elements that came after fluorine in the periodic table. 

This field of investigation appeared so fruitful that 
Enrico not only enlisted the help of Emilio Segre and of 
Edoardo Amaldi but felt justified in sending a cable to Rasetti 
[a colleague who had gone abroad], to inform him of the 
experiments and to advise him to come home at once. A 
short while later a chemist, Oscar D'Agostino, joined the 
group, and systematic investigation was carried on at a fast 

With the help of his colleagues, Fermi's work at the laboratory was 
pursued with high spirit, as Laura Fermi's account shows: 

Follow the story rather than worrying 
about the techniques of the 

. . . Irradiated substances were tested for radioactivity with 
Geiger counters. The radiation emitted by the neutron source 
would have disturbed the measurements had it reached the 


counters. Therefore, the room where substances were 
irradiated and the room with the counters were at the two 
ends of a long corridor. 

Sometimes the radioactivity produced in an element was 
of short duration, and after less than a minute it could no 
longer be detected. Then haste was essential, and the time to 
cover the length of the corridor had to be reduced by swift 
running. Amaldi and Fermi prided themselves on being the 
fastest runners, and theirs was the task of speeding short- 
lived substances from one end of the corridor to the other. 
They always raced, and Enrico claims that he could run 
faster than Edoardo. . . . 

And then, on the morning of October 22, 1934, a fateful discovery 
was made. Two of Fermi's co-workers were irradiating a hollow 
cylinder of silver with neutrons from a source placed at the center of the 
cylinder, to make it artificially radioactive. They found that the amount 
of radioactivity induced in the silver depended on other objects that 
happened to be present in the room! 

. . . The objects around the cylinder seemed to influence its 
activity. If the cylinder had been on a wooden table while being 
irradiated, its activity was greater than if it had been on a 
piece of metal. 

By now the whole group's interest has been aroused, and 
everybody was participating in the work. They placed the 
neutron source outside the cylinder and interposed objects 
between them. A plate of lead made the activity increase 
slightly. Lead is a heavy substance. "Let's try a light one 
next," Fermi said, "for instance, paraffin. " [The most plentiful 
element in paraffin is hydrogen.] The experiment with 
paraffin was performed on the morning of October 22. 

They took a big block of paraffin, dug a cavity in it, put 
the neutron source inside the cavity, irradiated the silver 
cylinder, and brought it to a Geiger counter to measure its 
activity. The counter clicked madly. The halls of the physics 
building resounded with loud exclamations: "Fantastic! 
Incredible! Black Magic! " Parrafin increased the artificially 
induced radioactivity of silver up to one hundred times. 

By the time Fermi came back from lunch, he had already formulated 
a theory to account for the strange action of paraffin. 

Paraffin contains a great deal of hydrogen. Hydrogen 
nuclei are protons, particles having the same mass as 
neutrons. When the source is enclosed in a paraffin block, the 
neutrons hit the protons in the paraffin before reaching the 
silver nuclei. In the collision with a proton, a neutron loses 
part of its energy, in the same manner as a billiard ball is 
slowed down when it hits a ball of its same size [whereas it 
loses little speed if it is reflected off a much heavier ball, or a 
solid wall]. Before emerging from the paraffin, a neutron will 
have collided with many protons in succession, and its 
velocity will be greatly reduced. This slow neutron will have 

tjtufron Source 

Silver cilindir- 

paraffin block 

Because of Fermi's earlier 
experiments, they knew the water 
would not become artifically 
radioactive. However, they now 
reasoned that it would slow down 
neutrons and so allow silver to 
become more strongly radioactive. 

The Language of Motion 


^^*^"*^ V f^^n 






The same process by which neutrons 
were slowed down in the fountain is 
used in todays large nuclear reactors. 
An example is the "pool" research 
reactor pictured above. 

a much better chance of being captured by a silver nucleus 
than a fast one, much as a slow golf ball has a better chance 
of making a hole than one which zooms fast and may 
bypass it. 

If Enrico's explanations were correct, any other 
substance containing a large proportion of hydrogen should 
have the same effect as paraffin. "Let's try and see what a 
considerable quantity of water does to the silver activity," 
Enrico said on the same afternoon. 

There was no better place to find a "considerable 
quantity of water " than the goldfish fountain ... in the garden 
behind the laboratory . . . 

In that fountain the physicists had sailed certain small 
toy boats that had suddenly invaded the Italian market. Each 
little craft bore a tiny candle on its deck. When the candles 
were lighted, the boats sped and puffed on the water like 
real motor-boats. They were delightful. And the young men, 
who had never been able to resist the charm of a new toy, 
had spent much time watching them run in the fountain. 

it was natural that, when in need of a considerable 
amount of water, Fermi and his friends should think of that 
fountain. On that afternoon of October 22, they rushed their 
source of neutrons and their silver cylinder to that fountain, 
and they placed both under water. The goldfish, I am sure, 
retained their calm and dignity, despite the neutron shower, 
more than did the crowd outside. The men's excitement was 
fed on the results of this experiment. It confirmed Fermi's 
theory. Water also increased the artificial radioactivity of 
silver by many times. 

This discovery- that slowed-down neutrons can produce much 
stronger effects in the transmutation of certain atoms than can fast 
neutrons- turned out to be a crucial step toward further discoveries 
that, years later, led Fermi and others to the controlled production of 
atomic energy from uranium. 

About this course: We will return to the study of nuclear physics later 
in the course. The reason for presenting a description of Fermi's 
discovery of slow neutrons here was not to instruct you now on the 
details of the nucleus, but to present a quick, almost impressionistic, 
view of scientists in action. Not every discovery in science is made in 
just the way Fermi and his colleagues made this one. Nevertheless, the 
episode does illustrate many of the major themes or characteristics of 
modern science— some of which are discussed below. Look for these 
themes as you read through this course; you will find them appearing 
over and over again in many varied situations. 

Progress in science over the years is the result of the work of many 
people in many lands — whether working alone, in pairs or small groups, 
or in large research teams. No matter how different the individual way 
of working, no matter where he works, each scientist expects to share 
his ideas and results with other scientists who will try independently to 
confirm and add to his findings. As important as such cooperation is, 
the most essential ingredient of science is individual thought and 


Fermi and his associates showed stubborn perseverance in the face 
of discouraging results, innagination in the invention of theories and 
experiments, alertness to the appearance of unexpected results, 
resourcefulness in exploiting the material resources at hand, and joy in 
finding out something new and important. Traits we usually think of as 
being distinctly humane are of value in pursuing scientific work no less 
than elsewhere in life. 

Scientists build on what has been found out and reported by other 
scientists in the past. Yet, every advance in science raises new scientific 
questions. The work of science is not to produce some day a finished 
book that can be regarded as closed once and for all, but to carry 
investigation and imagination on into fields whose importance and 
interest had not been realized before. 

Some work in science depends upon painstaking observation and 
measurement, which can sometimes stimulate new ideas and sometimes 
reveals the need to change or even completely discard existing theories. 
Measurement itself, however, is usually guided by a theory. One does 
not gather data just for their own sake. 

All these are characteristics of science as a whole and not of 
physics alone. This being a physics text, you may well wish to ask, "Yes, 
but just what is physics?" The question is fair enough, yet there is no 
simple answer. Physics can be thought of as an organized body of 
tested ideas about the physical world. Information about this world is 
accumulating ever more rapidly; the great achievement of physics has 
been to find a fairly small number of basic principles which help to 
organize and to make sense of certain parts of this flood of information. 
This course will deal with some, but not nearly all, of the ideas that 
together make up the content of physics. The purpose of this course is 
to provide you with the opportunity to become familiar with some of 
these ideas, to witness their birth and development, and to share in the 
pleasure that comes from using them to view the world in a new light. 

Physics is more than just a body of laws and an accumulation of 
facts. Physics is what each physicist does in his own way: It is a 
continuing activity- a process of search that sometimes leads to 
discovery. Look in on different physicists at work and you will see 
differences in problems being studied, in apparatus being used, in 
individual style, and in much more. Fermi has provided us with one 
example, but as the course proceeds, we will encounter other, sometimes 
very different examples. By the end of this course, you will have dealt 
with many of the ideas and activities which together comprise physics. 
You will not just have learned about it-you will have actually done 
some physics. 

Science gives us no final answers. But it has come upon wondrous 
things, and some of them may renew our childhood delight in the 
miracle that is within us and around us. Take, for example, so basic a 
thing as size ... or time. 

The Project Physics Course has 
made two documentary films that you 
might like to see. One is called The 
World of Enrico Fermi and includes 
the discovery described here. The 
other is entitled People and Particles 
and shows what it is like to be 
working now on a research problem 
in elementary particle physics. 

The Language of Motion 

Our place in space 

Physics deals with those laws of the universe that apply 
everywhere -fronn the largest to the smallest. 


Distance to the furthest observed galaxy 

Distance to the nearest galaxy 

Distance to the nearest star 

Distance to the sun 

Diameter of the earth 

One mile 

Human height 

Finger breadth 

Paper thickness 

Large bacteria 

Small virus 

Diameter of atom 

Diameter of nucleus 

10-*^ meters 












A globular star cluster 

The estimated size of the universe 
now is of the order of 100 million, 
million, million, million times a man's 
height (man's height x 10.000,000, 

The smallest known constituent 
units of the universe are less in size 
than a hundreth of a millionth of a 
millionth of a man's height (mans 
height x 0.000,000,000,000,01). 


Our place in time 

Physicists study phenomena in the extremes of time-space 
and the whole region between the longest and shortest. 


Age of universe 
Precession of the earth's axis 
Human life span 
One year 
One day 
Light from sun to earth 
Time between heartbeats 
One beat of fly's wings 
Duration of strobe flash 
Short laser pulse 
Time for light to cross an atom 
Shortest-lived subatomic particles 

10'^ seconds 





Particle tracks in a bubble chamber 

Fossilized trilobites 


The history of the universe has 
been traced back as far into the past 
as a hundred million times the length 
of a man's life (man's life x 100,000, 

Events have been recorded that 
last only a few millionths of a millionth 
of a millionth of a millionth of a 
man's heartbeat (man's heartbeat x 

It is hard to resist the temptation to say more about these intriguing 
extremes; however, this is not where physics started. Physics started 
with the human-sized world-the world of horse-drawn chariots, of 
falling rain, and of flying arrows. It is with the physics of phenomena on 
this scale that we shall begin. 

1.1 The motion of things 

1.2 A motion experiment that does not quite work 

1.3 A better experiment 

1.4 Leslie's "50" and the meaning of average speed 

1.5 Graphing motion and finding the slope 

1.6 Time out for a warning 

1.7 Instantaneous speed 

1.8 Acceleration -by comparison 


1.1 The motion of things 


The Language of Motion 

The world is filled with things in motion: things as small as 
dust and as large as galaxies, all continually moving. Your textbook 
may seem to be lying quietly on the desk, but each of its atoms is 
incessantly vibrating. The "still" air around it consists of molecules 
tumbling chaotically, at various speeds, most of them moving as fast 
as rifle bullets. Light beams dart constantly through the room, 
covering the distance from wall to wall in about a hundred-millionth 
of a second, and making about ten million vibrations during that 
time. Even the whole earth, our majestic spaceship, is moving at 
about 18 miles per second around the sun. 

There is a very old maxim: "To be ignorant of motion is to be 
ignorant of nature." Of course we cannot investigate all motions. 
So, from this swirling, whirling, vibrating world of ours let us choose 
just one moving object for our attention, something interesting and 
typical, but above all, something manageable. Then let us describe 
its motion. 

But where shall we start? A machine, such as a rocket or a car? 
Though made and controlled by man, they or their parts move in 
fast and complicated ways. We really ought to start with something 
simpler and slower, something that our eyes can follow in detail. 
Then how about a bird in flight? Or a leaf falling from a tree? 

Surely, in all of nature there is no motion more ordinary than 
that of a leaf fluttering down from a branch. Can we describe how 
it falls or explain why it falls? As we think about it we quickly 
realize that, while the motion may be "natural," it is very 
complicated. The leaf twists and turns, sails to the right and left, 

Study for "Dynamism of a Cyclist" 
(1913) by Umberto Boccioni. Courtesy 
Yale University Art Gallery. 


' y 

Section 1.2 


back and forth, as it floats down. Even a motion as ordinary as this 
may turn out, on closer examination, to be more complicated than 
the motion of machines. And even if we could describe it in detail, 
what would we gain? No two leaves fall in quite the same way; 
therefore, each leaf would seem to require its own detailed 
description. Indeed, this individuality is typical of most events 
occurring spontaneously on earth. 

And so we are faced with a dilemma. We want to describe 
motion, but the motions we encounter under ordinary circumstances 
appear too complex. What shall we do? The answer is that we 
should go, at least for a while, into the physics laboratory -because 
the laboratory is the place to separate the simple ingredients that 
make up all complex natural phenomena and to make those 
phenomena more easily visible to our limited human senses. 

1.2 A motion experiment that does not quite worit 

A billiard ball, hit squarely in the center, speeds easily across a 
tabletop in a straight line. An even simpler motion (simpler because 
there is no rolling) can be obtained in this way: Take a disk of what 
is called "dry ice" (really frozen carbon dioxide), put it on a 
smooth floor, and give it a gentle push. It will move slowly and 
with very little friction, supported on its own vapor. We did this in 
front of a camera to get a photograph that would "freeze" the action 
for easier measurement later. While the dry ice disk was moving, 

Laboratory setup 

Time exposure of the disk in motion 

Close-up of 
a dry ice disk 


The Language of Motion 

The speed of an object, is, of course, 
how fast it moves from one place 
to another. A more formal way to 
say the same thing is: Speed is the 
time rate of change of position. 

From time to time you will be 
referred to items in the Study Guide, 
a few pages found at the end of 
each chapter. Usually the letters SG 
plus a number will indicate this. See 
SG 1.1 on page 31 for more 
information on how to study for this 
course and on the use of the Study 

the shutter of the camera was kept open; the resulting time- 
exposure shows the path taken by the disk. 

What can we learn about the disk's motion by examining the 
photographic record? Our question is easy enough to answer: as 
nearly as we can judge by placing a ruler on the photograph, the 
disk moved in a straight line. This is a very useful result, and we 
shall see later that it is really quite surprising. It shows how 
simplified the laboratory can be: the kinds of motion one ordinarily 
sees are almost never that simple. But did it move steadUy, or did it 
slow down? From this photograph we cannot tell. Let us improve 
our experiment. Before we do so, however, we must be clear on 
just how we might expect to measure the speed. 

Why not use something like an automobile speedometer? A 
speedometer is supposed to tell us directly the speed at which the 
car is moving at any time. Everyone knows how to read that most 
popular of all meters, even though few of us have a clear notion of 
how it works. Think of how speeds are expressed. We say, for 
example, that a car is moving at 60 miles per hour. This means 
that if the car continues to move with the same speed it had at the 
instant the speed reading was taken, the car would move a distance 
of 60 miles in a time interval of 1.0 hour. Or we could say that the 
car would move 1.0 mile in 1/60 of an hour, or 6.0 miles in 1/10 of 
an hour — or any distance and time intervals for which the ratio of 
distance to time is 60 miles per hour. 

Unfortunately, an automobile speedometer cannot be hooked to 
a disk of dry ice, or to a bullet, or to many other objects whose 
speed we might wish to measure. (See SG 1.2.) However, there is a 
way to measure speeds in most cases that would interest us. 

As a clue, remember what you would have to do if the 
speedometer in your car were broken and you still wanted to know 
your speed as you moved along a turnpike. You would do one of two 
things (the result is the same in either case): you would count the 
number of mile markers passed in one hour (or some fraction of it) 
and find the average speed by getting the ratio of miles and hours; 
or, you would determine the fraction of an hour it takes to go from 
one mile marker to the next (or to another marker a known number 
of miles away) and find again the average speed as a ratio of miles 
to hours. 

Either method gives, of course, only the average speed for the 
interval during which speed is measured. That is not the same as 
the speed at any given instant, which a speedometer registers, but 
it is good enough for a start. After we get average speeds clear, we 
shall see a simple way of getting instantaneous speeds. 

Therefore, to find the speed of an object, we measure the 
distance it moves and the time it takes to move that distance. Then 
we divide the distance by the time, and the speed comes out in 
miles per hour, or feet per second, or meters per second, depending 
upon the units used to measure distance and time. With this plan of 
attack, we return to the experiment with the dry ice disk. Our task 
now is to find the speed of the disk as it moves along its straight-line 
path. If we can do it for the disk, we can do it for many other 
objects as well. 

Section 1.3 


There will usually be one or more brief questions at the end of 
each section in a text chapter. Ql below is the first. Use these to 
check on your own progress. Answer the questions before continuing 
to the next section. Check your answers to the end-of- section 
questions at the back of this book (page 197); whenever you 
find you did not get the correct answers, study through the section 
again. And of course, if anything is still unclear after you have 
tried to study it on your own or together with other students, then 
ask your teacher! 

Ql Why is it not possible to determine the speed of the dry ice 
puck in the time-exposure photograph on page 11? 

1.3 A better experiment 

To find speed, we need to be able to measure both distance and 
time. So let's repeat the experiment with the dry ice disk after first 
placing a meter stick (100 cm) on the table parallel to the expected 
path of the disk. This is the photograph we obtain: 

We now have a way of measuring the distance traveled by the 
disk, but we still need a way to measure the time it takes the disk 
to travel a given distance. 

This can be done in various ways but here is a fine trick that 
you can try in the laboratory. The camera shutter is again kept open 
and everything else is the same as before, except that the only 
source of light in the darkened room comes from a stroboscopic 
lamp. This lamp produces bright flashes of light at a frequency 
which can be set as we please. Since each pulse or flash of light 
lasts for only about 10 millionths of a second (10 microseconds), the 
moving disk appears in a series of separate, sharp exposures, rather 
than as a continuous blur. The photograph below was made by 
using such a stroboscopic lamp flashing 10 times a second, after the 
disk had been gently pushed as before. 


The Language of Motion 

See the articles "Motion in Words" 
and "Representation of Movement" 
in Project Physics Reader 1. 

Now we're getting somewhere. Our special setup enables us to 
record accurately a series of positions of the moving object. The 
meter stick helps us to measure the distance moved by the front 
edge of the disk between successive light flashes. The time interval 
between images is, of course, equal to the time interval between 
stroboscopic lamp flashes (which is 0.10 second in these photos). 

We can now determine the speed of the disk at the beginning 
and end of its photographed path. The front edge of the first clear 
image of the disk at the left is 6 cm from the zero mark on the 
meter stick. The front edge of the second image from the left is at 
the 19-cm position. The distance traveled during that time was the 
difl"erence between those two positions, or 13 cm. The corresponding 
time interval was 0.01 second. Therefore, the speed at the start must 
have been 13 cm/0.10 sec, or 130 cm/sec. 

Turning now to the two images of the disk farthest to the right 
in the photograph, we find that the distance traveled during 0.10 
sec was 13 cm. Thus the speed at the right end was 13 cm/0.10 sec, 
or 130 cm/sec. 

The disk's motion was not measurably slower at the right end 
than at the left end. Its speed was 130 cm/sec near the beginning 
of the path — and 130 cm/sec near the end of the path. However, 
that does not yet prove that the speed was constant all the way. 
We might well suspect that it was, and you can easily check for 
yourself that this suspicion is justified. Since the time intervals 
between images are equal, the speeds will be equal if the distance 
intervals are equal to one another. Is the distance between images 
always 13 cm? Did the speed stay constant, as far as you can tell 
from the measurements? 

When you think about this result, there is something really 
unusual in it. Cars, planes, and ships do not move in neat, straight 
lines with precisely constant speed even when they go under 
power. Yet this disk did it, coasting along on its own, without 
anything to keep it moving. You might well think it was just a 
rare event and it would not happen again. In any case, you should 
try it. The equipment you will use for this study of physics will 
include cameras, strobe lamps (or mechanical strobes, which work 
just as well), and low-friction disks of one sort or another. Repeat the 
experiment several times at diff"erent initial speeds, and then 
compare your results with those we found above. 

You may have a serious reservation about the experiment. If 
you ask, "How do you know that the disk didn't slow down an 
amount too small to be detected by your measurements?" we can 
only answer that we don't know. All measurements involve some 
uncertainty which one can usually estimate. With a meter stick we 
can measure distances reliably to the nearest 0.1 cm. If we had 
been able to measure to the nearest 0.01 cm or 0.001 cm. we might 
have detected some slowing down. But if we again found no change 
in speed, you could still raise the same objection. There is no way 
out of this. We must simply admit that no physical measurements 
are ever infinitely precise. Thus it is wise to leave open to question 

Section 1.4 


the results of any set of measurements and the findings based on 
them if increased precision could reveal other results. 

Let us briefly review the results of our experiment. We devised 
a way to measure the successive positions of a moving dry ice disk 
at known time intervals. From this we calculated first the distance 
intervals and then the speed between selected positions. We soon 
discovered that (within the limits of accuracy of our measurement) 
the speed did not change. Objects that move in such a manner are 
said to have uniform speed or constant speed. We know now how 
to measure uniform speed. But, of course, actual motions are 
seldom uniform. What about the more usual case of nonuniform 
speed? That is our next concern. 

Q2 Suppose the circles below represent the successive positions 
of a moving object as photographed stroboscopically. Did the object 
move with uniform speed? How do you know? 

o o o o o o 

Q3 Describe uniform speed without referring to dry ice pucks 
and strobe photography or to any particular object or technique of 

Some practice problems dealing 
with constant speed are given in 
Study Guide 1.3 (a, b, c, and d). 

1.4 Leslie's "50" and the meaning of average speed 

Consider the situation at a swimming meet. At the end of each 
race, the name of the winner is announced — the swimmer with the 
shortest time; but since in a given race — say the 100-yard back- 
stroke—every swimmer goes the same distance, the swimmer with 
the shortest time is the one having the highest average speed while 
covering the measured distance. The ratio of the distance traveled 
to the elapsed time is the measure of average speed. This relation- 
ship is expressed in the following equation: 

distance traveled 

average speed 

elapsed time 

What information does a knowledge of the average speed give us? 
We shall answer this question by studying a real example. 

Leslie is not the fastest girl freestyle swimmer in the world, but 
Olympic speed is not necessary for our purposes. One day after 
school, Leslie was timed while swimming two lengths of the 
Cambridge High School pool. The pool is 25.0 yards long, and it took 
her 56.1 seconds to swim the two lengths. Thus her average speed 
over the whole 50-yard distance was 

50.0 yd 

56.1 sec 

= 0.89 yd/sec, or nearly 2.7 ft/sec 

Did Leslie swim the 50 yards at uniform (or constant) speed? If 
not. which length did she cover more quickly? What was her 
greatest speed? her least speed? How fast was she moving when 
she passed the 10-yard, or 18-yard or 45-yard mark? These are 

2.7 ft/sec is the equivalent of 1.8 
miles per hour. No great speed! A 
sailfish can do over 40 mph. But 
man is a land animal. For short 
distances he can run better than 
20 mph. 


The Language of Motion 

useful things to know when training for a meet. But so far we do 
not have a way to answer any of these questions. The value 0.89 
yd/sec probably comes closer than any other one value to describing 
the whole event. 

To compare Leslie's speed at different parts of the swim, we 
must observe the times and distances covered as we did in 
experimenting with the dry ice disk. That is why we arranged the 
event as shown on the photograph below. 

Observers stationed at 5-yard intervals from the mark along 
the length of the pool started their stopwatches when the starting 
signal was given. Each observer had two watches, one which he 
stopped as Leslie passed his mark going down the pool, and another 
which he stopped as she passed on her return trip. The data are 
tabulated in the margin. 
























From these data we can determine Leslie's average speed for 
the first 25 yards and for the last 25 yards separately. 

Average speed fov first 25 yards = 

Average speed for the last 25 yards = 

distance traveled 
elapsed time 
_ 25.0 yards 

22.0 seconds 
= LIO yd/sec 

distance traveled 

elapsed time 
25.0 yards 
56.0 sec - 22.0 sec 
25.0 yd 

34.0 sec 

= 0.735 yd/sec 

It is now clear that Leslie did not swim with uniform speed. She 
swam the first length much faster (1.10 yd/sec) than the second 
length (0.74 yd/sec). Notice that the overall average speed (0.89 
yd/sec) does not describe either lap very well. Here and elsewhere 

Section 1.4 


in our study of motion, the more we refine our measurements to look 
at detail, the more variation we find. 

In a moment we shall continue our analysis of the data we have 
obtained for Leslie's swim — mostly because the concepts we are 
developing here, to discuss this everyday type of motion, will be 
needed later to discuss other motions, ranging from that of planets 
to that of atoms. First, we shall introduce some shorthand notation 
with which the definition of average speed can be simplified from 

average speed = 

distance traveled 

elapsed time 

to the more concise statement that says exactly the same thing: 


In this equation v,„. is the symbol for the average speed. Ad is the 
symbol for change in position, and At is the symbol for an elapsed 
interval of time. The symbol A is the fourth letter in the Greek 
alphabet and is called delta. When A precedes another symbol, it 
means "the change in. . . ." Thus, Ad does not mean "A multiplied 
by d" but rather "the change in d" or "the distance interval." 
Likewise, At stands for "the change in t" or "the time interval." 

We can now go back to the data to see what we can learn about 
Leslie's average speed for each 5-yard interval, from beginning to 
end. This calculation is easily made, especially if we reorganize our 
data as in the table on page 19. The values of v,„. calculated at 
5-yard intervals for the first lap are entered in the right-hand column. 
(The computations for the second lap are left for you to complete.) 

Much more detail is emerging from the picture. Looking at the 
speed column, we see that Leslie's speed was at its greatest right 
near the beginning. Her racing jump into the water gave her extra 
speed at the beginning. In the middle of her first length she was 
swimming at a fairly steady rate, and she slowed down coming into 
the turn. Use your own figures to see what happened after the turn. 

Although we have determined Leslie's speeds at various 
intervals along the path, we are still dealing with average speeds. 
The intervals are smaller — 5 yards rather than the entire 50 — but 
we do not know the details of what happened within any of the 
5-yard intervals. Thus, Leslie's average speed between the 15- and 
20-yard marks was 1.0 yd/sec, but we don't know yet how to 
compute her speed at the very instant and point when she was, say, 
18 yards or 20 yards from the start. Even so, we feel that the average 
speed computed over the 5-yard interval between the 15- and 
20-yard marks is probably a better estimate of her speed as she went 
through the 18-yard mark than is the average speed computed 
over the whole 50 yards, or over either 25-yard length. We shall 
come back to this problem of the determination of "speed at a 
particular instant and point" in Sec. 1.7. 

Q4 Define average speed. 

Practice problems on average speed 
can be found in Study Guide 1.3 
(e, f, g, and h.) Study Guide 1.4, 1.5, 
1.6, and 1.7 offer somewhat more 
challenging problems. Some 
suggestions for average speeds to 
measure are listed in Study Guide 
1.8 and 1.9. An interesting activity 
is suggested in Study Guide 1.10. 


The Language of Motion 

Q5 If you have not already completed the table on page 19, do 
so now before going on to the next section. 

1.5 Graphing motion and finding the slope 

What can we learn about motion by graphing the data rather 
than just tabulating them? Let us find out by preparing a distance- 
versus-time graph, using the data from Leslie's 50-yard swim. As 
shown in the first graph on the next page, all we really know are the 
data points. Each point on the graph shows the time when Leslie 
was at a particular position along her path. In the second graph, 
dotted lines have been drawn to connect the points. We don't 
actually know what the values are between the data points — the 
straight-line connections are just a very simple way of suggesting 
what the overall graph might look like. In fact, the straight lines are 
not likely to be a very good approximation, because the resulting 
broken-line graph would indicate very abrupt changes. If we believe 
that Leslie changed speed only gradually, we can get a better 
approximation by drawing the smoothest curve possible through the 
data points. One experimenter's idea of a smooth curve is shown 
in the last graph. 

Now let us "read" the graph. Notice that the line is steepest at 
the start. This means that there was a comparatively large change 
in position during the first seconds — in other words, she got off to a 
fast start! The steepness of the graph line is an indication of how 
fast she was moving. From 10 yards to 20 yards the line appears to 
be straight, becoming neither more nor less steep. This means that 
her speed in this stretch was constant. Reading the graph further, 
we see that she slowed down noticeably before she reached the 
25-yard mark, but gained in speed right after the turn. The steepness 
decreases gradually from the 30-yard mark to the finish as Leslie 
was slowing down. There was no final spurt over the last 5 yards. 
(She could barely drag herself out of the pool after the trial.) 

Looked at in this way, a graph provides us at a glance with a 
visual representation of motion. But this way of representing 
motion so far does not help us if we want to know actual values of 
her speed at various times. For this, we need a way of measuring 
the steepness of the graph line. Here we can turn to mathematics 
for help, as we often shall. There is an old method in geometry for 
solving just this problem. The steepness of a graph at any point is 
related to the change in the vertical direction (Ai/) and the change 
in the horizontal direction (Ax). By definition, the ratio of these two 
changes (Ay I Ax) is the slope: 


slope = - — 


Slope is a widely-used mathematical concept, and can be used to 
indicate the steepness of a line in any graph. In a distance-time 
graph like the one for Leslie's swim, distance is usually plotted on 

Section 1.5 


Ad At 

0.0 yd 























2.5 sec 

















2.0 i/o/scc 





^ 301 

"I 201 

/0| © 





^% ?0 20 X W 

time (seco/x/^) 

50 60 









^ /D 2D 30 ?D '^ 50 



5 ^ 

^ 201 









10 20 30 ?D 50 60 

time (.5&:onds) 

the vertical axis (d replaces y) and time on the horizontal axis (t 
replaces x). Therefore, in such a graph, the slope of a straight line 
is given by 

slope = 


But this reminds us of the definition of average speed, Va,. = Ad/At. 
Therefore, Va,- = slope! In other words, the slope of any straight-line 
part of a graph of distance versus time gives a measure of the 
average speed of the object during that interval. What we do when 
we measure slope on a graph is basically the same thing that 
highway engineers do when they specify the steepness of a road. 
They simply measure the rise in the road and divide that rise by the 
horizontal distance one must go in order to achieve the rise. The 
only difference between this and what we have done is that the 

Above are shown four ways of repre- 
senting Leslie's swim: a table of 
data, a plot of the data points, broken 
straight-line segments that connect 
the points, and a smooth curve that 
connects the points. 

If this concept is new to you or if 
you wish to review it, turn now to 
Study Guide 1.11 before continuing 


The Language of Motion 

highway engineers are concerned with an actual physical slope: on 
a graph of their data the vertical axis and horizontal axis would 
both show distance. We, on the other hand, are using the 
mathematical concept of slope as a way of expressing distance 
measured against tim.e. 

We can get a numerical value quickly and directly for the slope 
of each straight-line segment in the graph on p. 19. so we will have 
the value of the average speed for each of the 5-yard intervals 
between data points. For example, we used our data table to 
calculate Leslie's average speed between the 5- and 10-yard 
markers as 1.4 yd/sec. She moved 5 yards on the vertical 
(distance) axis during a lapse of 3.5 seconds on the horizontal 
(time) axis. Therefore, the slope of the hne segment connecting the 
5-yard and 10-yard points is equal to 5 yards divided by 3.5 seconds, 
or 1.4 yd/sec. 

The slope, as we have defined it here, is not exactly the same 
thing as the steepness of the line on the graph paper. If we had 
chosen a different scale for either the distance or time axis (making 
the graph, say, twice as tall or twice as wide), then the apparent 
steepness of the entire graph would be different. The slope, however, 
is measured by the ratio of the distance and time units — a Ad of 10 
meters in a At of 5 seconds gives a ratio of 2 meters/second, no 
matter how much space is used for meters and seconds on the 

But the graph is more useful than just leading us back again 
to the values in the table. We can now ask questions that cannot 
be answered directly from the original data: What was Leslie's 
speed 10 seconds after the start? What was her speed as she 
crossed the 37-yard mark? Questions like these can be answered 
by finding the slope of a fairly straight portion of the graph line 
around the point of interest. Two examples are worked out on the 

MiSL_ = 0.7O yVjec 
■1.0 sec "^ 

- 0.S5 i^lscc 

20 30 40 

timt (stconds) 



Section 1.6 


graph at the bottom of page 20. For each example, At was chosen The 4-sec value for f is just for 
to be a 4-sec interval -from 2 sec before the point in question to 2 convenience; some other value 

sec after it; then the Ad for that At was measured. ^°"'^ ^^"^ ^^^" ^,^^^- ^'' ^^ ^°"'^ 

have chosen a value for Ad and then 
The reasonableness of usmg the graph in this way can be measured the corresponding At. 

checked by comparing the results with the values listed in the table 
on p. 19. For example, the speed near the 10-second mark is found 
from the graph to be about 3.0 yd/4.0 sec = 0.75 yd/sec. This is 
somewhat less than the value of 0.9 yd/sec given in the table for 
the average speed between 6 and 11 seconds; and that is just what 
we would expect, because the smooth-curve graph does become 
momentarily less steep around the 10-second point. If the smooth 
curve that was drawn really is a better description of Leslie's 
swimming than the broken line is, then we can get more information 
out of the graph than we put into it. 

Q^ Turn back to p. 13 and draw a distance- time graph for the 
motion of the dry ice disk. 

Which of the two graphs below has the greater slope? 



timt (I toon) 

Q8 Where was Leslie swimming most rapidly? Where was she 
swimming most slowly? 

Q9 From the graph, find Leslie's speed at the 47-yard mark. 
From the table on p. 19, calculate her average speed over the last 
5 yards. How do the two values compare? 

1.6 Time out for a warning 

Graphs are useful — but they can also be misleading. You must 
always be aware of the limitations of any graph you use. The only 
actual data in a graph are the plotted points. There is a limit to the 
precision with which the points can be plotted, and a limit to how 
precisely they can be read from the graph. 

The placement of a line through a series of data points, as in 
the graph on page 19, depends on personal judgment and 
interpretation. The process of estimating values between data 
points is called interpolation. That is essentially what you are 
doing when you draw a line between data points. Even more risky 
than interpolation is extrapolation, where the graph line is 

The Language of Motion 

19 minutes 

17 minutes 

27 minutes 

The Language of Motion 



These photographs show a stormy 
outburst of incandescent gas at the 
edge of the sun, a developing chive 
plant and a glacier. From these pic- 
tures and the time intervals given 
between pictures, you can determine 
the average speeds (1) of the growth 
of the solar flare with respect to the 
sun's surface (radius of sun is about 
432,000 mi), (2) of the growth of one 
of the chive shoots with respect to the 
graph paper behind it (large squares 
are one inch), (3) of the moving glacier 
with respect to its "banks." 

17 hours 

33 hours 

4 years 

Section 1.7 


extended to provide estimated points beyond the known data. 

An example of a high-altitude balloon experiment carried out in 
Lexington, Massachusetts, nicely illustrates the danger of 
extrapolation. A cluster of gas-filled balloons carried cosmic ray 
detectors high above the earth's surface, and from time to time a 
measurement was made of the height of the cluster. The graph on 
the right shows the data for the first hour and a half. After the first 20 
minutes the balloons seem to be rising in a cluster with unchanging 
speed. The average speed can be calculated from the slope: speed 
of ascent = Ad/ At = 27,000 ft/30 min = 900 ft/min. If we were asked 
how high the balloons would be at the very end of the experiment 
(500 min), we might be tempted to extrapolate, either by extending 
the graph or by computing from the speed. In either case we would 
obtain the result 500 min x 900 ft/min = 450,000 ft, which is over 
90 miles high! Would we be right? Turn to Study Guide 1.12 to see 
for yourself. (The point is that mathematical aids, including graphs, 
can be a splendid help, but only within the limits set by physical 

Q10 What is the difference between extrapolation and 

Q11 Which estimate from the graph would you expect to be 
less accurate: Leshe's speed as she crossed the 30-yard mark, or 
her speed at the end of an additional lap? 

Jimt (min) 

SG 1.13 

1.7 Instantaneous speed 

Now let us wrap up the chief lessons of this first chapter. In 
Sec. 1.5 we saw that distance-time graphs could be very helpful in 
describing motion. When we reached the end of the section, we 
were speaking of specific speeds at particular points along the path 
(like "the 14-yard mark") and at particular instants of time (like "the 
instant 10 seconds after the start"). You probably were bothered by 
this manner of talking, since at the same time we admitted that the 
only kind of speed we can actually measure is average speed. To 
find average speed we need a ratio of distance and time intervals. 
A particular point on the path, however, does not have any 
interval. Nevertheless, it makes sense to speak about the speed at a 
point. We will summarize what reasons there are for using "speed" 
in this way, and see how well we can get away with it. 

You remember that our answer to the question (page 20), 
"How fast was Leslie swimming at time t = 10 sec?" was 0.85 yd/sec. 
That answer was obtained by finding the slope of a small portion 
of the curve encompassing the point P when t = 10 sec. That 
section of the curve has been reproduced in the margin here. Notice 
that the part of the curve we used appears to be nearly a straight 
line. As the table under the graph shows, the value of the slope 
for each interval changes very little as we decrease the time interval 
At. Now imagine that we closed in on the point where t = 10 sec 

^ 20 


At Ad 


G.O sec 5A()d 0.90y'^/5C£ 
4.0 3.4 0.85 
2.0 \n 0-95 


The Language of Motion 

until the amount of curve remaining became vanishingly small. 
Could we not safely assume that the slope of that infinitesimal part 
of the curve would be the same as that on the straight line of which 
it seems to be a part? We think so. That is why we took the slope 
of the straight line from t = 8 sec to t = 12 sec, and called it the 
speed at the midpoint, the speed at t = 10 sec, or to use the correct 
term, "the instantaneous speed" at t = 10 sec. 

In estimating a value for Leslie's instantaneous speed at a 
particular time, we actually measured the average speed over a 
4.0-sec interval. We then made the conceptual leap that we have 
described. We decided that the instantaneous speed at a particular 
instant can be equated to an average speed Ad/At provided: 1) that 
the particular instant is included in At, and 2) that the ratio Ad/At 
is obtained for a small enough part of the curve, one which is 
nearly a straight-line segment, so that it does not change appreciably 
when we compute it over a still smaller time interval. 

A second concrete example will help here. In the oldest known 
study of its kind, the French scientist de MontbeOlard periodically 
recorded the height of his son during the period 1759-1777. A graph 
of height versus age for his son is shown in the margin. 

From the graph, we can compute the average growth rate 
(v,a^ over the entire 18-year interval or over any shorter time 
interval within that period. Suppose, however, we wanted to know 
how fast the boy was growing just as he reached his fifteenth 
birthday. The answer becomes evident if we enlarge the graph in 
the vicinity of the fifteenth year. (See the second graph.) His height 
at age 15 is indicated as point P, and the other letters designate 







1 — 

i i 








( , 

1 1 


!■ - 



, i 
















' — 1 








1 i 


\ i 


























Q^e ( {jrs ) 

I 2 3+ 56 1 S> 9 10 It IZ li If li lb IT Id 

l^c (yr,5) 

age (jfr-s) 

Section 1.7 


instants of time on either side of P. The boy's average growth rate 
over a two-year interval is given by the slope of the line segment 
AB in the enlarged figure in the margin. Over a one-year interval 
this average growth rate is given by the slope of CD. (See the third 
graph.) The slope of EF gives the average growth rate over six 
months, etc. The four lines, AB, CD, EF, GH, are not parallel to each 
other and so their slopes are different. However, the difference in 
slope gets smaller and smaller. It is large when we compare AB and 
CD, less if we compare CD and EF, less still between EF and GH. 
For intervals less than At = 1 yr, the lines appear to be more nearly 
parallel to each other and gradually to merge into the curve, 
becoming nearly indistinguishable from it. For very small intervals, 
you can find the slope by drawing a straight line tangent to this 
curve at P, placing a ruler at P (approximately parallel to line GH), 
and extending it on both sides as in Study Guide 1.11. 

The values of the slopes of the straight-line segments in the 
middle and lower graphs have been computed for the corresponding 
time intervals and are tabulated at the right. 

We note that values of Vav calculated for shorter and shorter time 
intervals approach closer and closer to 6.0 cm/yr. In fact, for any 
time interval less than 2 months, the average speed Va,- will be 6.0 
cm/yr within the limits of accuracy of measuring height. Thus we 
can say that, on his fifteenth birthday, young de Montbeillard was 
growing at a rate of 6.0 cm/yr. At that instant in his life, t = 15.0 yr, 
this was his instantaneous growth rate (or if you will, the 
instantaneous speed of his head with respect to his feet!) 

Average speed over a time interval At, we have said, is the ratio 
of distance traveled to elapsed time, or in symbols, 




We now have added the definition of instantaneous speed at an 
instant as the final limiting value approached by the average 
speeds when we compute Vgv for smaller and smaller time intervals 
including the instant t. In almost all physical situations, such a 
limiting value can be accurately and quickly estimated by the 
method described on the previous page. 

From now on we will use the letter v without the subscript m- to 
mean the instantaneous speed defined in this way. You may wonder 
why we have used the letter "z;" instead of "s" for speed. The 
reason is that speed is closely related to velocity. We shall reserve 
the term "velocity" for the concept of speed in a specified direction 
(such as 50 mph to the north) and denote it by the symbol v. When 
the direction is not specified and only the magnitude (50 mph) is of 
interest, we remove the arrow and just use the letter v, calling the 
magnitude of the velocity "speed." This crucial distinction between 
speed and velocity, and why velocity is more important in physics, 
will be discussed in more detail in later sections. 

Q12 Define instantaneous speed, first in words and then in 


Growth rate 





^"' = aF 


2 yr 

19.0 cm 

9.5 cm/year 


1 yr 




6 mo 




4 mo 




2 mo 



SG 1.14 

See SG 1.15, 1.16, and 1.17 for 
problems that check your under- 
standing of the chapter up to this 

1 . Paris street scene, 1 839. A daguerro- 
type made by Louis Daguerre himself. 

2. American street scene, 1859 

3. Boys on skateboards 

Photography 1839 to the Present 

1. Note the lone figure in the otherwise empty street. He was getting his 
shoes shined. The other pedestrians did not remain in one place long 
enough to have their images recorded. With exposure times several 
minutes long, the outlook for the possibility of portraiture was gloomy. 

2. However, by 1859, due to improvements in photographic emulsions and 
lenses, it was not only possible to photograph a person at rest, but one 
could capture a bustling crowd of people, horses and carriages. Note the 
slight blur of the jaywalker's legs. 

3. Today, one can "stop" action with an ordinary camera. 

4. A new medium-the motion picture. In 1873 a group of California 
sportsmen called in the photographer Eadweard Muybridge to settle the 
question, "Does a galloping horse ever have all four feet off the ground at 
once?" Five years later he answered the question with these photos. The 
five pictures were taken with five cameras lined up along the track, each 
camera being triggered when the horse broke a string w'hich tripped the 
shutter. The motion of the horse can be restructured by making a flip pad 
of the pictures. 

With the perfection of flexible film, only one camera was needed to 
take many pictures in rapid succession. By 1895, there were motion 
picture parlors throughout the United States. Twenty-four frames each 
second were sufficient to give the viewer the illusion of motion. 












? - -^ 


4. Muybridge's series, 1878 

5. Stroboscopic photo of golfer's 
swing. (See the article "The Dynamics 
of a Golf Club" in Project Physics 
Reader 1 .) 

5. A light can be flashed successfully at a controlled rate and a multiple 
exposure (similar to the strobe photos in this text) can be made. In this 
photo of a golfer, the light flashed 100 times each second. 

6. It took another ninety years after the time the crowded street was 
photographed before a bullet in flight could be "stopped." This remarkable 
picture was made by Harold Edgerton of MIT, using a brilliant electric 
spark which lasted for about one millionth of a second. 

7. An interesting offshoot of motion pictures is the high-speed motion 
picture. In the frames of the milk drop series shown below, 1000 pictures 
were taken each second (by Harold Edgerton). The film was whipped past 
the open camera shutter while the milk was illuminated with a flashing 
light (similar to the one used in photographing the golfer) synchronized 
with the film. When the film is projected at the rate of 24 frames each 
second, action which took place in 1 second is spread out over 42 seconds. 

It is clear that the eye alone could not have seen the elegant details of 
this event. This is precisely why photography of various kinds is used in 
the laboratory. 

6. Bullet cutting through a playing 

7. Action shown in high-speed film of milk drop. 


The Language of Motion 

Q13 Explain the difference in meaning between average speed 
and instantaneous speed. 

Unless noted otherwise, "rate of 
change" will always mean "rate of 
change with respect to time." 

1.8 Acceleration -by comparison 

You can tell from the photograph below of a rolling baseball 
that it was changing speed — accelerating. The increasing distances 
between the instantaneous images of the ball give you this informa- 
tion, but how can you tell how much acceleration the ball has? 

To answer this question we have only one new thing to learn — 
the definition of acceleration. The definition itself is simple; our 
task is to learn how to use it in situations like the one above. For 
the time being, we will define acceleration as rate of change of 
speed. Later this definition will have to be modified somewhat when 
we encounter motion in which change in direction becomes an 
important additional factor. But for now, as long as we are dealing 
only with straight-line motion, we can equate the rate of change 
of speed with acceleration. 

Some of the effects of acceleration are familiar to everyone. It 
is acceleration, not speed, that you notice when an elevator 
suddenly starts up or slows down. The flutter in one's stomach 
comes only during the speeding up and slowing down, not during 
most of the ride when the elevator is moving at a steady speed. 
Likewise, much of the excitement of the roller coaster and other 
rides at amusement parks is a result of their unexpected 
accelerations. Speed by itself does not cause these sensations. 
Otherwise they would occur during a smooth plane ride at 650 mph, 
or even just during the continuous motion of the earth around the 
sun at 65,000 mph. 

Simply stated, speed is a relationship between two objects, one 
of which is taken to be the reference object while the other moves 
with respect to it. Some examples are the speed of the earth with 
respect to the stars, the speed of the swimmer with respect to the 
pool edge, the speed of the top of the growing boy's head with 
respect to his feet. . . In a perfectly smooth-riding train, we could 
tell that we were moving at a high speed only by seeing the 
scenery whizzing by. We would have just the same experience if 
the train were somehow fixed and the earth with rails, etc., were to 
whiz by in the other direction. And if we "lost the reference object " 
(by pulling down the shades, say) we might not know at all 
whether we were moving or not. In contrast, we "feel" accelerations 
and do not need to look out the train window to realize that the 
engineer has suddenly started the train or has slammed on the 

Section 1.8 


brakes. We might be pushed against the seat, or the luggage might 
fly from the rack. 

All this suggests a profound physical diff^erence between 
motion at constant speed and motion with acceleration. While it is 
best to learn about acceleration at first hand (in the laboratory and 
through the film loops), we can summarize the main ideas here. For 
the moment let us focus on the similarities between the concepts 
speed and acceleration; for motion in a straight line: 

The rate of change of position 
is called speed. 

The rate of change of speed 
is called acceleration. 

This similarity of form will enable us to use what we have just 
learned about the concept of speed as a guide for making use of the 
concept of acceleration. For example, we have learned that the 
slope of the line of a distance-time graph is a measure of the 
instantaneous speed. The slope of a speed-time graph is a measure 
of the instantaneous acceleration. 

This section concludes with a list of six statements about 
motion along a straight line. The list has two purposes: 1) to help 
you review some of the main ideas about speed presented in this 
chapter, and 2) to present the corresponding ideas about 
acceleration. For this reason, each statement about speed is 
immediately followed by a parallel statement about acceleration. 

1. Speed is the rate of change of position. Acceleration is the 
rate of change of speed. 

2. Speed is expressed in units of distance/time. Acceleration is 
expressed in units of speed/time. 

3. Average speed over any time interval is the ratio of the 
change of position Ad and the time interval At: 


Var = 


Average acceleration over any time interval is the ratio of the 
change of speed Ar and the time interval At : 


4. Instantaneous speed is the value approached by the 
average speed as At is made smaller and smaller. Instantaneous 
acceleration is the value approached by the average acceleration 
as At is made smaller and smaller. 

5. On a distance-time graph, the instantaneous speed at any 
instant is the slope of the straight line tangent to the curve at the 
point of interest. On a speed-time graph, the instantaneous 
acceleration at any instant is the slope of the straight line tangent 
to the curve at the point of interest. 

6. For the particular case of constant speed, the distance-time 
graph is a straight line; everywhere on it the instantaneous speed 
has the same value, equal to the average speed computed for the 
whole trip. For the particular case of constant acceleration, the 
speed- time graph is a straight line; everywhere on it the 

For example, if an airplane changes 
its speed from 500 mph to 550 mph 
in 10 minutes, its average 
acceleration would be 

Ay 550 mi/hr - 500 mi/hr 

_ 50 mi/hr 
10 min 

_ 5 mi/hr 

That is, its speed changed at a rate 
of 5 mph per minute. (If the speed 
was decreasing, the value of the 
acceleration would be negative.) 



a = 



Constant speed and constant 
acceleration are often called 
"uniform" speed and "uniform" 
acceleration. In the rest of this 
course, we will use the terms 


The Language of Motion 

SG 1.18 provides an opportunity to 
worl< with distance-time and 
speed-time graphs and to see their 
relationship to one another. 
Transparencies T3 and T4 may be 
helpful also. 

SG 1.19 to 1.21 are review problems 
for this chapter. Some of these will 
test how thoroughly you grasp the 
language used for describing 
straight-line motion. 

instantaneous acceleration has the same value, equal to the 
average acceleration computed for the whole trip. When speed is 
constant, its value can be found from any corresponding Ad and At. 
When acceleration is constant, its value can be found from any 
corresponding At; and At. (This is useful to remember because 
constant acceleration is the kind of motion we shall encounter 
most often in the following chapters.) 

We now have most of the tools needed to get into some real 
physics problems. The first of these is the accelerated motion of 
bodies caused by gravitational attraction. It was by studying motion 
of falling objects that Galileo, in the early 1600's, was first able to 
shed light on the nature of accelerated motion. His work remains 
to this day a wonderful example of how scientific theory, 
mathematics, and actual measurements can be combined to develop 
physical concepts. More than that. Galileo's work was one of the 
early and most crucial battlegrounds of the scientific revolution. 
The specific ideas he introduced are even now fundamental to the 
science of mechanics, the study of bodies in motion. 

Q14 What is the average acceleration of an airplane which 
goes from to 60 mph in 5 seconds? 

Q15 What is your average acceleration if, while walking, you 
change your speed from 4.0 miles per hour to 2.0 miles per hour in 
an interval of 15 minutes? Is your answer affected by how your 
change of speed is distributed over the 15 minutes? 


11 This book is probably different in many ways 
from textbooks you have had in other courses. 
Therefore we feel it might help to make some 
suggestions about how to use it. 

1. Do not write in this book unless your 
teacher gives you permission to do so. In many 
schools the books must be used again next year by 
other students. However, if you are fortunate 
enough to be in a situation in which the teacher 
can permit you to mark in the book, we encourage 
you to do so. You will note that there are wide 
margins. One of our reasons for leaving that 
much space is to enable you to record questions 
or statements as they occur to you when you are 
studying the material. Mark passages that you do 
not understand so that you can seek help from 
your teacher. 

2. If you may not write in the textbook itself, 
try keeping a notebook keyed to the text chapters. 
In this study notebook jot down the kinds of 
remarks, questions and answers that you would 
otherwise write in the textbook as suggested 
above. Also, you ought to write down the questions 
raised in your mind by the other learning 
materials you will use, by the experiments 

you do, by demonstrations or other observations, 
and by discussions you may have with 
fellow students and others with whom you talk 
physics. Most students find such an informal 
notebook to be enormously useful when studying, 
or when seeking help from their teachers (or, for 
that matter, from advanced students, parents, 
scientists they may know, or anyone else whose 
understanding of physics they have confidence in). 

3. You will find answers to all of the end-of- 
section review questions on page 197. Always try 
to answer the questions yourself first and then 
check your answers. If your answer agrees with 
the one in the book, it is a good sign that you 
understand the main ideas in that section — 
although it is true that you can sometimes get the 
right answer for the wrong reason, and also that 
there may sometimes be other answers as good 
(or better than!) those given in the book. 

4. There are many different kinds of items in 
the Study Guide at the end of each chapter. 
Brief answers to some of them are given on page 
199. It is not intended that you should do every 
item. Sometimes we include material in the Study 
Guide which we think will especially interest only 
some students. Notice also that there are several 
kinds of problems. Some are intended to give 
practice in the use of a particular concept, 
while others are designed to help you bring 
together several related concepts. Still other 
problems are intended to challenge those students 
who particularly like to work with numbers. 

5. This text is only one of the learning 
materials of the Project Physics course. The 
course includes several other materials such as 
film loops, programmed instruction booklets, and 
transparencies. Use those. Be sure to familiarize 
yourself also with the Handbook, which de- 
scribes outside activities and laboratory 
experiments, and with the Reader, in which we 

have collected interesting articles related to 
physics. Each of these learning aids makes its 
own contribution to an understanding of physics, 
and all are designed to be used together. 

The Project Physics learning materials particularly 
appropriate for Chapter 1 include: 

Experiments (in the Handbook) 
Naked Eye Astronomy 
Regularity and Time 
Variations in Data 
Measuring Uniform Motion 

Activities (in the Handbook) 
Using the Electronic Stroboscope 
Making Frictionless Pucks 

Reader Articles 

Motion in Words 
Representation of Motion 
Motion Dynamics of a Golf Club 
Bad Physics in Athletic 


Analyzing a Stroboscopic Photograph 
Stroboscopic Measurements 
Graphs of Various Motions 
Instantaneous Speed 
Instantaneous Rate of Change 

In addition the following Project Physics materials 
can be used with Unit 1 in general: 

Reader Articles 

The Value of Science 
Close Reasoning 
How to Solve It 
Four Pieces of Advice 
to Young People 
On Being the Right Size 
The Vision of Our Age 
Becoming a Physicist 
Chart of the Future 

1.2 One type of automobile speedometer is a 
small electric generator driven by a flexible cable 
run off" the drive shaft. The current produced 
increases with the rate at which the generator is 
turned by the drive shaft. The speedometer needle 
indicates the current. Until the speedometer is 
calibrated it cannot indicate actual speeds in 



miles per hour. Try answering the questions 
below. If you have trouble you may want to try 
again after you have studied through Sec. 1.9. 

(a) How would you calibrate the speedometer 
in a car if the company had forgotten to 
do the job? 

(b) If you replaced the 24"-diameter rear 
wheels with 28"-diameter wheels, what 
would your actual speed be if the 
speedometer read 50 mph? 

(c) Would the speedometer read too high or 
too low if you loaded down the rear end of 
your car and had the tire pressure too low? 

(d) Does the operation of the speedometer 
itself affect the motion of the car? 

(e) How would you test to see if a bicycle 
speedometer affects the speed of a bike? 

(f ) Can you invent a speedometer that has no 
effect on the motion of the vehicle that 
carried it? 

1.3 Some practice problems: 




Speed uniform, dis- 
tance = 72 cm, 
time= 12 sec 



Speed uniform at 
45 miles per hour 

Distance traveled 
in 20 minutes 

Speed uniform at 
36 ft/min 

Time to move 
9.0 feet 


d, = f, = 

dz = 1 5 cm fa = 5.0 sec 

d;, = 30 cm f3= 10 sec 

Speed and posi- 
tion at 8.0 sec 


You drive 240 miles in 
6.0 hr 

Average speed 


Same as e 

Speed and 
position after 
3.0 hr 


Average speed is 76 
cm/sec, computed 
over a distance 
of 418 cm 

Time taken 


Average speed is 44 
m/sec, computed 
over time interval 
of 0.20 sec 

Distance moved 

1. 1 A tsunami caused by an earthquake occurring 
near Alaska in 1946 consisted of several sea 
waves which were found to travel at the average 
speed of 490 mph. The first of the waves reached 

Hawaii 4 hrs and 34 min after the earthquake 
occurred. From these data, calculate how far the 
origin of the tsunami was from Hawaii. 

1.5 Light and radio waves travel through a 
vacuum in a straight line at a speed of very nearly 
3 X IQs m/sec. 

(a) How long is a "light year" (the distance 
light travels in a year)? 

(b) The nearest star, Alpha Centauri, is 
4.06 X 10'« m distant from us. If this star 
possesses planets on which highly 
intelligent beings live, how soon, at the 
earliest, could we expect to receive a reply 
after sending them a radio or light signal 
strong enough to be received there? 

1.6 If you traveled one mile at a speed of 1000 
miles per hour and another mile at a speed of 1 
mile per hour, your average speed would not be 
1000 mph + 1 mph/2 nr 500.5 mph. What would 
be your average speed? (Hint: What is the total 
distance and total time?) 

1.7 What is your average speed in each of these 

(a) You run 100 m at a speed of 5.0 m/sec 
and then you walk 100 m at a speed of 
1.0 m/sec. 

(b) You run for 100 sec at a speed of 5.0 m/sec 
and then you walk for 100 sec at a speed 
of 1.0 m/sec? 

1.8 Design and describe experiments to enable 
you to make estimates of the average speeds for 
some of the following objects in motion. 

(a) A baseball thrown from outfield to home 

(b) The wind 

(c) A cloud 

(d) A raindrop 

(e) A hand moving back and forth as fast as 

(f ) The tip of a swinging baseball bat 

(g) A person walking on level ground, 
upstairs, downstairs 

(h) A bird flying 
(i) An ant walking 

(j) A camera shutter opening and closing 

(k) An eye blinking 
( 1 ) A whisker growing 
(m) The center of a vibrating guitar string 

1-9 What problems arise when you attempt to 
measure the speed of light? Can you design an 
experiment to measure the speed of light? 

1.10 Sometime, when you are a passenger in an 
automobile, compare the speed as read from the 
speedometer with the speed calculated from Id! At. 
Explain any differences. Refer again to SGI. 2. 
(For other activities see your Project Physics 



l.n Take a look at the graph of y versus x 
shown below: 

Although in this particular graph the 
steepness of the line increases as x increases, the 
method presented below would also hold for a 
curve of any other shape. One way to indicate the 
steepness of the line at a point P is by means of 
its "slope." The numerical value of the slope at a 
point P is obtained by the following procedure 
(diagrammed above): At a very short distance 
along the line from point P to either side of it, 
mark 2 points, A and B. Choose these points so 
close to P that although they also lie on the curve, 
the line APB is a straight line as nearly as one 
can determine with a ruler. Measure Ay Uhe 
change in y) in going from A to B. In this example 
y = 0.6. Measure Ax (the corresponding change in 
x) in going from A to B. Ax here is 0.3. The slope 
of the segment AB is defined as the ratio of Ay 
to Ax of the short straight-line-segment APB. By 
definition, the slope of the curve at point P is 
taken to be equal to the slope of the straight-line- 
segment APB. 

slope = — 


In this example, 

slope = -T-^ = 


Q. What are the dimensions or units for the 

A. The dimensions are just those of y/x. For 
example, if y represents a distance in meters and 
X represents a time in seconds, then the units for 
slope will be meters per second (or m/sec). 

Q. In practice, how close must A and B be to 
point P? (Close is not a very precise adjective. 
Baltimore is close to Washington if you are flying 

over both by jet. If you are walking, it is not close.) 

A. Choose A and B near enough to point P so 
that a straight line drawn carefully to connect A 
and B also goes through point P. 

Q. Suppose A and B are so close together that 
you cannot adequately read Ax or Ay from your 
graph. How would you try to calculate the slope? 

A. Extend the straight line AB in both 
directions, as shown in the figure, as far as you 
wish, and compute its slope. What you are then 
doing is putting a tangent line to the curve at the 
chosen point between A and B. Notice that the 
small triangle is similar to the large triangle, and, 

Ay/ Ax = AY/ AX 

Problem : 

(a) Determine the slope of this graph of 
distance versus time (y in meters, t in 
seconds) at four different points or 
instants, namely when t = 1, 2, 3, and 4 

(b) Find the instantaneous speed at these 4 
points, and plot a graph of speeds vs. time. 

1.12 (Answer to question in text, page 23.) 

Indeed the prediction based upon the first 
hour and a half would be vastly wrong. A 
prediction based on an extrapolation from the first 
labour's observation neglects all the factors 
which limit the maximum height obtainable by 
such a cluster of balloons, such as the bursting of 
some of the balloons, the change in air pressure 
and density with height and many others. 
Actually, at the end of 500 minutes the cluster 
was not 450,000 feet high but had come down 
again, as the distance-time graph for the entire 
experiment shows. See top of next page. For 
another extrapolation problem, see SG 1.13. 



100 zoo 300 400 500 
iime (sec) 

1.13 World's 400-nieter swimming records in 
minutes and seconds for men and women 
(numbers in parentheses are ages): 
















Johnny Weissmuller (18) 
Gertrude Ederle (17) 
Syozo Makino (17) 
Helene Madison (18) 
(1936 record unbroken) 
R. Hveger (18) 
Hironoshin Furuhashi (23) 
Lorraine Crapp (18) 
Frank Weigand (23) 
Martha Randall (18) 
By about how many meters would Martha Randall 
have beaten Johnny Weissmuller if they had 
raced each other? Could you predict the 1976 
records for the 400-meter race by extrapolating 
the graphs of world's records vs. dates up to the 
year 1976? 

1.14 How can we justify defining instantaneous 
speed as we have on p. 25? How can we be sure 
the definition is right? 

1.15 Using the graph on p. 20 find the 
instantaneous speeds v at several points (0, 10, 
20, 30, 40, and 50 sec, and near 0, or at other 
points of your choice) by finding the slopes of lines 
tangent to the curve at each of those points. Make 
a graph of v vs. t. Use your graph to describe her 

1.1 (i Turn back to p. 28. At the bottom of this page 
there is a multiple-exposure photograph 
of a baseball rolling to the right. The time interval 
between successive flashes was 0.20 sec. The 
distance between marks on the meter stick was 
1 centimeter. You might tabulate your 
measurements of the ball's progress between 
flashes and construct a distance-time graph. From 
the distance-time graph, you can determine the 
instantaneous speed at several instants and 

construct a speed-time graph. You can check your 
results by referring to the answer page at the end 
of this unit. 

1.17 Careful analysis of a stroboscopic photograph 
of a moving object yielded information which 
was plotted on the graph below. By placing your 
ruler tangent to the curve at appropriate points 
estimate the following: 














^ 7 




1 _^ 




u n 

4 6 

t\n^e (sec) 


(a) At what moment or interval was the speed 
greatest? What was the value of the speed 
at that time? 

(b) At what moment or in which interval was 
the speed least? What was it at that time? 

(c) What was the speed at time t = 5.0 sec? 

(d) What was the speed at time t = 0.5 sec? 

(e) How far did the object move from time 
t = 7.0 sec to t = 9.5 sec? 

1.18 The data below show the instantaneous 
speed in a test run of a car starting from rest. Plot 
the speed-vs-time graph, then derive data from 
it and plot the acceleration-vs-time graph. 

(a) What is the speed at t = 2.5 sec? 

(b) What is the maximum acceleration? 

Time (sec) 

Speed (m/ 


Time (sec) 

Speed (m/sec) 























1.19 The electron beam in a typical TV set 
sweeps out a complete picture in 1/30 sec and 
each picture is composed of 525 lines. If the 
width of the screen is 20 inches, what is the speed 
of that beam over the surface of the screen? 

1.20 Suppose you must measure the 
instantaneous speed of a bullet as it leaves the 
barrel of a rifle. Explain how you might do this. 



1.21 Discuss the motion of the cat in the 
following series of photographs, "Cat in trot 
changing to gallop." The numbers on each 

photograph indicate the number of inches 
measured from the fixed line marked "0." The 
time interval between exposures is 0.030 sec. 


10 20 ^iP 20 I 10 I 20 

^^^ ^^^ ^^^ 

20 1 ; 20 30 j 20 30 1 2l) 30 

^ --fr /^y" 1^^^ 

50 : 40 

50 40 

♦*- vtlStr 


2.1 The Aristotelian theory of motion 37 

2.2 Galileo and his times 43 

2.3 Galileo's Two New Sciences 43 

2.4 Why study the motion of freely falling bodies? 47 

2.5 Galileo chooses a definition of uniform acceleration 47 

2.6 Galileo cannot test his hypothesis directly 49 

2.7 Looking for logical consequences of Galileo's hypothesis 50 

2.8 Galileo turns to an indirect test 53 

2.9 Doubts about Galileo's procedure 56 

2.10 Consequences of Galileo's work on motion 57 

Portrait of Galileo by Ottavio Leoni, a 
contemporary of Galileo. 


Free Fall- 

Galileo Describes Motion 

2.1 The Aristotelian theory of motion 

In this chapter we shall follow the development of an important 
piece of basic research: Galileo's study of freely falling bodies. 
While the physical problem of free fall is interesting in itself, our 
emphasis will be on the way Galileo, one of the first modern 
scientists, presented his argument. His view of the world, his way 
of thinking, his use of mathematics, and his reliance upon 
experimental tests set the style for modem science. These aspects of 
his work, therefore, are as important to us as the actual results of his 

To understand the nature of Galileo's work and to appreciate its 
significance, we must first examine the previous system of 
physical thought that it eventually replaced. In medieval physical 
science, as Galileo learned it at the University of Pisa, a sharp 
distinction was thought to exist between the objects on the earth 
and those in the sky. All terrestrial matter, the matter within our 
physical reach, was believed to contain a mixture of four "elements" 
-Earth, Water, Air, and Fire. These elements were not thought of 
as identical with the natural materials for which they were named. 
Ordinary water, for example, was thought to be a mixture of all 
four elements, but mostly the element Water. Each of the four 
elements was thought to have a natural place in the terrestrial 
region. The highest place was allotted to Fire. Beneath Fire was Air, 
then Water, and finally, in the lowest position, Earth. Each was 
thought to seek its own place. Thus Fire, if displaced below its 
natural position, would tend to rise through Air. Similarly, Air would 
tend to rise through Water, whereas Earth would tend to fall 
through both Air and Water. The movement of any real object 
depended on its particular mixture of these four elements, and on 
where it was in relation to the natural places of these elements. 

SG 2.1 

A sketch of a medieval world-system. 



Free Fall — Galileo Describes Motion 

A good deal of common-sense 
experience supports this natural- 
place view. See SG 2.2 

From quinta essentia, meaning 
fifth essence. In earlier Greek 
writings the term for it was aether 
(also written ether). 

The painting entitled "School of 
Athens, " was done by Raphael in the 
beginning of the sixteenth century. 
It reflects a central aspect of the 
Renaissance, the rebirth of interest in 
classical Greek culture. The central 
figures are Plato (on the left, pointing 
to the heavens) and Aristotle (pointing 
to the ground). 

When water boiled, for example, the element Water would be joined 
by the element Fire, whose higher natural place would cause the 
mixture to rise as steam. A stone, on the other hand, being 
composed primarily of the element Earth, would fall when released 
and would pass through Fire, Air, and Water until it came to rest on 
the ground, its natural place. 

The medieval thinkers also believed that the stars, planets, and 
other celestial bodies differed in composition and behavior from 
objects on or near the earth. The celestial bodies were believed to 
contain none of the four ordinary elements, but instead to consist 
solely of a fifth element, the quintessence. The natural motion of 
objects composed of this element was neither rising nor falling, but 
endless revolution in circles around the center of the universe. That 
center was considered to be identical with the center of the earth. 
Heavenly bodies, although moving, were at all times in their 
natural places. Thus heavenly bodies were altogether different from 
terrestrial objects, which displayed natural motion only as they 
returned to their natural places from which they had been displaced. 

This theory, so widely held in Galileo's time, had originated 
almost 2000 years before, in the fourth century B.C. We find it stated 
clearly in the writings of the Greek philosopher Aristotle. This 
physical science, built on order, class, place, and purpose, fits well 
many facts of everyday observation. It seemed particularly plausible 
in societies like those in which Aristotle and Galileo lived, where 
rank and order were dominant in human experience. Moreover, 
these conceptions of matter and motion were part of an all- 
embracing universal scheme or "cosmology." In his cosmology 
Aristotle sought to relate ideas which are nowadays discussed 
separately under such headings as science, poetry, politics, ethics, 
and theology. 

Not very much is known of Aristotle's physical appearance or 
life. It is thought that he was bom in 384 B.C. in the Greek 
province of Macedonia. His father was the physician to the King of 
Macedonia, and so Aristotle's early childhood was spent in an 
environment of court life. He completed his education in Athens 
and later returned to Macedonia to become the private tutor to 
Alexander the Great. In 335 B.C., Aristotle came back to Athens 
and founded the Lyceum, a school and center of research. 

500 BC 

400 BC 384 BC ^^^tt2 BC 300 BC 



















200 BC 


PTOLEMY I of Egy pt 




















40 Free Fall -Galileo Describes Motion 

After the decline of the ancient Greek civilization, the writings 
of Aristotle remained virtually unknown in Western Europe for 
1500 years. They were rediscovered in the thirteenth century A.D. 
and were later incorporated into the works of Christian scholars and 
theologians. Aristotle became such a dominant influence in the late 
Middle Ages that he was referred to simply as "The Philosopher." 

The works of Aristotle make up almost an encyclopedia of 
ancient Greek thought. Some of it was summarized from the work 
of others, but much of it seems to have been created by Aristotle 
himself. Today it is hard to believe that one man could have been 
so well informed on such different subjects as logic, philosophy, 
theology, physics, astronomy, biology, psychology, politics, and 
literature. Some scholars doubt that it was all the work of one man. 

Unfortunately, Aristotle's physical theories had serious 
limitations. (This does not, of course, detract from his great 
achievements in other fields.) According to Aristotle, the fall of a 
heavy object toward the center of the earth is an example of 
"natural" motion. He evidently thought that any object, after 
release, quickly achieves some final speed of fall at which it 
continues to move to the end of its path. What factors determine the 
final speed of a falling object? It is a common observation that a 
rock falls faster than a leaf. Therefore, he reasoned, weight is a 
factor that governs the speed of fall. This fitted in well with his idea 
that the cause of weight was the presence of the element Earth, 
whose natural tendency was to the center of the earth. Thus a 
heavier object, having a greater content of Earth, has a greater 
tendency to fall to its natural place, and hence develops a greater 
speed in falling. 

The same object falls more slowly in water than in air, so it 
seemed to Aristotle that the resistance of the medium must also be 
a factor. Other factors, such as the color or temperature of the 
falling object, could conceivably affect the rate of fall, but 
Aristotle: rate of fall is proportional Aristotle decided that their influence could not be significant. He 

to weight divided by resistance. concluded that the rate of fall must increase in proportion to the 

weight of the object and decrease in proportion to the resisting 
force of the medium. The actual rate of fall in any particular case 
would be found by dividing the weight by the resistance. 

Aristotle also discussed "violent" motion — that is. any motion 
SG 2.3 of ^n object other than going freely toward its "natural place." 

Such motion, he argued, must always be caused by a force, and the 
speed of the motion will increase as the force increases. When the 
force is removed, the motion must stop. This theory agrees with 
our common experience, say in pushing a chair or a table across the 
floor. It doesn't work quite so well for objects thrown through the 
air, since such projectiles keep moving for a while even after we 
have stopped exerting a force on them. To account for this kind of 
motion, Aristotle proposed that the air itself somehow exerts a force 
that keeps the object moving. 

Later scientists proposed some modifications in Aristotle's 

Section 2.1 


theory of motion. For example, in the fifth century A.D. John 
Philoponus of Alexandria argued that the speed of an object in 
natural motion should be found by subtracting the resistance of 
the medium from the weight of the object, rather than dividing by 
the resistance. Philoponus claimed that his experimental work 
supported his theory, though he did not report the details; he simply 
said that he dropped two weights, one of which was twice as heavy 
as the other, and observed that the heavy one did not reach the 
ground in half the time taken by the light one. 

There were still other difficulties with Aristotle's theory of 
motion. However, the realization that his teachings concerning 
motion had limitations did little to modify the importance given 
to them in the universities of France and Italy during the fifteenth 
and sixteenth centuries. Aristotle's theory of motion did, after all, 
fit much of ordinary experience in a general -if qualitative- way. 
Besides, the study of motion through space was of major interest to 
only a few scholars, just as it had been only a very small part of 
Aristotle's own work. 

Two other influences stood in the way of radical changes in 
the theory of motion. First, Aristotle believed that mathematics was 
of little value in describing terrestrial phenomena. Second, he put 
great emphasis upon direct, qualitative observation as the basis for 
theorizing. Simple qualitative observation was very successful in 
Aristotle's biological studies. But as it turned out, real progress in 
physics began only when the value of mathematical prediction and 
detailed measurement was recognized. 

A number of scholars in the fifteenth and sixteenth centuries 
had a part in this change to a new way of doing science. But of 
all these, Galileo was by far the most eminent and successful. He 
showed how to describe mathematically the motions of simple, 
ordinary objects -falling stones and balls rolling on an incline. This 
work not only paved the way for other men to describe and explain 
the motions of everything from pebbles to planets, it also began an 
intellectual revolution which led to what we now consider modem 

John Philoponus: rate of fall is 
proportional to weight minus 


Qualitative refers to quality - 
the sort of thing that happens. 
Quantitative refers to quantity - 
the measurement or prediction of 
numerical values. This distinction 
will appear often in the course. 

Q1 Describe two ways in which, according to the Aristotelian 
view, terrestrial and celestial bodies differ from each other. 

Q2 Which of these statements would be accepted in the 
fifteenth and sixteenth centuries by persons who believed in the 
Aristotelian system of thought? 

(a) Ideas of motion should fit in with poetry, politics, theology 
and other aspects of human thought and activity. 

(b) Heavy objects fall faster than light ones. 

(c) Except for motion toward their natural location, objects will 
not move unless acted on violently by a force. 

(d) Mathematics and precise measurement are especially 
important in developing a useful theory of motion. 


Sections 2.2 and 2.3 


2.2 Galileo and his times 

Galileo Galilei was bom in Pisa in 1564 -the year of 
Michelangelo's death and Shakespeare's birth. Galileo was the son 
of a nobleman from Florence, and he acquired his father's active 
interest in poetry, music, and the classics. His scientific inventive- 
ness also began to show itself early. For example, as a young 
medical student at the University of Pisa, he constructed a simple 
pendulum-type timing device for the accurate measurement of 
pulse rates. 

Lured from medicine to physical science by reading Euclid and 
Archimedes, Galileo quickly became known for his unusual ability 
in science. At the age of 26, he was appointed Professor of 
Mathematics at Pisa. There he showed an independence of spirit 
unmellowed by tact or patience. Soon after his appointment, he 
began to challenge the opinions of his older colleagues, many of 
whom became his enemies. He left Pisa before his term was 
completed, apparently forced out by financial difficulties and by his 
enraged opponents. Later, at Padua in the Republic of Venice, he 
began his work in astronomy. His support of the sun-centered theory 
of the universe eventually brought him additional enemies, but it 
also brought him immortal fame. We shall deal with that part of 
his work in Unit 2. 

Drawn back to his native province of Tuscany in 1610 by a 
generous offer of the Grand Duke, Galileo became Court 
Mathematician and Philosopher, a title which he chose himself. 
From then until his death at 78, despite illness, family troubles, 
occasional brushes with poverty, and quarrels with his enemies, he 
continued his research, teaching and writing. 





Italy about 1600 

2.3 Galileo's Two New Sciences 

Galileo's early writings on mechanics (the study of the 
behavior of matter under the influence of forces) were in the 
tradition of the standard medieval theories of physics, although he 
was aware of some of the shortcomings of those theories. During 
his mature years his chief interest was in astronomy. However, 
when his important astronomical book. Dialogue on the Two Great 
World Systems (1632), was condemned by the Roman Catholic 
Inquisition and he was forbidden to teach the "new" astronomy, 
Galileo decided to concentrate again on mechanics. This work led to 
his book Discourses and Mathematical Demonstrations Concerning 
Two New Sciences Pertaining to Mechanics and Local Motion 
(1638), usually referred to as Two New Sciences. This treatise 
signaled the beginning of the end, not only of the medieval theory 
of mechanics, but also of the entire Aristotehan cosmology which it 

Galileo was old, sick, and nearly blind at the time he wrote 
Two New Sciences. Yet, as in all his writings, his style is spritely 

Title page of Dialogue on Two Great 
World Systems (1632). 


Free Fall -Galileo Describes Motion 

D I S C O R S I 



intorno a due nuoue fcicnzf 

Ancncnci alia 




Filofofo e Matemacico primario del Screnilllmo 

Granjd Duca di Tofcana. 

Cm vnt Afftniice iclctntrt digrtuiti itUnni Stliii. 


IN L E I D A, 
Apprcdb gli Eifcvicii. m. d. c. xxxviii. 

Title page of Discourses and Mathe- 
matical Demonstrations Concerning 
Two New Sciences Pertaining to Me- 
chanics and Local Motion (1638). 

SG 2.5 

DEL Galileo. tfj 

ftcM Htafiftrtbhc il moto , Upcfizica del rtcu> dj/ituttmrnte 
frtfi, entn in relticione tl mote , nan vien dtUrultt , mi per dire 
qutlche ftt tu»entMr*folr<bher ri^ondcre (\itegii anlichi , tceti 
mtglitfifiorga,autnlo (tndudi U dimfffrtzionc d' AriihieU.mi 
fir chejifiirebht tndir ctniro i gli afunli di ijuttlo , nrgtridogli 
tmindiu. E ijHtHit itprimi^io frtndtmenle dahiio, (he Arilfo- 
tele Ktitjferimentiiffe mti ifatntefu vero , the due fieire -vnt fii 
grille di/fulirt died velie Ufiitte net medefma inittnte c*der 
dtvH tUet.t,t , V. gr. di eenio iraccid fujfer ttlmente differeali 
we Be lor veliciii , the tit arriuc dellt mtggior iit tern [tUrtfitre- 
Mtffe Hen htuere ni tneefeji died true ci*^ 

Simp. Si vede ftre dtUe fue f*rele,tlfei meHrt ^hmerlt fbe- 
rimenino, perche ei dice: yeggiamt ilfiHgrtne : her quel' vederfi 
tccennt I'hiuernefattt tefferienT^. 

Sajir. MiieS.Sim». chen'hhfttteUpreiiiMtlpcurcehevH* 
fttldttdrliglieri* yche pefi tenia, dugente,eineepilitihl)re,n<in 
enliciperidi vnpilmo feUmenie ttrnm in terrt dellipelUdua 
mefthetto , the nepefi -unt mez,i,t , venende tnta delt tltez,t.t di 
dugento hretdi. 

Salu. Mifent,' tltre e^erienT^ ten breue , e ccndudente dime- 
Hriz,iene poffttme thitrtmente preutrentn ejfervere^thevn me- 
tile piu greueji mucin piii veletemenie d'un'iltre men grtue , in - 
tendende di mobiti dell" ittejft meierid ) cf injimmd MqteSi de i 
qudlipdrld AriU elite. Peri dilemi S. vei dmmettete,the 
di ddfthedtino terpo grdue tddintefid vnd dd netitrd determindtd 
velecitk \(ithe I'dttreJierglieU ,ediminuirglieldnenfipejfdfe Hen 
ten ffirgli vielent,d , i tpporgU qudlche impedimente. 

Simp. Nen/ipuidiibitdre,thel'iilejfemebite neWifteJfemez- 
zt hdbbid vnd fldtuild , e dd ndturd determinttd veledti. Id qud- 
U nen ft gli pejfd dttrefeere fe nen (en nueue impete tenferite- 
gU , i diminuiiglieUfilM the ten qudlche impedimente the le ri- 

Salu. SsfiteU eUmqne lui htHejJimt Jin mtbili, U ntturdli 


A page from the original Italian edition 
of Two New Sciences, showing state- 
ments that are translated in this text. 

and delightful. He used the dialogue form to allow a lively 
conversation among three 'speakers": Simplicio, who competently 
represents the Aristotelian view; Salviati, who presents the new 
views of Galileo; and Sagredo, the uncommitted man of good will 
and open mind, eager to learn. Eventually, of course, Salviati leads 
his companions to Galileo's views. Let us listen to Galileo's three 
speakers as they discuss the problem of free fall: 

Salviati: I greatly doubt that Aristotle ever tested by 
experiment whether it is true that two stones, one 
weighing ten times as much as the other, if allowed to 
fall at the same instant from a height of, say, 100 cubits, 
would so differ in speed that when the heavier had 
reached the ground, the other would not have fallen 
more than 10 cubits. [A "cubit" is equivalent to about 20 

Simplicio: His language would indicate that he had tried 
the experiment, because he says: We see the heavier; 
now the word see shows that he had made the 

Sagredo: But, I, Simplicio, who have made the test can 
assure you that a cannon ball weighing one or two 
hundred pounds, or even more, will not reach the ground 
by as much as a span [hand-breadth] ahead of a musket 
ball weighing only half a pound, provided both are 
dropped from a height of 200 cubits. 

Here, perhaps, one might have expected to find a detailed report 
on an experiment done by Galileo or one of his colleagues. Instead, 
Galileo uses a "thought experiment" -an analysis of what would 
happen in an imaginary experiment -to cast grave doubt on 
Aristotle's theory of motion: 

Salviati: But, even without further experiment, it is 
possible to prove clearly, by means of a short and 
conclusive argument, that a heavier body does not move 
more rapidly than a lighter one provided both bodies are 
of the same material and in short such as those mentioned 
by Aristotle. But tell me, Simplicio, whether you admit ' 
that each falling body acquires a definite speed fixed by 
nature, a velocity which cannot be increased or 
diminished except by the use of violence or resistance? 

Simplicio: There can be no doubt but that one and the 
same body moving in a single medium has a fixed velocity 
which is determined by nature and which cannot be 
increased except by the addition of impetus or diminished 
except by some resistance which retards it. 

Salviati: If then we take two bodies whose natural speeds 
are different, it is clear that on uniting the two. the more 
rapid one will be partly retarded by the slower, and the 
slower will be somewhat hastened by the swifter. Do you 
not agree with me in this opinion? 

Section 2.3 


Simplicio: You are unquestionably right. 

Salviati: But if this is true, and if a large stone moves 
with a speed of, say, eight, while a smaller moves with a 
speed of four, then when they are united, the system will 
move with a speed less than eight; but the two stones 
when tied together make a stone larger than that which 
before moved with a speed of eight. Hence the heavier 
body moves with less speed than the lighter one; an effect 
which is contrary to your supposition. Thus you see how, 
from your assumption that the heavier body moves more 
rapidly than the lighter one, I infer that the heavier body 
moves more slowly. 

Simplicio: I am all at sea. 
beyond my comprehension. 

. This is, indeed, quite 

SG 2.6 

Simplicio retreats in confusion when Salviati shows that the 
Aristotelian theory of fall is self-contradictory. But while Simplicio 
cannot refute Galileo's logic, his own eyes tell him that a heavy 
object does fall faster than a light object: 

Simplicio: Your discussion is really admirable; yet I do 
not find it easy to believe that a birdshot falls as swiftly as 
a cannon ball. 

Salviati: Why not say a grain of sand as rapidly as a 
grindstone? But, Simplicio, I trust you will not follow the 
example of many others who divert the discussion from 
its main intent and fasten upon some statement of mine 
that lacks a hairsbreadth of the truth, and under this hair 
hide the fault of another that is as big as a ship's cable. 
Aristotle says that "an iron ball of one hundred pounds 
falling from a height of 100 cubits reaches the ground 
before a one-pound ball has fallen a single cubit." I say 
that they arrive at the same time. You find, on making 
the experiment, that the larger outstrips the smaller by 
two fingerbreadths. . . . Now you would not hide behind 
these two fingers the 99 cubits of Aristotle, nor would you 
mention my small error and at the same time pass over 
in silence his very large one. 

This is a clear statement of an important principle: even in 
careful observation of a common natural event, the observer's 
attention may be distracted by what is really a minor effect, with the 
result that he fails to see a much more significant regularity. 
Different bodies falling in air from the same height, it is true, do 
not reach the ground at exactly the same time. However, the 
important point is not that the times of arrival are slightly different, 
but that they are very nearly the same\ Galileo regarded the 
failure of the bodies to arrive at exactly the same time as a minor 
effect which could be explained by a deeper understanding of motion 
in free fall. Galileo himself correctly attributed the observed results 
to differences in the effect of the resistance of the air on bodies of 

A stroboscopic photograph of two 
freely falling balls of unequal weight. 
The balls were released simultane- 
ously. The time interval between 
images is 1/30 sec. 


Free Fall — Galileo Describes Motion 

The phrase "free fall" as now used 
in physics generally refers to fall 
when the only force acting is gravity; 
that is, when air friction is 

different size and weight. A few years after Galileo's death, the 
invention of the vacuum pump allowed others to show that Galileo 
was right. Once the effect of air resistance was eliminated — for 
example, when a feather and a heavy gold coin were dropped from 
the same height at the same time inside an evacuated container— 
the different bodies fell at the same rate and struck the bottom of 
the container at the same instant. Long after Galileo, it became 
possible to formulate the laws of air resistance, so one could 
understand exactly why and by how much a light object falls 
behind a heavier one. 

Learning what to ignore has been almost as important in the 
growth of science as learning what to take into account. In the case 
of falling bodies, Galileo's explanation depended on his being able 
to imagine how an object would fall if there were no air resistance. 
This may be easy for us who know of vacuum pumps, but in 
Galileo's time it was an explanation that was difficult to accept. For 
most people, as for Aristotle, common sense said that air resistance 
is always present in nature. Thus a feather and a coin could never 
fall at the same rate. Why should one talk about hypothetical 
motions in a vacuum, when a vacuum could not be shown to exist? 
Physics, said Aristotle and his followers, should deal with the world 
all around us that we can readily observe, not with some imaginary 
world which might never be found. 

Aristotle's physics had dominated Europe since the thirteenth 
century, mainly because many intelligent scientists were convinced 
that it offered the most rational method for describing natural 
phenomena. To overthrow such a firmly established doctrine 
required much more than writing reasonable arguments, or simply 
dropping heavy and light objects from a tall building, as Galileo is 
often said to have done (but probably did not) at the Leaning Tower 
of Pisa. It demanded Galileo's unusual combination of mathematical 
talent, experimental skill, literary style, and tireless campaigning 
to discredit Aristotle's theories and to begin the era of modem 

A chief reason for Galileo's success was that he exposed the 
Aristotelian theory at its weakest point: he showed that physics can 
deal better with the world around us if we realize that the world of 
common observation is not the simple starting point the 
Aristotelians thought it to be. On the contrary, the world as we 
ordinarily observe it is usually quite complex. For example, in 
observing the fall of bodies you see the effects of both the law of 
fall and the law of resistance on objects moving through air. To 
understand what you see, you should start from a simple case (such 
as fall without resistance), even if this has to be "seen" only in your 
mind or by a mathematical model. Or you may turn to an 
experiment in the laboratory, where the usual conditions of 
observation can be changed. Only after you understand each of the 
different effects by itself should you go back to face the complexities 
of the ordinary case. 

Sections 2.4 and 2.5 


Q3 If a nail and a toothpick are simultaneously dropped from 
the same height, they do not reach the ground at exactly the same 
instant. (Try it with these or similar objects.) How would Aristotelian 
theory explain this? What was Galileo's explanation? 

2.4 Why study the motion of freely falling bodies? 

In Galileo's attack on the Aristotelian cosmology, few details 
were actually new. However, his approach and his findings together 
provided the first coherent presentation of the science of motion. 
Galileo realized that, out of all the observable motions in nature, 
free-fall motion is the key to the understanding of all motions of all 
bodies. To decide which is the key phenomenon to study is the real 
gift of genius. But Galileo is also in many ways typical of scientists 
in general. His approach to the problem of motion makes a good 
"case" to be used in the following sections as an opportunity to 
discuss strategies of inquiry that are still used in science. 

These are some of the reasons why we study in detail Galileo's 
attack on the problem of free fall. Galileo himself recognized 
another reason — that the study of motion which he proposed was 
only the starting phase of a mighty field of discovery: 

My purpose is to set forth a very new science dealing 
with a very ancient subject. There is, in nature, perhaps 
nothing older than motion, concerning which the books 
written by philosophers are neither few nor small; 
nevertheless, I have discovered some properties of it that 
are worth knowing that have not hitherto been either 
observed or demonstrated. Some superficial observations 
have been made, as for instance, that the natural motion 
of a heavy falling body is continuously accelerated; but to 
just what extent this acceleration occurs has not yet 
been announced. . . . 

Other facts, not few in number or less worth knowing 
I have succeeded in proving; and, what I consider more 
important, there have been opened up to this vast and 
most excellent science, of which my work is merely the 
beginning, ways and means by which other minds more 
acute than mine will explore its remote comers. 

2.5 Galileo chooses a definition of uniform acceleration 

Two New Sciences deals directly with the motion of freely 
falling bodies. In studying the following paragraphs from it, we 
must be alert to Galileo's overall plan. First, he discusses the 
mathematics of a possible, simple type of motion (which we now 
call uniform acceleration or constant acceleration). Then he 
proposes that heavy bodies actually fall in just that way. N'ext, on 
the basis of this proposal, he derives a prediction about balls rolling 
down an incline. Finally, he shows that experiments bear out these 

By Aristotelian cosmology is meant 
the whole interlocking set of ideas 
about the structure of the physical 
universe and the behavior of all the 
objects in it. This was briefly 
mentioned in Sec. 2.1. Other aspects 
of it will be presented in Unit 2. 

In fact, more than mere "superficial 
observations" had been made long 
before Galileo set to work. For 
example, Nicolas Oresme and others 
at the University of Paris had by 
1330 discovered the same distance- 
time relationship for falling bodies 
that Galileo was to announce in 
the Two New Sciences. Some of 
their reasoning is discussed in 
SG 2.7. 

It will help you to have a plan clearly 
in mind as you progress through the 
rest of this chapter. As you study 
each succeeding section, ask 
yourself whether Galileo is 

— presenting a definition 

— stating an assumption (or 

-deducing predictions from his 

-experimentally testing the 



Free Fall — Galileo Describes Motion 

This is sometimes l<nown as the 
Rule of Parsimony: unless forced to 
do otherwise, assume the simplest 
possible hypothesis to explain 
natural events. 

Rephrasing Galileo and using our 
symbols: for uniform speed v, the 
ratio Ad/Af is constant. Similarly, 
recall that for accelerated motion, 
as we saw in Chapter 1, we defined 
uniform acceleration as 

a = -rr = constant 

Other ways of expressing this 
relationship are discussed in SG 2.8 
and 2.9. 

The first part of Galileo's presentation is a thorough discussion 
of motion with uniform speed, similar to our discussion in Chapter 1. 
That leads to the second part, where we find Salviati saying: 

We pass now to . . . naturally accelerated motion, such 
as that generally experienced by heavy falling bodies. 

... in the investigation of naturally accelerated 
motion we were led, by hand as it were, in following the 
habit and custom of nature herself, in all her various 
other processes, to employ only those means which are 
most common, simple and easy . . . 

When, therefore, I observe a stone initially at rest 
falling from an elevated position and continually 
acquiring new increments of speed, why should I not 
believe that such increases take place in a manner which 
is exceedingly simple and rather obvious to everybody? If 
now we examine the matter carefully we find no addition 
or increment more simple than that which repeats itself 
always in the same manner. This we readily understand 
when we consider the intimate relationship between 
time and motion; for just as uniformity of motion is 
defined by and conceived through equal times and equal 
spaces (thus we call a motion uniform when equal 
distances are traversed during equal time-intervals), so 
also we may, in a similar manner, through equal time- 
intervals, conceive additions of speed as taking place 
without complication. . . . 

Hence the definition of motion which we are about to 
discuss may be stated as follows: 

A motion is said to be uniformly accelerated when, 
starting from rest, it acquires during equal time- 
intervals, equal increments of speed. 

Sagredo: Although I can offer no rational objection to 
this or indeed to any other definition devised by any 
author whosoever, since all definitions are arbitrary, I 
may nevertheless without defense be allowed to doubt 
whether such a definition as the foregoing, established in 
an abstract manner, corresponds to and describes that 
kind of accelerated motion which we meet in nature in 
the case of freely falling bodies .... 

Here Sagredo questions whether Galileo's arbitrary definition of 
acceleration actually corresponds to the way real objects fall. Is 
acceleration, as defined, really useful in describing their observed 
change of motion? Sagredo wonders about a further point, so far 
not raised by Galileo: 

From these considerations perhaps we can obtain an 
answer to a question that has been argued by philosophers, 
namely, what is the cause of the acceleration of the 
natural motion of heavy bodies .... 

But Salviati. the spokesman of Galileo, rejects the ancient 
tendency to investigate phenomena by looking first for their causes. 
It is premature, he declares, to ask about the cause of any motion 
until an accurate description of it exists: 

Section 2.6 


Salviati: The present does not seem to be the proper 
time to investigate the cause of the acceleration of 
natural motion concerning which various opinions have 
been expressed by philosophers, some explaining it by 
attraction to the center, others by repulsion between the 
very small parts of the body, while still others attribute it 
to a certain stress in the surrounding medium which 
closes in behind the falling body and drives it from one 
of its positions to another. Now, all these fantasies, and 
others, too, ought to be examined; but it is not really worth 
while. At present it is the purpose of our Author merely 
to investigate and to demonstrate some of the properties 
of accelerated motion, whatever the cause of this 
acceleration may be. 

Galileo has now introduced two distinct propositions: 1) 
"uniform" acceleration means equal speed increments Ai; in equal 
time intervals At; and 2) things actually fall that way. Let us first 
look more closely at Galileo's proposed definition. 

Is this the only possible way of defining uniform acceleration? 
Not at all! Galileo says that at one time he thought a more useful 
definition would be to use the term uniform acceleration for motion 
in which speed increased in proportion to the distance traveled. Ad, 
rather than to the time At. Notice that both definitions met Galileo's 
requirement of simplicity. (In fact, both definitions had been 
discussed since early in the fourteenth century.) Furthermore, both 
definitions seem to match our common sense idea of acceleration 
about equally well. When we say that a body is "accelerating," we 
seem to imply "the farther it goes, the faster it goes," and also "the 
longer time it goes, the faster it goes." How should we choose 
between these two ways of putting it? Which definition will be more 
useful in the description of nature? 

This is where experimentation becomes important. Galileo chose 
to define uniform acceleration as the motion in which the change 
of speed Av is proportional to elapsed time At, and then demonstrate 
that this matches the behavior of real moving bodies, in laboratory 
situations as well as in ordinary, "un-arranged," experience. As you 
will see later, he made the right choice. But he was not able to 
prove his case by direct or obvious means, as you shall also see. 

Q4 Describe uniform speed without referring to dry ice pucks 
and strobe photography or to any particular object or technique of 

Q5 Express Galileo's definition of uniformly accelerated 
motion in words and in the form of an equation. 

Q6 What two conditions did Galileo want his definition of 
uniform acceleration to meet? 

Here Salviati refers to the 
Aristotelian assumption that air 
propels an object moving through it 
(see Sec. 2.1). 

2.6 Galileo cannot test his hypothesis directly 

After Galileo defined uniform acceleration so that it would 
match the way he believed freely falling objects behaved, his next 

50 Free Fall -Galileo Describes Motion 

task was to devise a way of showing that the definition for uniform 
acceleration was useful for describing observed motions. 

Suppose we drop a heavy object from several different heights — 
say, from windows on different floors of a building. We want to 
check whether the final speed increases in proportion to the time it 
The symbol a: means "directly takes to fall — that is, whether At; cc At, or what amounts to the same 

proportional to." thing, whether Az;/At is constant. In each trial we must observe the 

time of fall and the speed just before the object strikes the ground. 
But there's the rub. Practically, even today, it would be very 
SG 2.10 difficult to make a direct measurement of the speed reached by an 
object just before striking the ground. Furthermore, the entire 
time intervals of fall (less than 3 seconds even from the top of a 
10-story building) are shorter than Galileo could have measured 
accurately with the clocks available to him. So a direct test of 
whether Aiy/At is constant was not possible for Galileo. 

Q7 Which of these are valid reasons why Galileo could not test 
directly whether the final speed reached by a freely falling object is 
proportional to the time of fall? 

(a) His definition was wrong. 

(b) He could not measure the speed attained by an object just 
before it hit the ground. 

(c) There existed no instruments for measuring time. 

(d) He could not measure ordinary distances accurately enough. 

(e) Experimentation was not permitted in Italy. 

2.7 Looking for logical consequences of Galileo's hypothesis 

Galileo's inability to make direct measurements to test his 
hypothesis -that Av/At is constant in free fall -did not stop him. He 
turned to mathematics to derive from this hypothesis some other 
relationship that could be checked by measurement with 
equipment available to him. We shall see that in a few steps he 
came much closer to a relationship he could use to check his 

Large distances of fall and large time intervals for fall are, of 
course, easier to measure than the small values of Ad and At that 
would be necessary to find the final speed just before the falling 
body hits. So Galileo tried to find, by reasoning, how total fall 
distance ought to increase with total fall time if objects did fall with 
uniform acceleration. You already know how to find total distance 
from total time for motion at constant speed. Now we will derive a 
new equation that relates total fall distance to total time of fall for 
motion at constant acceleration. In this we shall not be following 
Galileo's own derivation exactly, but the results will be the same. 
First, we recall the definition of average speed as the distance 
traversed Ad divided by the elapsed time At : 


Section 2.7 51 

This is a general definition and can be used to compute the average 
speed from measurement of Ad and At, no matter whether Ad and 
At are small or large. We can rewrite the equation as 

Ad = Vav X At 

This equation, still being really a definition of !;„,., is always true. 
For the special case of motion at a constant speed v, then Vav = v 
and therefore, Ad = i; x At. When the value of v is known (as, for 
example, when a car is driven with a steady reading of 60 mph on 
the speedometer), this equation can be used to figure out how far 
(Ad) the car would go in any given time interval (At). But in 
uniformly accelerated motion the speed is continually changing — so 
what value can we use for Vav"^ 

The answer involves just a bit of algebra and some plausible 
assumptions. Galileo reasoned (as others had before) that for any 
quantity that changes uniformly, the average value is just halfway 
between the beginning value and the final value. For uniformly 
accelerated motion starting from rest (where t^initiai ^ and ending 

at a speed Vanau this rule tells us that the average speed is halfway More generally the average speed 

between and rnnai - that is, t;„,. = y t'nnai. If this reasoning is would be 

correct, it follows that w ^ yjninai + Vfinai 

" av 2 

Ad = Jl^final X ^t 

for uniformly accelerated motion starting from rest. SG 2.11 and 2.12 

This relation could not be directly tested either, because the last 
equation still contains a speed factor. What we are trying to arrive 
at is an equation relating total distance and total time, without any 
need to measure speed. 

Now we look at Galileo's definition of uniform acceleration: 
a = Az;/At. We can rewrite this relationship in the form Av= aX At. 
The value of Ai; is just L'finai - ^^initiai; and i^initiai = for motion that 
begins from rest. Therefore we can write 

Az;= a X At 

l^final ~ ^initial — « X At 
X^final ^ a X At 

Now we can substitute this expression for Vami into the equation 
for Ad above. Thus if the motion starts from rest, and if it is 
uniformly accelerated (and if the average rule is correct, as we have 
assumed) we can write 

Ad = iVfinal X ^t 

= ^(a X At) X At 

Or. regrouping terms, 

Ad = ja(My 

This is the kind of relation Galileo was seeking -it relates total 
distance Ad to total time At, without involving any speed term. 

Before finishing, though, we will simplify the symbols in the 
equation to make it easier to use. If we measure distance and time 
from the position and the instant that the motion starts (dmitiai ^ 


Free Fall — Galileo Describes Motion 

SG 2.13 and 2.14 

SG 2.15 

Because we will use the expression 
t'fitu./f"fin;,i many times, it is simpler 
to write it as d/r--it is understood 
that d and f mean total distance and 
time interval of motion, starting 
from rest. 

and tinitiai = 0), then the intervals Ad and At have the values given 
by dfinai and tfinai- The equation above can therefore be written more 
simply as 

"final ~ 2"^^ final 

Remember that this is a very specialized equation — it gives the 
total distance fallen as a function of total time of fall but only if the 
motion starts from rest (t'lniuai ^ 0), if the acceleration is uniform 
(a = constant), and if time and distance are measured from the 
start (tinitiai = and di„i,iai = 0). 

Galileo reached the same conclusion, though he did not use 
algebraic forms to express it. Since we are dealing only with the 
special situation in which acceleration a is constant, the quantity 
■ja is constant also, and we can cast the conclusion in the form of 
a proportion: in uniform acceleration from rest, the distance 
traveled is proportional to the square of the time elapsed, or 

"final °~ t final 

For example, if a uniformly accelerating car starting from rest 
moves 10 m in the first second, in twice the time it would move 
four times as far, or 40 m in the first two seconds. In the first 3 
seconds it would move 9 times as far— or 90 m. 

Another way to express this relation is to say that the ratio 
cifinai to t^finai has a constant value, that is, 

^^= constant 

t final 

Thus a logical result of Galileo's original proposal for defining 
uniform acceleration can be expressed as follows: if an object 
accelerates uniformly from rest, the ratio dlt' should be constant. 
Conversely, any motion for which this ratio of d and t'^ is found to 
be constant for different distances and their corresponding times, 
we may well suppose to be a case of motion with uniform, 
acceleration as defined by Galileo. 

Of course, we still must test the hypothesis that freely falling 
bodies actually do exhibit just such motion. Recall that earlier we 
confessed we were unable to test directly whether Av/At has a 
constant value. Galileo showed that a logical consequence of a 
constant value of Av/At would be a constant ratio of dfi„a\ to t-fmai- 
The values for total time and distance of fall would be easier to 
measure than the values of short intervals Ad and At needed to find 
All. However, measuring the time of fall still remained a difficult 
task in Galileo's time. So, instead of a direct test of his hypothesis, 
Galileo went one step further and deduced an ingenious, indirect 

Q8 Why was the equation d = ^at- more promising for Galileo 
than a = Ai^/At in testing his hypothesis? 

Q9 If you simply combined the two equations Ad = i;At and 
Ai; = aAt it looks as if one might get the result Ad = aAt-. What is 
wrong with doing this? 

Section 2.8 

2.8 Galileo turns to an Indirect test 


Realizing that a direct quantitative test with a rapidly and 
freely falling body would not be accurate, Galileo proposed to make 
the test on an object that was moving less rapidly. He proposed a 
new hypothesis: if a freely falling body has an acceleration that is 
constant, then a perfectly round ball rolling down a perfectly 
smooth inclined plane will also have a constant, though smaller, 
acceleration. Thus Galileo claimed that if dit- is constant for a body 
falling freely from rest, this ratio will also be constant, although 
smaller, for a ball released from rest and rolling different distances 
down a straight inclined plane. 

Here is how Salviati described Galileo's own experimental test 
in Two New Sciences: 

A piece of wooden moulding or scantling, about 12 cubits 
long, half a cubit wide, and three finger- breadths thick, 
was taken; on its edge was cut a channel a little more 
than one finger in breadth; having made this groove very 
straight, smooth, and polished, and having lined it with 
parchment, also as smooth and polished as possible, we 
rolled along it a hard, smooth, and very round bronze ball. 
Having placed this board in a sloping position, by lifting 
one end some one or two cubits above the other, we rolled 
the ball, as I was just saying, along the channel, noting, 
in a manner presently to be described, the time required 
to make the descent. We repeated this experiment more 
than once in order to measure the time with an accuracy 
such that the deviation between two observations never 
exceeded one-tenth of a pulse beat. Having performed 
this operation and having assured ourselves of its 
reliability, we now rolled the ball only one-quarter of 
the length of the channel; and having measured the time 
of its descent, we found it precisely one-half of the 
former. Next we tried other distances, comparing the 
time for the whole length with that for the half, or with 
that for two-thirds, or three-fourths, or indeed for any 
fraction; in such experiments, repeated a full hundred 
times, we always found that the spaces traversed were to 
each other as the squares of the times, and this was true 
for all inclinations of the . . . channel along which we 
rolled the ball .... 

Note the careful description of the 
experimental apparatus. Today an 
experimenter would add to his 
verbal description any detailed 
drawings, schematic layouts or 
photographs needed to make it 
possible for other competent 
scientists to duplicate the 

This picture painted in 1841 by G. 
Bezzuoii, attempts to reconstruct an 
experiment Galileo is alleged to have 
made during his time as lecturer at 
Pisa. Off to the left and right are men 
of ill will: the blase Prince Giovanni 
de Medici (Galileo had shown a dredg- 
ing-machine invented by the prince to 
be unusable) and Galileo's scientific 
opponents. These were leading men 
of the universities; they are shown 
here bending over a book of Aristotle, 
where it is written in black and white 
that bodies of unequal weight fall 
with different speeds. Galileo, the 
tallest figure left of center in the 
picture, is surrounded by a group of 
students and followers. 



Free Fall-Galileo Describes Motion 

For each angle, the acceleration is 
found to be a constant. 

Galileo's technique for measuring 
time is discussed in the next section. 

SG 2.16 

Spheres rolling down planes of in- 
creasingly steep inclination. At 90° the 
inclined plane situation matches free 
fall. (Actually, the ball will start slip- 
ping instead of rolling long before the 
angle has become that large.) 

Galileo has packed a great deal of information into these lines. 
He describes his procedures and apparatus clearly enough to allow 
other investigators to repeat the experiment for themselves if they 
wished. Also, he gives an indication that consistent measurements 
can be made, and he restates the two chief experimental results 
which he believes support his free-fall hypothesis. Let us examine 
the results carefully. 

(a) First, he found that when a ball rolled down an incline at a 
fixed angle to the horizontal, the ratio of the distance covered to the 
square of the corresponding time was always the same. For 
example, if d^, d^, and d^ represent distances measured from the 
same starting point on the inclined plane, and t,, tj, and tg the 
corresponding times taken to roll down these distances, then 


In general, for each angle of incline, the value of dlt^ was 
constant. Galileo did not present his experimental data in the full 
detail which has become the custom since. However, his experiment 
has been repeated by others, and they have obtained results which 
parallel his (see data in SG 2.16). This is an experiment which you 
can perform yourself with the help of one or two other students. 
(The photographs on the next page show students in the Project 
Physics course doing this experiment and also show some of their 

(b) Galileo's second experimental finding relates to what happens 
when the angle of inclination of the plane is changed. He found 
that whenever the angle changed, the ratio dit- took on a new value, 
although for any one angle it remained constant regardless of 
distance of roll. GalUeo confirmed this by repeating the experiment 
"a full hundred times" for each of many different angles. After 
finding that the ratio d/t- was constant for each angle of inclination 
for which measurements of t could be carried out conveniently, 
Galileo was willing to extrapolate. He concluded that the ratio dlt^ 
is a constant even for larger angles, where the motion of the ball is 
too fast for accurate measurements of t to be made. Finally, Galileo 
reasoned that in the particular case when the angle of inclination 
became 90°, the ball would move straight down — and so becomes 
the case of a falling object. By his reasoning, d/t- would still be 
some constant in that extreme case (even though he couldn't say 
what the numerical value was.) 

Because Galileo had deduced that a constant value of dIt- was 
characteristic of uniform acceleration, he could conclude at last 
that free fall was uniformly accelerated motion. 

Q10 In testing his hypothesis that free fall motion is uniformly 
accelerated, Galileo made the unproved assumption that (check one 
or more): 

(a) dlt^ is constant. 


Free Fall — Galileo Describes Motion 

For problems that will check and 
extend your understanding of 
uniform acceleration. See SG 2.17 
through 2.24. 

(b) the acceleration has the same value for all angles of 
inclination of the plane. 

(c) the results for small angles of inclination can be 
extrapolated to large angles. 

(d) the speed of the ball is constant as it rolls. 

(e) the acceleration of the rolling ball is constant if the 
acceleration in free fall is constant, though the value of the 
two constants is not the same. 

Q11 Which of the following statements best summarizes the 
work of Galileo on free fall when air friction is negligible? (Be 
prepared to defend your choice.) Galileo: 

(a) proved that all objects fall at exactly the same speed 
regardless of their weight. 

(b) proved that for any freely falling object the ratio dlt^ is 
constant for any distance of fall. 

(c) proved that an object rolling down a smooth incline 
accelerates in the same way as (although more slowly than) 
the same object falling freely. 

(d) supported indirectly his assertion that the speed of an object, 
falling freely from rest is proportional to the elapsed time. 

(e) made it clear that until a vacuum could be produced, it 
would not be possible to settle the free-fall question once 
and for all. 

2.9 Doubts about Galileo's procedure 

This whole process of reasoning and experimentation looks long 
and involved on first reading, and some doubts may well arise 
concerning it. For example, was Galileo's measurement of time 
precise enough to establish the constancy of dlt^ even for the case 
of a slowly rolling object? In his book, Galileo tries to reassure 
possible critics by providing a detailed description of his 
experimental arrangement (thereby inviting any skeptics to try it 
for themselves): 

For the measurement of time, we employed a large 
vessel of water placed in an elevated position; to the 
bottom of this vessel was soldered a pipe of small 
diameter giving a thin jet of water, which we collected 
in a small cup during the time of each descent, whether 
for the whole length of the channel or for a part of its 
length; the water thus collected was weighed on a very 
accurate balance; the differences and ratios of these 
weights gave us the differences and ratios of the time 
intervals, and this with such accuracy that, although the 
operation was repeated many, many times, there was no 
appreciable discrepancy in the results. 

The water clock described by Galileo was not invented by him. 
Indeed, there are references to water clocks in China as early as the 

Section 2.9 


sixth century B.C., and they were probably used in Babylonia and 
India even earlier. In the early 16th century a good water clock was 
the most accurate of the world's instruments for measuring short 
time intervals. It remained so until shortly after Galileo's death, 
when the work of Christian Huygens and others led to practical 
pendulum clocks. When better clocks became available, Galileo's 
results on inclined-plane motion were confirmed. 

Another reason for questioning Galileo's results is related to the 
great difference between free fall and rolling motion on a slight 
incline. Galileo does not report what angles he used in his 
experiment. However, as you may have found out from doing a 
similar experiment, the angles must be kept rather small. As the 
angle increases, the speed of the ball soon becomes so great that it 
is difficult to measure the times involved. The largest usable angle 
reported in a recent repetition of Galileo's experiment was only 
6°. (See SG 2.15) It is not hkely that Galileo worked with much 
larger angles. This means that the extrapolation to free fall (90° 
incline) is a large one, perhaps much too large for a cautious 
person — or for one not already convinced of Galileo's argument. 

Still another reason for questioning Galileo's results is the 
observation that, as the angle of incline is increased, there comes 
a point where the ball starts to slide as well as roll. This change in 
behavior could mean that the motion is very different at large 
angles. Galileo does not discuss these cases. It is surprising that he 
apparently did not repeat the experiment with blocks which would 
slide, rather than roll, down a smooth incline. If he had, he would 
have found that for accelerated sliding motion the ratio dit- is also 
a constant, although the constant has a different numerical value 
than for rolling at the same angle. 

01 2 Which of the following statements could be regarded as 
major reasons for doubting the validity of Galileo's procedure? 

(a) His measurement of time was not sufficiently accurate. 

(b) He used too large an angle of inclination in his experiment. 

(c) It is not clear that his results apply when the ball can slide 
as well as roll. 

(d) In Galileo's experiment the ball was rolling, and therefore 
he could not extrapolate to the case of free fall where the 
ball did not roll. 

(e) dlt^ was not constant for a sliding object. 

Early water clock 

SG 2.25 

2.10 Consequences of Galileo's work on motion 

Galileo seems to have been well aware that one cannot get the 
correct numerical value for the acceleration of a body in free fall 
simply by extrapolating the results to increasingly large angles of 
inclination. He did not attempt to calculate a numerical value for 
the acceleration of freely falling bodies. But for his purposes it was 
enough that he could support the hypothesis that the acceleration is 
constant for any given body, whether rolling or falling. This is the 


Free Fall — Galileo Describes Motion 

We now know by measurement that 
the magnitude of the acceleration of 
gravity, symbol a^, is about 9.8 
m/sec per sec, or 32 ft/sec per sec, 
at the earth's surface. The Project 
Physics Handbook contains five 
different experiments for finding a 
value of a^. (For many problems, 
the approximate value 10 m/sec/sec 
is satisfactory.) 

SG 2.26 

You can derive this equation. (See 
SG 2.27) 

SG 2.28 and 2.29 

first consequence of Galileo's work, one that has been fully borne 
out by all subsequent tests. 

Second, if spheres of different weights are allowed to roll down 
an inclined plane set at a given angle, they turn out to have the 
same acceleration. We do not know how much experimental 
evidence Galileo himself had for this conclusion, but it is consistent 
with the observations for freely falling objects. It is consistent also 
with his "thought experiment" by which he argued that bodies of 
different weights fall at the same rate (aside from the comparatively 
small effects of air resistance). His results provided a decisive 
refutation of Aristotle's theory of motion. 

Third, Galileo developed a mathematical theory of accelerated 
motion from which other predictions about motion could be 
derived. We will mention just one example here, which will turn 
out to be very useful in Unit 3. Recall that Galileo chose to define 
acceleration as the rate at which the speed changes with time. He 
then found by experiment that falling bodies actually do experience 
equal changes of speed in equal times, and not in equal distances 
as some had supposed. Still, the idea of something changing by 
equal amounts in equal distances has an appealing simplicity, too. 
One might ask if there isn't something that does change in that way 
during uniform acceleration. In fact, there is. It follows without 
any new assumptions that, during uniform acceleration from rest, 
the square of the speed changes by equal amounts in equal 
distances. There is a mathematical equation which expresses this 
result: If z/inuiai = 0, and a = constant, then 

In words: if an object starts from rest and moves with uniform 
acceleration, then the square of its speed at any point is equal to 
twice the product of its acceleration and the distance it has moved. 
(We shall see the importance of this relation in Unit 3.) 

These consequences of Galileo's work, important as they are to 
the development of physics, would scarcely have been enough to 
bring about a revolution in science by themselves. No sensible 
scholar in the seventeenth century would have given up his belief 
in the Aristotelian cosmology only because some of its predictions 
had been refuted in the case of falling (or rolling) bodies. But 
Galileo's work on free-fall motion helped to prepare the way for the 
development of a new kind of physics, and indeed a new cosmology, 
by planting the seeds of doubt about the crucial assumptions of 
Aristotelian science. For example, when it was recognized that all 
bodies fall with equal acceleration if air friction is negligibly small, 
then the whole Aristotelian explanation of falling motion (Section 
2.1) broke down. 

The most agitating scientific problem during Galileo's lifetime 
was not in mechanics but in astronomy. A central question in 
cosmology was whether the earth or the sun is the center of the 
universe. Galileo supported the view that the earth and other 
planets revolve around the sun, a view entirely contrary to 

Section 2.10 59 

Aristotelian cosmology. But to support such a view required a 
physical theory of why and how the earth itself moved. Galileo's 
work on free fall and other motions turned out to be just what was 
needed to begin to construct such a theory. His work did not have its 
full effect, however, until it had been combined with the 
investigations of forces and motion by the English scientist Isaac 
Newton. But as Newton acknowledged, Galileo was the pioneering 
pathfinder. (In the next chapter we will consider Newton's work on 
force and motion. In Chapter 8, after studying about motion in the 
heavens, we will return to Newton's laws and the revolution they 
began in science.) 

Galileo's work on motion introduced a new and significant 
method of doing scientific research, a method as applicable today 
as when GalHeo demonstrated it. The basis of this procedure is a 
cycle, repeated as often as necessary, entirely or in part, until a 
satisfactory theory has emerged: general observation -^ hypothesis 
-* mathematical analysis or deduction from hypothesis ->■ 
experimental test of deduction -* modification of hypothesis in light 
of test, and so forth. 

While the steps in the mathematics are often determined 
mainly by "cold logic," this is not so for the other parts of the SG 2.30 

process. A variety of paths of thought can lead to the hypothesis in 
the first place. A new hypothesis can come from an inspired hunch 
based on general knowledge of the experimental facts, or from a 
desire for mathematically simple statements, or from modifying a 
previous hypothesis that failed. Moreover, there are no general 
rules about exactly how well the experimental data must agree 
with the theoretical predictions. In some areas of science, a theory 
is expected to be accurate to better than one 1/ 1000th of one 
percent; in other areas, or at an early stage of any new work, one 
might be delighted to find a theory from which he could make 
predictions with an error of only 50 percent. Finally note that while 
experiment has an important place in this process, it is not at all 
the only or even the main element. On the contrary, experiments 
are worthwhile only in conjunction with the other steps in the 

The general cycle of observation, hypothesis, deduction, test, 
modification, etc., so skillfully demonstrated by Galileo in the 
seventeenth century, commonly appears in the work of scientists 
today. Though there is no such thing as the scientific method, some 
form of this cycle is almost always present in scientific research. It 
is used not out of respect for Galileo as a towering figure in the 
history of science, but because it works so well so much of the 

Galileo himself was aware of the value of both the results and 
the methods of his pioneering work. He concluded his treatment of 
accelerated motion by putting the following words into the mouths 
of the commentators in his book: 

Salviati: ... we may say the door is now opened, for the 

60 Free Fall -Galileo Describes Motion 

first time, to a new method fraught with numerous and 
wonderful results which in future years will command 
the attention of other minds. 

Sagredo: I really believe that . . . the principles which are 
set forth in this little treatise will, when taken up by 
speculative minds, lead to another more remarkable 
result; and it is to be believed that it will be so on account 
of the nobility of the subject, which is superior to any 
other in nature. 

During this long and laborious day, I have 
enjoyed these simple theorems more than their proofs, 
many of which, for their complete comprehension, would 
require more than an hour each; this study, if you will 
be good enough to leave the book in my hands, is one 
which I mean to take up at my leisure after we have 
read the remaining portion which deals with the motion 
of projectiles; and this if agreeable to you we shall take 
up tomorrow. 

Salviati: I shall not fail to be with you. 

Many details of physics, mathe- Q13 which one of the following was not a result of Galileo's 

matics and history have appeared , ^. o 

• ».-• u * .- ■ s .^ work on motion .'' 

in this chapter. For a review of the 

most important ideas, see SG 2.31, ^^^ The correct numerical value of the acceleration in free fall 

2.32, and 2.33. was obtained by extrapolating the results for larger and 

larger angles of inclination. 

(b) If an object starts from rest and moves with uniform 
acceleration a through a distance d, then the square of its 
speed will be proportional to d. 

(c) Bodies rolling on a smooth inclined plane are uniformly 
accelerated (according to Galileo's definition of acceleration). 


2.1 Note that at the beginning of each chapter in 
this book there is a Hst of the section titles. This 
is a sort of road map you can refer to from time to 
time as you study the chapter. It is important, 
expecially in a chapter such as this one, to know 
how the part you are studying relates to what 
preceded it and to have some idea of where it is 
leading. For this same reason, you will find it very 
helpful at first to skim through the entire chapter, 
reading it rapidly and not stopping to puzzle out 
parts that you do not quickly understand. Then 
you should return to the beginning of the chapter 
and work your way through it carefully, section 
by section. Remember also to use the end-of-section 
questions to check your progress. 
The Project Physics learning materials particularly 
appropriate for Chapter 2 include: 


A Seventeenth-Century Experiment 

Twentieth Century Version of Galileo's 


Measuring the Acceleration Due to 

Gravity, Og 

When is Air Resistance Important? 
Measuring Your Reaction Time 
Falling Weights 

Reader Article 
On the Scientific Method 

Film Loops 

Acceleration Due to Gravity — Method I 
Acceleration Due to Gravity — Method II 


Derivation of d 

Vit + jat^ 

2.2 Aristotle's theory of motion seems to be 
supported to a great extent by common sense 
experience. For example, water bubbles up 
through earth at springs. When sufficient fire is 
added to water by heating it, the resulting mixture 
of elements (what we call steam) rises through 
the air. Can you think of other examples? 

2.3 Drop sheets of paper with various degrees of 
"crumpling." Try to crumple a sheet of paper 
tight enough that it will fall at the same rate as a 
tennis ball. Can you explain the results with 
Aristotle's theory? 

2.4 Compare Aristotle's hypothesis about falling 
rate (weight divided by resistance) with 
PhUoponus' (weight minus resistance) for some 
extreme cases: a very heavy body with no 
resistance, a very light body with great resistance. 
Do the two hypotheses suggest very different 

2.5 Consider Aristotle's statement "A given 
weight moves [falls] a given distance in a given 
time; a weight which is as great and more moves 
the same distances in less time, the times being 
in inverse proportion to the weights. For instance. 

if one weight is twice another, it will take half as 
long over a given movement." (De Caelo) 

Indicate what Simplicio and Salviati each 
would predict for the falling motion in these 

(a) A 2-pound rock falls from a cliff and, 
whUe dropping, breaks into two equal 

(b) A hundred-pound rock is dropped at the 
same time as one hundred 1-pound 
pieces of the same type of rock. 

(c) A hundred 1-pound pieces of rock, falling 
from a height, drop into a draw-string 
sack which closes, pulls loose and falls. 

2.6 Tie two objects of greatly different weight 
Gike a book and a pencil) together with a piece of 
string. Drop the combination with different 
orientations of objects. Watch the string. In a few 
sentences summarize your results. 


2.7 A good deal of work preceded that of Galileo 
on the topic of motion. In the period 1280-1340, 
mathematicians at Merton College, Oxford, 
carefully considered different quantities that 
change with the passage of time. One result that 
had profound influence was a general theorem 
known as the "Merton Theorem" or "Mean Speed 

This theorem might be restated in our 
language and applied to uniform acceleration as 
follows: the distance an object goes during some 
time while its speed is changing uniformly is the 
same distance it would go if it went at the average 
speed the whole time. 

(a) First show that the total distance traveled 
at a constant speed can be expressed as 
the area under the graph line on a speed- 
time graph. ("Area" must be found in speed 
units X time units.) 

(b) Assume that this area represents the 
total distance even when the speed is not 
constant. Draw a speed vs. time graph for 
uniformly increasing speed and shade in 
the area under the graph line. 




(c) Prove the "Merton Rule" by showing that 
the area is equal to the area under a 
constant-speed line at the average speed. 

2.H According to Galileo, uniform acceleration 
means equal Av's in equal At's. Which of the 
following are other ways of expressing the same 

(a) Av is proportional to At 

(b) AvIAt = constant 

(c) the speed-time graph is a straight line 

(d) V is proportional to t 

2.9 In the Two New Sciences Galileo states, ". . . 
for so far as I know, no one has yet pointed out 
that the distances traversed, during equal intervals 
of time, by a body falling from rest, stand to one 
another in the same ratio as the odd numbers 
beginning with unity (namely 1:3:5:7 ...)...." 

The area beneath the curve in a speed-time 
graph represents the distance traveled during 
some time interval. Using that idea, give a proof 
that the distances an object falls in successive 
equal time intervals will be in the ratios of the odd 

2.10 Using whatever modem equipment you 
wish, describe how you could find an accurate 
value for the speed of a falling object just before 
striking the ground. 

2.11 Show that the expression 

,, _ ^Initial + ^final 
Vav 2 

is equivalent to the "Merton Rule" discussed in 
SG 2.7. 

2.12 For any quantity that changes uniformly, 
the average is the sum of the initial and final 
values divided by two. Try it out for any quantity 
you may choose -for example: what is the 
average age in a group of five people having 
individually the ages of 15, 16, 17, 18, and 19 
years? What is your average earning power over 
five years if it grows steadily from $5000 

per year at the start to $9000 per year at the end? 

2.13 Several special assumptions have been 
made in arriving at the equation d = jat^. What 
is the "unwritten text" behind it? 

2.1 I Lt. Col. John L. Stapp achieved a speed of 
632 mph (284 m/sec) in an experimental rocket 
sled at the Holloman Air Base Development 
Center, Alamogordo, New Mexico, on March 19, 
1954. Running on rails and propelled by nine 
rockets, the sled reached its top speed within 5 
seconds. Stapp survived a maximum acceleration 
of 22 g's in slowing to rest during a time interval 
of I7 seconds (one g is an acceleration equal in 
magnitude to that due to gravity; 22 g's means 
22 X Og.) 

(a) Find the average acceleration in reaching 
maximum speed. 

(b) How far did the sled travel before 
attaining maximum speed? 

(c) Find the average acceleration while 

2.1.T Derive the expression dlt^ = constant from 
the expression d = jat^. 

2.1(i Table 2.1 reports results from a recent 
repetition of Galileo's experiment in which the 
angle of inclination was 3.73° (Science, 133, 19-23, 
June 6, 1961). A water clock with a constant-level 
reservoir was used. 

TABLE 2.1 

TIME (measured in 



liliters of water) 























Do these data really support Galileo's 
assertion that d/t^ is constant? Explain your 

2.17 Indicate whether the following statements 
are true or false when applied to the strobe photo 
below : 

(a) The speed of the ball 
is greater at the 
bottom than at the top. 

(b) This could be a freely 
falling object. (Make 
measurements on 

(c) This could be a ball 
thrown straight 

(d) If (b) is true, the 
speed increases with 
time because of the 
acceleration due to 

(e) If (c) is true, the speed 
decreases with time 
because of the effect 
of gravity; this effect 
could still be called 
acceleration due to 



2.18 (a) Show by means of equations that 

Galileo's statement in SG 2.9 follows from 
dlt^ = constant for free fall from rest, 
(b) The time interval between strobe flashes 
was 0.35 sec. Use this information to 
make a rough graph of d vs. t, also one of 
V vs. t, and find the acceleration of the 

2.19 The photograph in the figure below is of a 
ball thrown upward. The acceleration due to 
gravity increases the speed of the ball as it goes 
down from its highest point (like any free-falling 
object), if air friction is negligible. But the 
acceleration due to gravity, which does not change, 
acts also whUe the ball is still on its way up, and 
for that portion of the path causes the baU to slow 
down as it rises. 

Stroboscopic photograph of a ball 
thrown into the air. 

When there is both up and down motion, it 
will help to adopt a sign convention, an arbitrary 
but consistent set of rules, similar to designating 
the height of a place with respect to sea level. To 
identify distances measured above the point of 
initial release, give them positive values, for 
example, the distance at B or at D, measured from 

the release level, is about +60 cm and +37 cm, 
respectively. If measured below the release level, 
give them negative values; for example, E is at 
—23 cm. Also, assign a positive value to the speed 
of an object on its way up to the top (about +3 
m/sec at A) and a negative value to a speed a 
body has on the way down after reaching the top 
(about —2 m/sec at D and —6 m/sec at E). 
(a) Fill in the table with + and — signs. 











(b) Show that it foUows from this convention 
and from the definition of a = Az;/At that 
the value or sign given to the acceleration 
due to gravity is negative, and for both 
parts of the path. 

(c) What would the sign of acceleration due 
to gravity be in each case if we had 
chosen the + and — sign conventions just 
the other way, that is associating — with 
up, + with down? 

2.20 Draw a set of points (as they would appear 
in a strobe photo) to show the successive positions 
of an object that by our convention in SG 2.19 
had a positive acceleration, that is, "upward." Can 
you think of any way to produce such an event 

2.21 Memorizing equations will not save you 
from having to think your way through a problem. 
You must decide if, when and how to use 
equations. This means analyzing the problem to 
make certain you understand what information 
is given and what is to be found. Test yourself 
on the following problem. Assume that the 
acceleration due to gravity is nearly enough equal 
to 10 m/sec/sec. 

Problem: A stone is dropped from rest from 
the top of a high cliff". 

(a) How far has it fallen after 1 second? 

(b) What is the stone's speed after 1 second 
of fall? 

(c) How far does the stone fall during the 
second second? (That is, from the end of 
the first second to the end of the second 

2.22 From the definition for a, show it follows 
directly that t^nnai = ^'initial + at for motion with 
constant acceleration. Using this relation, and the 
sign convention in SG 2.19. answer the questions 
below. (Assume Og = 10 m/sec/sec.) An object is 
thrown straight upward with an initial speed of 
20 m/sec. 

(a) What is its speed after 1.0 sec? 

(b) How far did it go in this first second? 



(c) How long did the object take to reach its 
maximum height? 

(d) How high is this maximum height? 

(e) When it descends, what is its final speed 
as it passes the throwing point? 

If you have no trouble with this, you may wish 
to try problems SG 2.23 and 2.24. 

2.23 A batter hits a pop fly that travels straight 
upwards. The ball leaves his bat with an initial 
speed of 40 m/sec. (Assume a„ = 10 m/sec/sec) 

(a) What is the speed of the ball at the end of 
2 seconds? 

(b) What is its speed at the end of 6 seconds? 

(c) When does the ball reach its highest point? 

(d) How high is this highest point? 

(e) What is the speed of the ball at the end of 
10 seconds? (Graph this series of speeds.) 

(f ) What is its speed just before it is caught 
by the catcher? 

2.24 A ball starts up an inclined plane with a 
speed of 4 m/sec, and comes to a halt after 2 

(a) What acceleration does the ball 

(b) What is the average speed of the ball 
during this interval? 

(c) What is the ball's speed after 1 second? 

(d) How far up the slope will the ball travel? 

(e) What will be the speed of the ball 3 
seconds after starting up the slope? 

(f ) What is the total time for a round trip to 
the top and back to the start? 

2.25 As Director of Research in your class, you 
receive the following research proposals from 
physics students wishing to improve upon Galileo's 
free-fall experiment. Would you recommend 
support for any of them? If you reject a proposal, 
you should make it clear why you do so. 

(a) "Historians believe that Galileo never 
dropped objects from the Leaning Tower 
of Pisa. But such an experiment is more 
direct and more fun than inclined plane 
experiments, and of course, now that 
accurate stopwatches are available, it can 
be carried out much better than in 
Galileo's time. The experiment involves 
dropping, one by one, different size spheres 
made of copper, steel, and glass from the 
top of the Leaning Tower and finding how 
long it takes each one to reach the 
ground. Knowing d (the height of the 
tower) and time of fall t, I will substitute 
in the equation d = jat' to see if the 
acceleration a has the same value for each 

(b) "An iron shot will be dropped from the 
roof of a 4-story building. As the shot falls, 
it passes a window at each story. At each 
window there will be a student who starts 
his stopwatch upon hearing a signal that 
the shot has been released, and stops the 
watch as the shot passes his window. 
Also, each student records the speed of the 

shot as it passes. From his own data, each 
student will compute the ratio vlt. I 
expect that all four students will obtain 
the same numerical value of the ratio." 
(c) "Galileo's inclined planes dilute motion 
all right, but the trouble is that there is 
no reason to suppose that a ball rolling 
down a board is behaving like a ball 
falling straight downward. A better way 
to accomplish this is to use light, fluffy, 
cotton balls. These will not fall as rapidly 
as metal spheres, and therefore it would 
be possible to measure the time of the 
fall t for different distances. The ratio dlt^ 
could be determined for different distances 
to see if it remained constant. The 
compactness of the cotton ball could then 
be changed to see if a different value was 
obtained for the ratio." 

2.26 A student on the planet Arret in another 
solar system dropped an object in order to 
determine the acceleration due to gravity at that 
place. The following data are recorded (in local 





(in surgs) 

(in welfs) 

(in surgs) 

(in welfs) 





















(a) What is the acceleration due to gravity on 
the planet Arret, expressed in welfs/surg*? 

(b) A visitor from Earth finds that one welf 
is equal to about 6.33 cm and that one 
surg is equivalent to 0.167 sec. What 
would this tell us about Arret? 

2.27 (a) Derive the relation v^ = 2ad from the 

equations d = ^at^ and v = at. What 
special conditions must be satisfied for the 
relation to be true? 
(b) Show that if a ball is thrown straight 
upward with an initial speed v it will rise 
to a height 

^ = 2^ 

2.28 Sometimes it is helpful to have a special 
equation relating certain variables. For example, 
for constant acceleration a, the final speed v, is 
related to initial speed v^ and distance traveled d 

Vf^ = Vi^ + 2ad 

Try to derive this equation from some others you 
are familiar with. 



2.29 Use a graph like the one sketched below, 
and the idea that the area under the graph line in 
a speed-time graph gives a value for the distance 
traveled, to derive the equation 

d = v,t + jaf 

can be summarized in the three equations listed 


2.30 List the steps by which Galileo progressed 
from his first definition of uniformly accelerated 
motion to his final confirmation that this definition 
is useful in describing the motion of a freely 
falling body. Identify each step as a hypothesis, 
deduction, observation, or computation, etc. What 
limitations and idealizations appear in the 

2.31 In these first two chapters we have been 
concerned with motion in a straight line. We have 
dealt with distance, time, speed and acceleration, 
and with the relationships among them. 
Surprisingly, most of the results of our discussion 


Ad ^v , I . 

The last of these equations can be applied only 
to those cases where the acceleration is constant. 
Because these three equations are so useful, they 
are worth remembering (together with the 
limitation on their use). 

(a) State each of the three equations in words. 

(b) Make up a simple problem to demonstrate 
the use of each equation. (For example: 
How long will it take a jet plane to travel 
3200 miles if it averages 400 mi/hr?) 
Then work out the solution just to be sure 
the problem can be solved. 

(c) Derive the set of equations which apply 
whether or not the initial speed is zero. 

2.32 Show to what extent the steps taken by 
Galileo on the problem of free fall, as described 
in Sections 2.5 through 2.8, follow the general 
cycle in the scientific process. 

2.33 What is wrong with the following common 
statements? "The Aristotelians did not observe 
nature. They took their knowledge out of old 
books which were mostly wrong. Galileo showed it 
was wrong to trust authority in science. He did 
experiments and showed everyone directly that 
the old ideas on free fall motion were in error. He 
thereby started science, and also gave us the 
scientific method." 


3.1 "Explanation" and the laws of motion 

3.2 The Aristotelian explanation of motion 

3.3 Forces in equilibrium 

3.4 About vectors 

3.5 Newton's first law of motion 

3.6 The significance of the first law 

3.7 Newton's second law of motion 

3.8 Mass, weight, and free fall 

3.9 Newton's third law of motion 

3.10 Using Newton's laws of motion 

3.11 Nature's basic forces 


*'^.^- '"^r^ 


The Birth of Dynamics — 
Newton Explains Motion 

3.1 "Explanation" and the laws of motion 

Kinematics is the study of how objects move, but not why they 
move. Galileo investigated many topics in kinematics with 
insight, ingenuity, and gusto. The most valuable part of that work 
dealt with special types of motion, such as free fall. In a clear and 
consistent way, he showed how to describe the motion of objects 
with the aid of mathematical ideas. 

When Isaac Newton began his studies of motion in the second 
half of the seventeenth century, Galileo's earlier insistence that 
"the present does not seem to be the proper time to investigate the 
cause of the acceleration of natural motion . . . ." was no longer 
appropriate. Indeed, because Galileo had been so effective in 
describing motion, Newton could turn his attention to dynamics, the 
study of why an object moves the way it does — why it starts to 
move instead of remaining at rest, why it speeds up or moves on a 
curved path, and why it comes to a stop. 

How does dynamics differ from kinematics? As we have seen in 
the two earlier chapters, kinematics deals with the description of 
motion. For example, in describing the motion of a stone dropped 
from a cliff, we can write an equation showing how the distance d 
through which the stone has dropped is related to the time t the 
stone has been falling. We can find the acceleration and the final 
speed attained during any chosen time interval. But when we have 
completed our description of the stone's motion, we are still not 
satisfied. Why, we might ask, does the stone accelerate rather than 
fall with a constant speed? Why does it accelerate uniformly as 
long as air friction is negligible? To answer these questions, we 
will have to add to our store of concepts those of force and mass; 
and in answering, we are doing dynamics. Dynamics goes beyond 
kinematics by taking into account the cause of the motion. 


SG 3.1 

Some kinematics concepts: position, 
time, speed, acceleration. 
Some dynamics concepts: mass, 
force, momentum (Ch. 9), energy 
(Ch. 10). 


The Birth of Dynamics- Newton Explains Motion 

In Chapter 4 we will take up motion 
also along curved paths. 

Newton's First Law: Every object 
continues in its state of rest or of 
uniform motion in a straight line 
unless acted upon by an unbalanced 

Newton's Second Law: The 
acceleration of an object is directly 
proportional to, and in the same 
direction as, the unbalanced force 
acting on it, and inversely pro- 
portional to the mass of the object. 

Newton's Third Law: To every action 
there is always opposed an equal 
reaction; or, mutual actions of two 
bodies upon each other are always 
equal and in opposite directions. 

In our study of kinematics in Chapters 1 and 2, we encountered 
four situations: an object may: 

(a) remain at rest; (b) move uniformly in a straight line; (c) speed 
up during straight-line motion; (d) slow down during straight-line 

Because the last two situations are examples of acceleration, the 
list could really be reduced to: 

(a) rest; (b) uniform motion; and (c) acceleration. 

Rest, uniform motion, and acceleration are therefore the 
phenomena we shall try to explain. But the word "explain" must 
be used with care. To the physicist, an event is "explained" when he 
can demonstrate that the event is a logical consequence of a law 
he has reason to believe is true. In other words, a physicist with 
faith in a general law "explains" an observation by showing that it 
is consistent with the law. In a sense, the physicist's job is to show 
that the infinite number of separate, different-looking occurrences 
all around and within us are merely different manifestations or 
consequences of some general rules which describe the way the 
world operates. The reason this approach to "explanation" works 
is still quite remarkable: the number of general rules or "laws" of 
physics is astonishingly small. In this chapter we shall learn three 
such laws. Taken together with the mathematical schemes of 
Chapters 1 and 2 for describing motion, they will suffice for our 
understanding of practically all motions that we can readily 
observe. And in Unit 2 we shall have to add just one more law (the 
law of universal gravitation), to explain the motions of stars, 
planets, comets, and satellites. In fact, throughout physics one sees 
again and again that nature has a marvelous simplicity. 

To explain rest, uniform motion, and acceleration of any 
object, we must be able to answer such questions as these: Why 
does a vase placed on a table remain stationary? If a dry-ice disk 
resting on a smooth, level surface is given a brief push, why does it 
move with uniform speed in a straight line rather than slow down 
noticeably or curve to the right or left? Answers to these (and 
almost all other) specific questions about motion are contained 
either directly or indirectly in the three general "Laws of Motion" 
formulated by Isaac Newton. These laws appear in his famous 
book, Philosophiae Naturalis Principia Mathematica (Mathe- 
matical Principles of Natural Philosophy, 1687), usually referred 
to simply as The Principia. They are among the most basic laws in 
physics to this day. 

We shall examine Newton's three laws of motion one by one. 
If your Latin is fairly good, try to translate them from the 
original. A modernized version of Newton's text of these laws, in 
English, is reproduced in the margin at the left. 

Before we look at Newton's contribution, it will be instructive 
to find out how other scientists of Newton's time, or earlier, might 
have answered questions about motion. One reason for doing this 
now is that many people who have not studied physics still show 
Intuitively a bit of the pre-Newtonian viewpoint! Let us look at 
what we must overcome. 

Section 3.2 







Cfrftu omae ftrftv<T<ire in ftatn fno jHiefcndi vcl movendi tmifof 
miter in dirtQwH, Kifi fiatemts aviribnf imfrtffu ccgltur fialiim 
iUwH tnHart. 

PRo^iKi perfeverant in moribusfuis ni/i quatenus a refiHcn- 
m. leris retardannir & vi gravitatis impelluntur dcoifum. 
Trochui, cujus partes cohaerendo perpetuo r«nihunt Me 
a motibus refiillneis , non Cfflae rotaii nil! quatcnus ab acre re- 
tardatur. Ma^ra auton Planctarum & Cometarum corpora n»o- 
tus fuos & progrcflllvos & circulares in fpatiis minus rcuftentibus 
6£los ci»Krvant duitHK. 

Lex. n. . 

UKtiHuMem iMtti/ profartimdiem effevi mctrici imfreff'^, ^ fieri fe- 
ewrdHM liaitm rtSiam qua vit ill* imfrimitur. 

Si vb aliqua oiotum qucmvis gcnerct, dupla diiplum, tripla tri- 
plum gcserabit, five fimul&fotml, fivcgradatim& fucccflivcim- 
preila iilcrk. Et hie motus quontitn in eaodcm fempcr plagam 
cumvigcneratiice dctcrminatur, ficofpiuantca movebatur, mo- 
tu3e)iisvc!confpirantiaddltur, vcl contrario fubducimr, vcl oblk 
^y> oblique ad;icitur, & cum (o i^uodumutriurqidetenninacio- 
aaa compooitui. Lex. Ilt- 


Lex. UL 

A &i> i u CMtTarUm femptr ^ itfiultm ejtertadkiuni •■fiuit , 
dtunim tHitnet in Je muino femfer ejje nputtt ^ nr fttUt etrtri 

Qlticquid pmnkvclcrafaic altenim, tanrundonab copmnitur 
vcl trabinir. Siquis bpidem digito prnnit, p tewk u r & lin)ui 
^iginua lapide. Ste(]uuslap«kin fimi aliegatum trahit, ratrahe- 
tur ctiam & equub xquaiiter in lapidein: nam funis iirri(t(}idHiciitus 
eodciD rcbicandi fc conatu urgeSit Equum verfus lapidcsi, ac la- 
pidem'ver&5 equum, tantumq, impedietprogrcfTumunius quan^ 
njm promovet progrcfTum aTtcrius. Si corpus afiquod in corpus 
afiud inipiflgcns, niotume)usvi fua quomcxjocunqt aiutaverit, i- 
deni quo^Ue vKifl^m in nioni proptk) eatid<uii inubtknnniii pat'; 
fern coBffariam vi iktam^th «qualitatnn pfcflto;iis anjr r ^ 
futnbit. Hisadion)buts<{uale5fiuiKuiuta(i6nc<DOO vcm 1.. 
iedmotuum, ( icilicct^in corporJbui non armndciiDfcJi^ 
tatiooet entna vclockanun, m contracus ttkkis part'et 60* 
AMtus zqualiter mutalttur, funt corpodbus rtc^tfooe proi 

cofo!. I. ; ;;• 

Corpiu virAtu aeiJHitQif iitgoKaltm ftriBtUgnmm 
defcriber*^ ^mo laHmftf^Hir. 

Si corpus dato tempore, vifola M, 
fcrretur ab /* ad B, & vi fola N, ab 
^ ad C, complcatur paralielograni- 
mum ABDC, tc vi utraoi feretur id 
eodein tempore ab A ti D. Nam 
quontim vb tf aeit fccunduin lineam 
.^C ipfi B D parallcliin, hxc vis nihil mutab^t velotitatera a«ce- 
dendi ad lineam illam B D a vi altera genitam. Accedet iginjr 
«<f pus eodem tempore ad lineam B D five vis N imprimatur, five 
afqi adeo in fine illius temporb xeperictur atcubi in linea 


Q1 A baseball is thrown straight upward. Which of these 
questions about the baseball's motion are kinematic and which 

(a) How high will the ball go before coming to a stop and starting 

(b) How long will it take to reach that highest point? 

(c) What would be the effect of throwing it upward twice as hard? 

(d) Which takes longer, the trip up or the trip down? 

(e) Why does the acceleration remain the same whether the ball 
is moving up or down? 

3.2 The Aristotelian explanation of motion 

The idea of force played a central role in the dynamics of 
Aristotle, twenty centuries before Newton. You will recall from 
Chapter 2 that in Aristotle's physics there were two types of motion 
— "natural" motion and "violent" motion. For example, a falling 
stone was thought to be in "natural" motion (towards its natural 
place), but a stone being steadily lifted was thought to be in 
"violent" motion (away from its natural place). To maintain this 
uniform violent motion, a force had to be continuously applied. 
Anyone lifting a large stone is very much aware of this as he 
strains to hoist the stone higher. 

The Aristotelian ideas were consistent with many common- 
sense observations. But there were also difficulties. Take a specific 
example — an arrow shot into the air. It cannot be in violent motion 
without a mover, or something pushing on it. Aristotelian physics 


The Birth of Dynamics— Newton Explains Motion 

Keeping an object in motion at uni- 
form speed. 

SG 3.2 

required that the arrow be continually propelled by a force; if the 
propelling force were removed, the arrow should immediately stop 
its flight and fall directly to the ground in "natural" motion. 

But of course the arrow does not fall to the ground as soon as 
it loses direct contact with the bowstring. What then is the force 
that propels the arrow? Here, the Aristotelians offered an ingenious 
suggestion; the motion of the arrow through the air was maintained 
by the air itself! A commotion is set up in the air by the initial 
movement of the arrow. That is; as the arrow starts to move, the air 
is pushed aside; the rush of air to fill the space being vacated by 
the arrow maintains it in its flight. 

More sophisticated ideas to explain motion were developed 
before the mid-seventeenth century. But in every case, a force was 
thought to be necessary to sustain uniform motion. The explanation 
of uniform motion depended on finding the force, and that was not 
always easy. There were also other problems. For example, a falling 
acorn or stone does not move with uniform speed — it accelerates. 
How is acceleration explained? Some Aristotelians thought the 
speeding up of a falling object was associated with its approaching 
anival at its natural place, the earth. In other words, a falling object 
was thought to be like the tired horse that starts to gallop as it 
approaches the barn. Others claimed that when an object falls, the 
weight of the air above it increases while the column of air below 
it decreases, thus offering less resistance to its fall. 

When a falling object finally reaches the ground, as close to the 
center of the earth as it can get, it stops. And there, in its "natural 
place," it remains. Rest, being regarded as the natural state of 
objects on earth, required no further explanation. The three 
phenomena — rest, uniform motion, and acceleration — could thus be 
explained in a more or less plausible fashion by an Aristotelian. 
Now, let us examine the Newtonian explanation of the same 
phenomena. The key to this approach is a clearer understanding of 
the concept of force. 

Q2 According to Aristotle, what is necessary to maintain 
uniform motion? 

Q3 Give an Aristotelian explanation of a dry-ice puck's uniform 
motion across a table top. 

3.3 Forces in equilibrium 

Our common-sense idea of force is closely linked with our own 
muscular activity. We know that a sustained effort is required to 
lift and support a heavy stone. When we push a lawn mower, row a 
boat, split a log, or knead bread dough, our muscles let us know 
we are applying a force to some object. Force and motion and 
muscular activity are naturally associated in our minds. In fact, 
when we think of changing the shape of an object, or moving it or 
changing its motion, we naturally think of the muscular sensation 

Section 3.3 


of applying a force to the object. We shall see that many -but not 
all -of the everyday common-sense ideas about force are useful in 

We know intuitively that forces can make things move, but 
they can also hold things still. The cable supporting the main span 
of the Golden Gate Bridge is under the influence of mighty forces, 
yet it remains at rest. Apparently, more is required to start motion 
than just the application of forces. 

Of course, this is not surprising. We have all seen children 
quarrelling over a toy. If each child pulls determinedly in his own 
direction, the toy may go nowhere. On the other hand, the tide of 
battle may shift if one of the children suddenly makes an extra 
effort, or if two children cooperate and pull side by side against the 

Likewise, in the tug-of-war between the two teams shown 
above, large forces were exerted on each side, but the rope remained 
at rest: one may say the forces balanced, or they "cancelled." A 
physicist would say that the rope was in equilibrium when the 
sum of the forces on each side of it were equally large and acting 
in opposite directions. Equally well, he might say the net force is 
zero. Thus a body in equilibrium would not start to move until a 
new, "unbalanced" force was added which destroyed the 

In all these examples, both the magnitude of the forces and 
their directions are important. The effect of a force depends on the 
direction in which it is applied. We can represent this directional 
nature of forces in a sketch by using arrows: The direction the 
arrow points represents the direction in which the force acts; the 
length of the arrow represents how large the force is (for example, 
a 10-lb force is shown by an arrow twice as long as a 5-lb force). 

Now we discover a surprising result. If we know separately each 
of the forces applied to any object at rest, we can predict whether 
it will remain at rest. It is as simple as this: The object acted on by 
forces will be in equilibrium under these forces and wUl remain at 
rest only if the arrows representing the forces all "add up to zero." 

How does one "add up" arrows? By a simple graphical trick. 
Take the tug-of-war as an example. Let us call the force exerted by 
the team pulling to the right ?,. (The httle arrow over the F 


The Birth of Dynamics— Newton Explains Motion 


■Pore* F2 










There are several ways of expressing 
the idea of unbalanced force: net 
force, resultant force, total force, 
vector sum of forces. All mean the 
same thing. 

indicates that we are dealing with a quantity for which direction is 
important.) The force of the second team is then called F-,. Figure 
(a) in the margin shows the two arrows corresponding to the two 
forces, each applied to the central part of the rope, but in opposite 
directions. Let us assume that these forces, F, and F2, were 
accurately and separately measured, for example, by letting each 
team in turn pull on a spring balance as hard as it can. The arrows 
for Ti and T2 are carefully drawn to a chosen scale, such as 
1" = 1000 lb, so that 750 lb of force in either direction would be 
represented by an arrow of 3/4" length. Next, in Figure (b), we take 
the arrows F, and ¥2 and draw them again in the correct direction 
and to the chosen scale, but this time we put them "head to tail." 
Thus F] might be drawn first, and then To is drawn with the tail of 
P2 starting from the head of Fj. (Since they would of course overlap 
in this example, we have drawn them a little apart in Figure (b) to 
show them both more clearly.) The trick is this: If the head end of 
the second arrow falls exactly on the tail end of the first, then we 
know that the effects of Fj and ¥2 balance each other. The two 
forces, acting in opposite directions and equally large, add up to 
zero. If they did not, the excess of one force over the other would 
be the net force and the rope would accelerate instead of being at 

To be sure, this was an obvious case, but the graphical 
technique turns out to work also for cases that are not simple. For 
example, apply the same procedure to the toy, or to a boat that is to 
be secured by means of three ropes attached to different moorings. 



We are defining equilibrium without 
worrying about whether the object 
will rotate. For example: The sum of 
the forces on the plank in the dia- 
gram below is zero, but it is obvious 
that the plank will rotate. 


'■^■^J^.g^L,..: ■'; 


^T F>F> 


Consider a situation where Ti is a force of 34 lb, ?, is 26 lb. f-^ is 
28 lb, each in the direction shown. (The scale for the magnitude 
of the forces here is 0.1 cm = 1 lb of force.) Is the boat in equilibrium 
under the forces? Yes, if the forces add up to zero. Let's see. With 
rule and protractor the arrows are drawn to scale and in exactly 
the right direction. Then, adding f*,. F... and ?., head to tail, we see 
that the head of the last arrow falls on the tail of the first. Yes. the 
forces cancel; they add up to zero; the net force is zero. Therefore 

Section 3.4 


the object is in equilibrium. This method tells us when an object is 
in equilibrium, no matter how many different forces are acting on it. 

We can now summarize our understanding of the state of rest 
as follows: if an object remains at rest, the sum of all forces 
acting on it must be zero. We regard rest as an example of the 
condition of equilibrium, the state in which all forces on the object 
are balanced. 

An interesting case of equilibrium, very different from the 
disputed toy or rope, is part of the "free fall" of a sky-diver. In fact 
his fall is "free" only at the beginning. The force of air friction 
increases with speed, and soon the upward frictional force on the 
sky-diver is great enough to balance the force of gravity 
downward. Under those circumstances he falls with constant speed, 
much like a badminton bird or falling leaf. The sensation is not of 
falling but, except for the wind, the same as lying on a soft bed. 
During part of a dive from an airplane you can be as much in 
equilibrium as lying in bed! In both cases the net force acting on 
you is zero. 

Q4 A vase is standing at rest on a table. What forces would you 
say are acting on the vase? Show how each force acts (to some 
scale) by means of an arrow. Can you show that the sum of the 
forces is zero? 

Q5 In which of these cases are the forces balanced? 

SG 3.3 







Q6 Does an object have to be at rest to be in equilibrium? 

3.4 About vectors 

Graphical construction with arrows really works. With it we can 
predict whether the forces balance and will leave the object in 
equilibrium or whether any net force is left over, causing the object 
to accelerate. Why can we use arrows in this way? The reason 
involves the precise mathematical definitions of displacement and 
of force, but you can demonstrate for yourself the reasonableness 
of the addition rule by trying a variety of experiments. For example, 
you could attach three spring scales to a ring and have some 
friends pull on the scales with forces that just balance, leaving the 
ring at rest. While they are pulling, you read the magnitudes of the 
forces on the scales and mark the directions of the pulls. You can 
then make a graphical construction with arrows representing the 
forces and see whether they add to zero. Many different experiments 
of this kind ought all to show a net force of zero. 


The Birth of Dynamics— Newton Explains Motion 

It is not obvious that forces should behave like arrows. But 
arrows drawn on paper happen to be useful for calculating how 
forces add. (If they were not, we simply would look for other 
symbols that do work.) Forces belong in a class of concepts called 
vector quantities, or just vectors for short. Some characteristics 
of vectors are well represented by arrows. In particular, vector 
quantities have magnitude which we can represent by the length 
of an arrow drawn to scale. They have direction which can be 
shown by the direction of an arrow. By experiment, we find that 
they can be added in such a way that the total effect of two or 
more, called the vector resultant, can be represented by the head- 
to-tail addition of arrows. 

In the example of the tug-of-war we talked about the effect of 
equally large, opposing forces. If two forces act in the same 
direction, the resultant force is found in essentially the same way, 
as shown below. 

If two forces act at some angle to each other, the same type of 
construction is still useful. For example, if two forces of equal 
magnitude, one directed due east and the other directed due north, 
are applied to an object at rest but free to move, the object will 
accelerate in the northeast direction, the direction of the resultant 
force. The magnitude of the acceleration will be proportional to the 
magnitude of the resultant force which is shown by the length of 
the arrow representing the resultant. 

You can equally well use a graphical 
construction called the "parallelo- 
gram method." It looks different 
from the "head-to-tail" method, but 
Is really exactly the same. In the 
parallelogram construction, the 
vectors to be added are represented 
by arrows joined tail-to-tail instead 
of head-to-tail, and the resultant is 
obtained by completing the diagonal 
of the parallelogram. 


The same adding procedure is used if the forces are of any magni- 
tude and act at any angles to each other. For example, if one force 
were directed due east and a somewhat larger force were directed 
northeast, the resultant vector sum could be found as shown below. 

Section 3.5 


To summarize, we can now define a vector quantity. It is a 
quantity which has both direction and magnitude and which can 
be added by the graphical construction of the head-to-tail 
representation of arrows, or by the equivalent parallelogram method. 
(It also has other properties which you will study if you take further 
physics courses.) By this definition, many important physical 
concepts are vectors — for example, displacement, velocity, and 
acceleration. Some other physical concepts, including volume, 
distance, and speed, do not require specification of direction, and so 
are not vector quantities; these are called scalar quantities. When 
you add 10 liters of water to 10 liters of water, the result is always 
20 liters, and direction has nothing to do with the result. Similarly, 
the term speed has no directional meaning; it is the magnitude of 
the velocity vector, as given by the length of the arrow, without 
regard to its direction. By contrast, when you add two forces of 10 
lb each, the resultant force may be anywhere between zero and 20 
lb, depending on the directions of the two individual forces. 

We shall soon have to correct an oversimplification we had to 
make in Sec. 1.8, where we defined acceleration as the rate of 
change of speed. That was only partly correct, because it was 
incomplete. We shall also want to consider changes in the direction 
of motion as well. The more useful definition of acceleration is the 
rate of change of velocity, where velocity is a vector having both 
magnitude and direction. In symbols. 

Any vector quantity is indicated by 
a letter with an arrow over it; for 
example, F^ a, or v^ 

-* Ax; 

where Ai; is the change in velocity. Velocity can change in two 
ways: by changing its magnitude (speed), and by changing its 
direction. In other words, an object is accelerating when it speeds 
up, or slows down, or changes direction. We shall explore this 
definition more fully in later sections. 

We shall use vectors frequently. To 
learn more about them you can use 
the Project Physics Programmed In- 
struction booklets on vectors. See 
also Reader 1 article "Introduction to 

Q7 List three properties of vector quantities. 
Q8 How does the new definition of acceleration given above 
differ from the one used in Chapter 1? 

3.5 Newton's first law of motion 

Were you surprised when you first watched a dry-ice disk or 
some other nearly frictionless device? Remember how smoothly it 
glides along after just the slightest nudge? How it shows no sign of 
slowing down or speeding up? Although our intuition and everyday 
experience tell us that some force is constantly needed to keep an 
object moving, the disk fails to hve up to our Aristotelian 
expectations. It is always surprising to see this for the first time. 

Yet the disk is behaving quite naturally. If the forces of friction 
were absent, a gentle, momentary push would make tables and 


The Birth of Dynamics— Newton Explains Motion 

chairs take off and glide across the floor just like a dry-ice disk. 
Newton's first law directly challenges the Aristotelian notion of 
what is "natural." It declares that the state of rest and the state of 
uniform, unaccelerated motion in a straight line are equally 
natural. Only the existence of some force, friction for example, 
keeps a moving object from moving forever \ Newton's first law of 
motion can be stated as follows in modem terminology: 

Because constant velocity means 
both constant speed and constant 
direction, we can write Newton's 
first law more concisely: 

r= constant 
if and only if 

This statement includes the 
condition of rest, since rest is a 
special case of unchanging velocity 
—the case where v'= 0. 

SG 3.5 

Every object continues in its state of rest or of uniform 
rectilinear motion unless acted upon by an unbalanced 
force. Conversely, if an object is at rest or in uniform 
rectilinear motion, the unbalanced force acting upon it 
must be zero. 

In order to understand the motion of an object, we must take 
into account all the forces acting on it. If all forces (including 
friction) are in balance, the body will be moving at constant v. 

Although Newton was the first to express this idea as a general 
law, Galileo had made similar statements fifty years before. Of 
course, neither Galileo nor Newton had dry-ice disks, and so they 
were unable to observe motion in which friction had been reduced 
so significantly. Instead, Galileo devised a thought experiment in 
which he imagined the friction to be zero. 

This thought experiment was based on an actual observation. If 
a pendulum bob on the end of a string is pulled back and released 
from rest, it will swing through an arc and rise to very nearly its 
starting height. Indeed, as Galileo showed, the pendulum bob will 
rise almost to its starting level even if a peg is used to change the 

It was from this observation that Galileo generated his thought 
experiment. He predicted that a ball released from a height on a 
frictionless ramp, would roll up to the same height on a similar 
facing ramp, regardless of the actual path length. For example, in 
the diagram at the top of the next page, as the ramp on the right is 
changed from position (a) to (b) and then to (c). the ball must roll 
further in each case to reach its original height. It slows down 
more gradually as the angle of the incline decreases. If the second 
ramp is exactly level as shown in (d). the ball can never reach its 
original height. Therefore, Galileo believed, the ball on this 
frictionless surface would roll on in a straight line and at an 

Section 3.5 


unchanged speed forever. This could be taken to be the same as 
Newton's first law. and some historians of science do give credit to 
Galileo for having come up with the law first. Other historians, 
however, point out that, for Galileo, rolling on forever meant staying 
at a constant height above the earth — not moving in a straight line 
through space. 

This tendency of objects to maintain their state of rest or of 
uniform motion is sometimes called "the principle of inertia." 
Newton's first law is therefore sometimes referred to as the "law of 
inertia." Inertia is a property of all objects. Material bodies have, so 
to speak, a stubborn streak so far as their state of motion is 
concerned. Once in motion, they continue to move with unchanging 
velocity (unchanging speed and direction) unless compelled by 
some externally applied force to do otherwise. If at rest, they remain 
at rest. This is why seat belts are so helpful when the car stops very 
suddenly, and also why a car may not follow an icy road around a 
turn, but travel a straighter path into a field or fence. The greater 
the inertia of an object, the greater its resistance to a change in its 
state of motion, and hence the greater is the force needed to 
produce a desired change in the state of its motion. This is why it is 
more difficult to start a train or a ship and to bring it up to speed 
than it is to keep it going once it is moving at the desired speed. (In 
the absence of friction, it would keep moving without any applied 
force at all.) But for the same reason it is difficult to bring it to a 
stop, and passengers and cargo keep going forward if the vehicle is 
suddenly braked. 

Newton's first law tells us that if we see an object moving with 
a constant speed in a straight line, we know at once that the forces 
acting on it must be balanced, that is, it is in equilibrium. In Sec. 
3.4 we established that an object at rest is in equilibrium. Does this 
mean that in Newtonian physics the state of rest and the state of 
uniform motion are equivalent? It does indeed. When we know that 
a body is in equilibrium, we know only that v = constant. Whether 
the value of this constant is zero or not depends in any case on 
which body is chosen as reference for measuring the magnitude of 
V. We can decide whether to say that it is at rest or that it is moving 
with constant t/ larger than zero only by reference to some other 

Take, for example, a tug-of-war. Suppose two teams were sitting 
on the deck of a barge that was drifting with uniform velocity down 
a lazy river. Two observers — one on the same barge and one on the 
shore — would each give a report on the incident as viewed from his 

Inside the laboratory there is no 
detectable difference between a 
straight (horizontal) line and a 
constant height above the earth. But 
on a larger scale, Galileo's eternal 
rolling would become motion in a 
circle around the earth. Newton 
made clear what is really important: 
that in the absence of the earth's 
gravitational pull or other external 
forces, the ball's undisturbed path 
would extend straight out into space. 

Galileo's idea of a straight 

Newton' s idea of a straight 


The Birth of Dynamics — Newton Explains Motion 

own frame of reference. The observer on the barge would observe 
that the forces on the rope were balanced and would report that it 
was at rest. The observer on the shore would report that the forces 
on the rope were balanced and that it was in uniform motion. 
Which observer is right? They are both right; Newton's first law of 
motion applies to both observations. Whether a body is at rest or in 
uniform motion depends on which reference frame is used to 
observe the event. In both cases the forces on the object involved 
are balanced. 

Q9 What is the net force on the body in each of the four cases 
sketched in the margin of the opposite page? 

Q10 What may have been a difference between Newton's 
concept of inertia and GalOeo's? 

3.6 The significance of the first law 

Of course, the idea of inertia does 
not explain why bodies resist change 
in their state of motion. It is simply 
a term that helps us to talk about 
this basic, experimentally observed 
fact of nature. (See SG 3.6 and 3.7.) 

The correct reference frame to use 
in our physics turns out to be any 
reference frame that is at rest or 
in uniform rectilinear motion with 
respect to the stars. The rotating 
earth is, therefore, strictly speaking 
not allowable as a Newtonian 
reference frame; but for most 
purposes the earth rotates so little 
during an experiment that the 
rotation can be neglected. (See 
SG 3.8.) 

You may have found Galileo's thought experiment convincing. 
But think how you might try to verify the law of inertia 
experimentally. You could start an object moving (perhaps a dry- 
ice disk) in a situation in which you believe there is no unbalanced 
force acting on it. Then you could observe whether or not the object 
continued to move uniformly in a straight line, as the first law 
claims it should. 

The experiment is not as simple as it sounds; in fact, Newton's 
laws involve some profound philosophical content (see SG 3.7); but 
we can see the significance of Newton's first law even without 
going into all these subtleties. For convenience let us list the 
important insights the first law provides. 

1. It presents the idea of inertia as a basic property of all material 
objects. Inertia is the tendency of an object to maintain its 
state of rest or uniform motion. 

2. It points up the equivalence of a state of rest for an object 
and a state of uniform motion in a straight line. Both states 
indicate that the net force is zero. 

3. It raises the whole issue of frame of reference. An object 
stationary for one observer might be in motion for another 
observer; therefore, if the ideas of rest or uniform motion are 
to have any significance, a frame of reference must be 
specified from which the observations of events are to be 

4. It purports to be a universal law. It emphasizes that a single 
scheme can deal with motion anywhere in the universe. For 
the first time no distinction is made between terrestrial and 
celestial domains. The same law applies to objects on earth as 
well as on the moon and the planets and the stars. And it 
applies to balls, dry-ice pucks, magnets, atomic nuclei, 
electrons — everything ! 

Section 3.7 


The first law describes the behavior of objects when no 
unbalanced force acts on them. Thus, it sets the stage for the 
question: precisely what happens when an unbalanced force 
does act on an object? 

3.7 Newton's second law of motion 

In Section 3.1 it was stated that a theory of dynamics must 
account for rest, uniform motion, and acceleration. So far we have 
met two of our three objectives: the explanation of rest and of 
uniform motion. In terms of the first law, the states of rest and 
uniform motion are equivalent; they are different ways of describing 
the state of equilibrium — that state in which no unbalanced force 
acts on an object. 

The last section concluded with a list of insights provided by 
the first law. You noticed that there was no quantitative relationship 
established between force and inertia. Newton's second law of 
motion enables us to reach our third objective — the explanation 
of acceleration — and also provides a quantitative expression, an 
equation for the relationship between force and inertia. We shall 
study separately the way in which force and inertia enter into the 
second law. Later in this section we will look more closely at how 
force and inertia are measured. But first we will take some time to 
be sure that Newton's statement is clear. First we consider the 
situation in which different forces act on the same object, and then 
the situation in which the same force acts on different objects. 

Force and Acceleration. To emphasize the force aspect, Newton's 
second law can be stated as follows : 

The net, unbalanced force acting on an object is directly 
proportional to, and in the same direction as, the acceleration 
of the object. 
More briefly, this can be written as: "acceleration is proportional 
to net force." If we let F^et stand for net force and a stand for 
acceleration, we can write this relationship precisely as: 

a ^ f net 

Both a and f^net are vectors; the statement that they are proportional 
includes the understanding that they also point in the same 

To say that one quantity is proportional to another is to make 
a precise mathematical statement. Here it means that if a given net 
force (Fnet) causes an object to move with a certain acceleration (a), 
then a new force equal to twice the previous force (2Fnet) will cause 
the same object to have a new acceleration equal to twice the 
earlier acceleration (or 2a); three times the net force will cause 
three times the acceleration; and so on. Using symbols, this 
principle can be expressed by a statement like the following: 

SG 3.9 
SG 3.10 


Apple falling- negligible friction 

Feather falling at nearly constant 

l^ili^^T^:^^ ""vS 

Kite held suspended in the wind 

Man running against the wind 

80 The Birth of Dynamics- Newton Explains Motion 

If a force f^net will cause a, then a force equal to 

2P^net will cause 2a 
sfnei will cause 3a 
2-Fnet will cause ja 
5.2Fnet will cause 5.2a 

and so on. 

One can readily imagine a rough experiment to test the 
validity of the law — more easily as a thought experiment than 
as a real one. Take a nearly frictionless dry-ice puck on a flat table, 
attach a spring balance, and pull with a steady force so that it 
accelerates continuously. The pull registered by the balance will be 
the net force since it is the only unbalanced force acting. Measure 
the forces and the corresponding accelerations in various tries, then 
compare the values of Fnet and a. We shall look into this method in 
detail in the next section. 

Mass and Acceleration. Now we can consider the inertia aspect of 
the second law, the effect of the same net force acting on different 
objects. In discussing the first law, we said inertia is the resistance 
an object exhibits to any change in its velocity. We know from 
experience and observation that some objects have greater inertia 
than others. For instance, if you were to throw a baseball and then 
put a shot with your full effort, you know that the baseball would 
be accelerated more and hence would reach a greater speed than 
the shot. Thus, the acceleration given a body depends as much on 
the body as it does on the force applied to it. The concept of the 
amount of inertia a body has is expressed by the term mass. 

Mass is a familiar word, but it becomes useful in physics only 
after it is disentangled from some aspects of its common sense 
meaning. For example, mass is often used as a synonym for weight. 
But although mass and weight are closely related, they are not 
at all the same thing. Weight is a force, the force with which 
gravity is acting on an object; mass, on the other hand, is a 
measure of an object's resistance to acceleration. It is true that 
on or near the surface of the earth, objects that are hard to 
accelerate are also heavy, and we will return to this relationship in 
SG 3.11 Sec. 3.8. 

If you supply the same force to several different objects, their 
What does it mean to say that mass accelerations will not be the same. Newton claimed that the 
is a scalar quantity? resulting acceleration of each object is inversely proportional to its 

mass. Using the symbol m for mass (a scalar quantity), and the 
symbol a for the magnitude of the vector acceleration a, we can 
write "a is inversely proportional to m," or what is mathematically 

the same, "a is proportional to — , " or 



a a. — 


This means that if a certain force makes a given object have a 
certain acceleration, then the same force will cause an object 
having twice the mass to have one-half the acceleration, an object 

Section 3.7 81 

having three times the mass to have one-third the acceleration, 
an object of one-fifth the mass to have five times the acceleration, 
and so on. This is why, for example, a truck takes much longer to 
reach the same cruising speed when it is full than when it is nearly 
empty. Using symbols, we can express this as follows: 

If a given force Fnet is applied, and an object 

of mass m experiences a, then an object 

of mass 2m will experience ^a, 

of mass 3m will experience ^a, 

of mass jm will experience 5a, 

of mass 2.5m will experience 0.4a, 

and so on. 

This can be demonstrated by experiment. Can you suggest how it 
might be done? 

The roles played by force and mass in Newton's second law can 
be combined in a single statement: 

The acceleration of an object is directly proportional to, and 
in the same direction as, the unbalanced force acting on it, and 
inversely proportional to the mass of the object. 

The ideas expressed in this long statement can be summarized by 

the equation SG 3.12 

Yn SG 3.13 

We can regard this equation as one possible way of expressing 
Newton's second law of motion. The same relation may of course 
be equally well written in the form 

FnPt — 


In either form, this is probably the most fundamental single 
equation in all of Newtonian mechanics. Like the first law, the 
second has an incredible range of application: It holds no matter 
whether the force is mechanical or electric or magnetic, whether 
the mass is that of a star or a nuclear particle, whether the 
acceleration is large or small. We can use the law in the easiest 
problems and the most sophisticated ones. By measuring the 
acceleration which an unknown force gives a body of known m.ass, 
we can compute a numerical value for the force from the equation 
Fnet ^ 'ma- Or, by measuring the acceleration that a known force 
gives a body of unknown mass, we can compute a numerical value 
for the mass from the equation (m = Fnet/a)- Clearly we must be 
able to measure two of the three quantities in order to be able to 
compute the other. 

Units of mass and force. Even before we can make such measure- 
ments, however, we must establish units for mass and force that 
are consistent with the units for acceleration (which have already 
been defined in terms of standards of length and time -for 
example, meters per second per second). 


The Birth of Dynamics- Newton Explains Motion 

1 kg corresponds to the mass of 
about 1 liter of water, or about 
2.2 lb (more precisely 2.205 lb). 
The 1/1000th part of 1 kg is 1 gram 


SG 3.14 

One way to do this is to choose some convenient object, perhaps 
a piece of corrosion-free metal, as the universal standard of mass, 
just as a meter is a universal standard of length. We can 
arbitrarily assign to this object a mass of one unit. Once this unit 
has been selected we can proceed to develop a measure of force. 

Although we are free to choose any object as a standard of 
mass, ideally it should be exceedingly stable, easily reproducible, 
and of reasonably convenient magnitude. Such a standard object 
has, in fact, been agreed on by the scientific community. By 
international agreement, the primary standard of mass is a 
cylinder of platinum-iridium alloy, kept near Paris at the 
International Bureau of Weights and Measures. The mass of this 
platinum cylinder is defined as exactly 1 kilogram (abbreviated 
1 kg). Accurately made copies of this international primary 
standard of mass are kept in the various standards laboratories 
throughout the world. Further copies have been made from these 
for distribution to manufacturers and research laboratories. 

The standard kilogram and meter at 
the U.S. Bureau of Standards. 

SG 3.15, 3.16 

SG 3.17,3.18 

In this equation we use only the 
magnitudes-the direction is not 
part of the definition of the unit of 

Now we can go on to answer the question of how much "push" 
or "pull" should be regarded as one unit of force. We define 1 unit 
of force as a force which, when acting alone, causes an object that 
has a mass of 1 kilogram to accelerate at the rate of exactly 
1 meter/second/per second. 

Imagine an experiment in which the standard 1-kg object is 
pulled with a spring balance in a horizontal direction across a level, 
frictionless surface. The pull is regulated to make the 1-kg object 
accelerate at exactly 1 m/secl The required force will by definition 
be one unit in magnitude: 

Fnei = 1 kg X 1 m/sec"^ = 1 kgm/sec- 


Section 3.8 


Thus, 1 kgm/sec^ of force is that quantity of force which causes a 
mass of 1 kg to accelerate 1 m/sec-. 

The unit kgm/sec^ has been given a shorter name, the newton 
(abbreviated as N). The newton is therefore a derived unit, defined 
in terms of a particular relationship between the meter, the 
feilogram, and the second. Thus the newton is part of the "mks" 
system of units, which is used almost universally in modem 
scientific work. 

The "hidden text" in Newton's second law involves both 
definitions and experimental facts. There are several possible ways 
of analyzing it: if you choose to define some part, you must prove 
others experimentally— or vice- versa. Textbooks do not all agree on 
how best to present the relation of definition and experiment in 
Newton's second law, and Newton himself may have not thought it 
through entirely. However, as a system of ideas (whichever way it 
is analyzed), it was powerful in leading to many discoveries in 

Newton did not "discover" the concepts of force and mass. But 
he did recognize that these concepts were basic to an understanding 
of motion. He clarified these concepts, and found a way to express 
them in numerical values, and so made a science of dynamics 

Q11 Which three fundamental units of distance, mass and 
time are used to define the unit of force? 

Q12 A net force of 10 N gives an object a constant acceleration 
of 4 m/sec^. What is the mass of the object? 

Q13 True or false? Newton's second law holds only when 
frictional forces are absent. 

Q14 A 2-kg object, shoved across the floor with a speed of 10 
m/sec, slides to rest in 5 sec. What was the magnitude of the force 
producing this acceleration? 

Q15 Complete the table in the margin which lists some 
accelerations resulting from applying equal forces to objects of 
diff"erent mass. 

The units of acceleration "m/sec 
per second" can be written as 
"m/sec/sec" or "m/sec-". The sec- 
means that division by time units 
occurs twice, not something like 
"square time." 

SG 3.19, 3.20, 3.21, 3.22, 3.23. 



30 m/sec^ 


15 m/sec^ 





3 m/sec^ 

75 m/sec^ 

3.8 Mass, weight, and free fall 

The idea of force has been generalized in physics to include 
much more than muscular pushes and pulls. Whenever we observe 
an acceleration, we infer that there is a force acting. Forces need 
not be "mechanical" or exerted by contact only; they can be due to 
gravitational, electric, magnetic, or other actions. Newton's laws 
are valid for all of them. 

The force of gravity acts without direct contact between objects 
that are separated not only by a few feet of air, as is the case with 
the earth and a falling stone, but also across empty space such as 
separates the earth from an artificial satellite in orbit. 

We shall use the symbol fg for gravitational force. The 


The Birth of Dynamics— Newton Explains Motion 

Is the boy weightless? Explain. 
SG 3.24.) 


magnitude of the gravitational pull T„ is, roughly speaking, the 
same anywhere on the surface of the earth for a particular object. 
When we wish to be very precise, we must take into account the 
facts that the earth is not exactly spherical, and that there are 
irregularities in the composition of the earth's crust. These factors 
cause slight differences — up to 1/2% — in the gravitational force 
on the same object at different places. An object having a constant 
mass of 1 kg will experience a gravitational force of 9.812 newtons 
in London, but only 9.796 newtons in Denver, Colorado. Geologists 
make use of these variations in locating oil and other mineral 

The term weight is often used in everyday conversation as if it 
meant the same thing as bulk or mass. In physics, we define the 
weight of an object as the gravitational force acting on the body. 
Weight is a vector quantity, as are all forces. Your weight is the 
downward force our planet exerts on you whether you stand or sit. 
fly or fall, orbit the earth in a space vehicle or merely stand on a 
scale to "weigh" yourself. 

Think for a moment what a scale does. The spring in it 
compresses until it exerts on you an upward force sufficient to hold 
you up. So what the scale registers is really the force with which 
it pushes up on your feet. When you and the scale stand still and are 
not accelerating, the scale must be pushing up on your feet with a 
force equal in magnitude to your weight. That is why you are in 
equilibrium — the sum of the forces on you is zero. 

Now imagine for a moment a ridiculous but instructive thought 
experiment: as you stand on the scale, the floor (which, sagging 
slightly, has been pushing up on the scale) suddenly gives way, and 
you and the scale are dropping into a deep well in free fall. At every 
instant, your fall speed and the scale's fall speed will be equal, 
since you started falling together and fall with the same 
acceleration. Your feet would now touch the scale only barely (if at 
all), and if you looked at the dial you would see that the scale 
registers zero. This does not mean you have lost your weight -that 
could only happen if the earth suddenly disappeared, or if you were 
suddenly removed to far, interstellar space. No, Pg still acts on you 
as before, accelerating you downward, but since the scale is 
accelerating with you, you are no longer pushing down on it -nor 
is it pushing up on you. 

You can get a fairly good idea of the difference between the 
properties of weight and mass by holding a big book: First, just lay 
the book on your hand; you feel the weight of the book acting down. 
Next, grasp the book and shake it back and forth sideways. You still 
feel the weight downwards, but you also feel how hard the book 
is to accelerate back and forth — its mass. You could make your 
sensation of the book's weight disappear by hanging the book on a 
string, but the sensation of its inertia as you shake it remains the 
same. This is only a crude demonstration, and it isn't clear that the 
shaking sensation doesn't still depend on the pull of the earth. More 
elaborate experiments would show, however, that weight can 

Section 3.8 85 

change without changing mass. Thus when an astronaut on the Consider SG 3.14 again, 

moon's surface uses a big camera, he finds it much easier to hold- 
its weight is only 1/6 of its weight on earth. But its mass or 
inertia is not less, and it is as hard to swing around suddenly 
into a new position as it is on earth. 

We can now understand the results of Galileo's experiment on 
falling objects in a more profound way. Galileo's discussion of 
falling objects showed that any given object (at a given locality) 
falls with uniform acceleration, a,,. What is responsible for its 
uniform acceleration? A constant net force — in this case of free fall, 
just Fg. Now Newton's second law expresses the relationship 
between this force and the resulting acceleration. Applying the 
equation Fnet = ma to this case, where Fnet ^ r „ and a = ay, we can 

fg = mag 

We can, of course, rewrite this equation as 

We conclude from Newton's second law that the reason why the 
acceleration of a body in free fall is constant is that for an object of 
given mass m the gravitational force Fg over normal distances of 
fall is nearly constant. 

Galileo, however, did more than claim that every object falls 
with constant acceleration: he found that all objects fall with the 
same uniform acceleration, which we now know has the value of 
about 9.8 m/sec at the earth's surface. Regardless of the mass m or 
weight Fg, all bodies in free fall (in the same locality) have the 
same acceleration a^. Is this consistent with the relation ag= Fglm7 
It is consistent only if for every object Fg is directly proportional to 
mass m: that is, if m is doubled, Fg must double; if m is tripled, 
Fg must triple. This is a significant result indeed. Weight and mass 
are entirely different concepts. Weight is the gravitational force on 
an object (hence weight is a vector). Mass is a measure of the 
resistance of an object to change in its motion, a measure of inertia 
(hence mass is a scalar). Yet the fact that different objects fall 
freely with the same acceleration means that the magnitudes of 
these two quite different quantities are proportional in any given 

Q16 An astronaut is orbiting the earth in a space vehicle. The 
acceleration due to gravity at that distance is half its value on the 
surface of the earth. Which of the following are true? SG 3.25, 3.26, 3.27, 3.28 

(a) His weight is zero. 

(b) His mass is zero. 

(c) His weight is half its original value. 

(d) His mass is half its original value. 

(e) His weight remains the same. 

(f ) His mass remains the same. 


The Birth of Dynamics- Newton Explains Motion 

Q17 A boy jumps from a table top. When he is halfway between 
the table top and the floor, which of the statements in Q16 are true? 

3.9 Newton's third law of motion 

He is, to be sure, pushing against 
the ground -but that is a force 
acting on the ground. 

In his first law, Newton described the behavior of objects when 
they are in a state of equilibrium; that is, when the net force acting 
on them is zero. His second law explained how their motion changes 
when the net force is not zero. Newton's third law added a new 
and surprising insight about forces. 

Consider this problem: In a 100-meter dash, an athlete will go 
from rest to nearly his top speed in less than a second. We could 
measure his mass before he makes the dash, and we could use 
high-speed photography to measure his initial acceleration. With his 
mass and acceleration known, we could use F = ma to find the force 
acting on him during the initial acceleration. But where does the 
force come from? It must have something to do with the runner 
himself. Is it possible for him to exert a force on himself as a 
whole? Can he lift himself by his own bootstraps? 

Newton's third law of motion helps us to understand just such 
puzzling situations. First, let us see what the third law claims. 
In Newton's words: 

To every action there is always opposed an equal reaction: or, 
mutual actions of two bodies upon each other are always equal 
and directed to contrary parts. 

SG 3.29 

This is a word-for-word translation from the Principia. It is 
generally agreed, however, that in Newton's statement the expression 
force on one object may be substituted for the word action, and 
the expression equally large force on another object for the words 
equal reaction. Read it over with this change. 

The most startling idea to come out of this statement is that 
forces always exist in mirror-twin pairs, and on two different 
objects. Indeed, the idea of a single force unaccompanied by another 
force acting somewhere else is without any meaning whatsoever. 
On this point Newton wrote: "Whatever draws or presses another 
is as much drawn or pressed by that other. If you press a stone 
with your finger, the finger is also pressed by the stone." This 
suggests that forces always arise as a result of interactions between 
objects: object A pushes or pulls on B. while at the same time 
object B pushes or pulls with precisely equal amount on A. These 
paired pulls and pushes are always equal in magnitude, opposite in 
direction, and on two different objects. 

Applying this idea to the athlete, we now see that his act of 
pushing his feet on the earth (one may call it here the action) is 
accompanied by a push of the earth on him (one can call it the 
reaction) -and the latter is what propels him forward. In this and 
all other cases it really makes no difference which we call the action 

Section 3.9 


and which the reaction, because they occur at exactly the same 
time. The action does not "cause" the reaction -if the earth could 
not "push back" on his feet, the athlete could not push on the earth 
in the first place, but would slide around — as on slippery ice. 
Action and reaction coexist. You can't have one without the other. 
And most important, the two forces are not acting on the same 
body. In a way, they are like debt and credit: one is impossible 
without the other; they are equally large but of opposite sign; and 
they happen to two different objects. 

Any body A that affects body B must itself be affected by B — 
equally and oppositely. We can use the efficient shorthand of algebra 
to express the idea that whenever bodies A and B interact: 

f^AB — ~^BA 

This is the equivalent of Newton's explanatory statement: 
Whenever two bodies interact, the forces they exert on each other 
are equal in magnitude and opposite in direction. 

A host of everyday observations illustrate Newton's third law: 
A boat is propelled by the water that pushes forward on the oar 
while the oar pushes back on the water. A car is set in motion by 
the push of the ground on the tires as they push back on the 
ground; when friction is not sufficient, the tires cannot start the car 
forward. While accelerating a bullet forward, a rifle experiences a 
recoil kick. A balloon jumps forward while the air spurts out the 
opposite direction. Many such effects are not easily observed; for 
example, when an apple falls, pulled down by its weight, the earth 
accelerates upward, pulled up by the attraction to the apple. 

Now note what the third law does not say — this, too, is 
important. The third law speaks of forces, not of the effects these 
forces produce. Thus in the last example, the earth accelerates 
upward as the apple falls down; the forces on each are equally 
large, but the accelerations produced by the forces are quite 
different; owing to the enormous mass of the earth, the earth's 
upward acceleration is insensibly small. The third law also 
does not describe how the push or pull is applied, whether by 
contact or by magnetic action or by electrical action. Nor does the 
law require that the force be either an attraction or repulsion. The 
third law really does not depend on any particular kind of force. It 
applies equally to resting objects and to moving objects, to 
accelerating objects as well as to objects in uniform motion. It 
applies whether or not there is friction present. Indeed, the 
universality of the third law makes it extremely valuable throughout 

In the collision between the ball and 
the club, the force the ball exerts on 
the club is equal and opposite to the 
force the club exerts on the ball. Both 
the club and the ball get defornned by 
the forces acting on them. 

Force on 
bail due 
V- to club 

is equal and 
opposite to 

Force on 
club due 
to ball 

foy-ce, on earih 

force on moon 

The force on the moon due to the 
earth is equal and opposite to the 
force on the earth due to the moon. 

88 The Birth of Dynamics — Newton Explains Motion 

Q18 According to Newton's third law, what are the four 
general characteristics of forces? 

Q19 Identify the forces that act according to Newton's third 
law when a horse accelerates; when a swimmer moves at constant 

Q20 A piece of fishing line breaks if the force exerted on it is 
greater than 500 N. Will the line break if two people at opposite 
ends of the line pull on it, each with a force of 300 N? 

Q21 State Newton's three laws of motion as clearly as you can 
SG 3.30, 3.31, 3.32 in your own words. 

3.10 Using Newton's laws of motion 

We have discussed each of Newton's three laws of motion in 
some detail. The first law emphasizes the modem point of view in 
the study of motion: What requires explanation is not motion itself, 
but change of motion. The first law stresses that one must account 
for why an object speeds up or slows down or changes direction. The 
second law asserts that the rate of change of velocity of an object 
is related to both the mass of the object and the net force applied to 
it. In fact, the very meanings of force and mass are shown by the 
second law to be closely related to each other. The third law is a 
statement of a force relationship between interacting objects. 

Despite their individual importance, Newton's three laws are 
most powerful when they are used together. So successful was the 
mechanics based on Newton's laws that until the late nineteenth 
century it seemed that all of creation must be understood as 
"matter in motion." Let us examine a specific example that 
illustrates the use of these laws. 

Example 1 

On September 12, 1966, a dramatic experiment based on 
Newton's second law was carried out high over the earth. In this 
experiment, the mass of an orbiting Agena rocket case was 
determined by accelerating it with a push from a Gemini spacecraft. 
After the Gemini spacecraft made contact with the Agena rocket 
case, the aft thrusters on the Gemini, calibrated to give an 
average thrusting force of 890 N, were fired for 7.0 sec. The change 
in velocity of the spacecraft and rocket case was found to be 0.93 
m/sec. The mass of the Gemini spacecraft was known to be about 
3400 kg. The question to be answered was: What is the mass of the 

(Actually, the mass of the Agena had already been measured 
independently. The purpose of the experiment was to develop a 
technique to find the unknown mass of a foreign satellite in orbit.) 

Section 3.10 


In this case, a known force of magnitude 890 N was acting on 
two objects in contact, with a total mass of m,otai, where 

Wtotal ~ mr 

+ ni; 

= 3400 kg + rriAgena 

The magnitude of the average acceleration produced by the thrust 
is found as follows: 


_ 0.93 m/sec 
7.0 sec 

= 0.13 m/sec2 

Newton's second law gives us the relation 

F = mtotai X « 


= (mAgena + 3400 kg) X a 

Solving for mAgena gives 

F ^,^^, 890 N 

mAgena--- 3400 kg = ^^3^^^^^ 

= 6900 kg - 3400 kg 
= 3500 kg 

3400 kg 

The actual mass of the Agena, as previously determined, was 
about 3660 kg. The technique of finding the mass by nudging the 
Agena while in orbit therefore gave a result that was accurate to 
within 5% — well within the margin of error expected in making 
this measurement. 


The Birth of Dynamics— Newton Explains Motion 

Example 2 

Imagine taking a ride on an elevator: (A) At first it is at rest on 
the ground floor; (B) it accelerates upward uniformly at Im/sec/sec 
for a few seconds; then (C) continues to go up at a constant speed 
of 5m/sec. 










SG 3.33 is an elaboration of a 
similar example. For a difficult 
worked-out example see SG 3.34. 

If a 100-kg man (whose weight would therefore be about 1000 
newtons) is standing in the elevator, with what force is the 
elevator floor pushing up on him during (A), (B), and (C)? 

Parts (A) and (C) are dynamically the same: Since he is not 
accelerating, the net force on him must be zero. So the floor must 
be pushing up on him just as hard as gravity is pulling him down. 
The gravitational force on him, his weight, is 1000 N. So the floor 
must be exerting an upward force of 1000 newtons. 

Part (B): Since the man is accelerating upward, there must be 
a net force upward on him; the unbalanced force is 

^net ~ TnUyjp 

= 100 kg X 2 m/sec/sec 
= 200 N 

So the floor must be pushing up on him with a force 200 N greater 
than what is required just to balance his weight; therefore, the total 
force upward on him is 1200 N, 

3.11 Nature's basic forces 

Our study of Newton's laws of motion has increased our 
understanding of objects at rest, moving uniformly, and 
accelerating. However, we have accomplished much more in the 
process. Newton's first law alerted us to the importance of frames 

Section 3.11 


of reference. A critical analysis of the relationship between 
descriptions of the same event seen from different frames of 
reference was in fact the necessary first step toward the theory of 

Newton's second law shows the fundamental importance of the 
concept of force. In fact, it presents us with a mandate: when you 
observe acceleration, find the force! This is how we were first 
directed to the gravitational force as an explanation of Galileo's 
kinematics: For all objects, at a given place, a*g is constant for all 
objects; since Ug = Tglm by Newton's second law, we must conclude 
that the magnitude of Tg is always proportional to m. 

But this is only a halfway solution. Now we want to know why 
Fg is proportional to m for all bodies at a given place and how Fg 
changes for a given body as it is moved to places more distant from 
the earth. Is there a law connecting Fg, m, and distance -a "force 
law"? As Unit 2 will show, there is indeed. Knowing that force 
law, we shall be able to claim to understand all gravitational 
interactions among objects. 

Gravitational attraction is not the only basic force by which 
objects interact. However, it is satisfying to realize that there 
appear to be very few such basic forces. In fact, physicists now 
believe that everything we observe in nature is the consequence of 
just four basic interactions. In terms of our present understanding, 
all the events of nature — subnuclear and nuclear, atomic and 
molecular, terrestrial and solar, galactic and extragalactic — are the 
manifestations of one or more of these few types of forces. 

There is, of course, nothing sacred about the number four. New 
discoveries or theoretical insights might increase or reduce the 
number. For example, two (or more) of the basic interactions might 
some day be seen as consequences of something even more basic. 

The first of the interactions is the gravitational force, which 
becomes important only on a relatively large scale, that is, when 
tremendous numbers of atoms of matter are involved. Between 
individual atoms, gravitational force is so weak so to be 
insignificant, but it is this weak force that literally holds the parts 
of the universe together. The second interaction involves electric 
and magnetic processes and is most important on the atomic and 
molecular scale. It is electromagnetic force that holds together 
objects in the range between the atom and the earth. 

We know the force laws governing gravitational and electro- 
magnetic interactions; therefore these interactions are fairly well 
"understood." The situation changes completely when we consider 
the two remaining basic interactions. They are the subject of 
vigorous research today. The third interaction (the so-called 
"strong" interaction) somehow holds the particles of the nucleus 
together. The fourth interaction (the so-called "weak" interaction) 
governs certain reactions among subnuclear particles. 

We do, of course, have other names for forces, but each of these 
belongs to one of the basic types. One of the most common is the 
"frictional" force; it is thought to be an electrical interaction — that is, 

Refer to K. Ford's: The World of 
Elementary Particles lor brief 
discussion of four forces. 

Einstein spent most of the latter 
half of his life seeking a theory that 
would express gravitational and 
electromagnetic effects in a unified 
way. A satisfactory "unified field 
theory" is still being sought. 


The Birth of Dynamics- Newton Explains Motion 

"The Starry Night, " by Vincent Van 


The intuitive feeling that all of nature's 

phenomena are interlinked on a grand 

scale is shared by scientists as well 

as artists. 

the atoms on the surfaces of the objects shding or rubbing against 
each other interact electrically. 

We shall be encountering these ideas again. We shall deal with 
the gravitational force in Unit 2, the electrical and magnetic forces 
in Units 4 and 5, and the forces between nuclear particles in Unit 6. 
In all these cases, an object subjected to the force will behave in 
accordance with Newton's laws of motion. 

The knowledge that there are so few basic interactions is both 
surprising and encouraging. It is surprising because at first glance 
the events all around us seem so varied and complex. It is 
encouraging because our elusive goal — an understanding of the 
events of nature — looks more attainable. 



4P SW*^ 


3.1 The Project Physics learning materials 
particularly appropriate for Chapter 3 include the 
following : 


Newton's Second Law 
Mass and Weight 


Checker Snapping 

Beaker and Hammer 

Pulls and Jerks 

Experiencing Newton's Second Law 

Make One of These Accelerometers 

Reader Articles 

Introduction to Vectors 

Newton's Laws of Dynamics 

The Scientific Revolution 

How the Scientific Revolution of the 17th 

Century Affected Other Branches of 


Film Loops 

Vector Addition- 

■ Velocity of a Boat 

3.2 The Aristotelian explanation of motion should 
not be dismissed lightly. Great intellects of the 
Renaissance period, such as Leonardo da Vinci, 
who among other things designed devices for 
launching projectiles, did not challenge such 
explanations. One reason for the longevity of 
these ideas is that they are so closely aligned with 
our common sense ideas. 

In what ways do your common sense notions 
of motion agree with the Aristotelian ones? 

3.3 Three ants are struggling with a crumb. One 
ant pulls toward the east with a force of 8 units. 
Another pulls toward the north with a force of 6 
units, and the third pulls in a direction 30° south 
of west with a force of 12 units. 

(a) Using the "head-to-taU" construction of 
arrows, find whether the forces balance, 
or whether there is a net (unbalanced) 
force on the crumb. 

(b) If there is a net force, you can find its 
direction and magnitude by measuring 
the line drawn from the tail of the first 
arrow to the head of the last arrow. What 
is its magnitude and direction? 

3.4 Show why the parallelogram method of adding 
arrows is geometrically equivalent to the head-to- 
tail method. 

3.5 There are many famihar situations in which 
the net force on a body is zero, and yet the body 
moves with a constant velocity. One example of 
such "dynamic equilibrium" is an automobile 
traveling at constant speed on a straight road: the 
force the road exerts on the tires is just balanced 
by the force of air friction. If the gas pedal is 
depressed further, the tires will push against the 
road harder and the road will push against the 
tires harder; so the car will accelerate forward — 
until the air friction builds up enough to balance 

the greater drive force. Give another example of a 
body moving with constant velocity under 
balanced forces. Specify the source of each force 
on the body and, as in the automobile example, 
explain how these forces could be changed to 
affect the body's motion. 

3.6 (a) You exert a force on a box, but it does not 

move. How would you explain this? How 
might an Aristotelian explain it? 

(b) Suppose now that you exert a greater 
force and the box moves. Explain this 
from your (Newtonian) point of view and 
from an Aristotelian point of view. 

(c) You stop pushing on the box and it 
quickly comes to rest. Explain this from 
both the Newtonian and the Aristotelian 
points of view. 

3.7 There are at least two drawbacks to an 
experimental test of Newton's law of inertia. 

(a) How can you really be sure that there is 
no unbalanced force acting on the object, 
even if you see that the object moves 
uniformly in a straight line? We can 
answer that we are sure because the 
object does continue to move uniformly in 
a straight line. But this answer is merely 
a restatement of the first law, which we 
wanted to prove by experiment. Surely we 
cannot use the first law to verify the first 
law! But we are not really caught in a 
circular argument. Practically, we can 
expect to find forces on an object only 
when other objects are in contact with it, 
or somewhere near it. The influences may 
be of unfamiliar kinds, and we may have 
to stretch what we mean by "near"; but 
whenever a force is detected we look for 
the source of the influence. If aU known 
influences on an object were balanced, 
and yet it didn't move uniformly, we 
would suspect an unknown influence and 
track it down — and we would find it. At 
least, that's how it has always turned out 
so far. As a practical example, consider 
the demonstration involving low friction 
pucks on a level surface. Without using 
Newton's first law, how could you be sure 
the surface was level? 

(b) What is meant by a straight line? 

3.8 (a) Assume that the floor of a laboratory 

could be made perfectly horizontal and 
perfectly smooth. A dry ice puck is placed 
on the floor and given a small push. 
Predict the way in which the puck would 
move. How would this motion differ if the 
whole laboratory were moving uniformly 
during the experiment? How would it 
differ if the whole laboratory were 
accelerating along a straight line? If the 
puck were seen to move in a curved path 
along the floor, how would you explain this? 
(b) A man gently starts a dry ice puck in 
motion while both are on a rotating 




platform. What will he report to be the 
motion he observes as the platform keeps 
rotating? How will he explain what he 
sees if he believes he can use Newton's 
first law to understand observations made 
in a rotating reference frame? Will he 
be right or wrong? 

3.9 In terms of Newton's first law, explain: 

(a) Why people in a moving car lurch forward 
when the car suddenly slows down; 

(b) What happens to the passengers of a car 
that makes a sharp, quick turn; 

(c) When a coin is put on a phonograph 
turntable and the motor started, does the 
coin fly off when the turntable reaches a 
certain speed? Why doesn't it fly off 

3.10 A balloon-like object stands before you, 
unmoving, suspended in mid-air. What can you 
say about the forces that may be acting on it? 
Suddenly it moves off in a curved path. Give two 
diff'erent explanations. How can you test which is 

3. 11 In an actual experiment on applying the 
same force to different masses, how would you 
know it was the "same force"? 

3.12 Several proportionalities can be combined 
into an equation only if care is taken about the 
units in which the factors are expressed. When 
we wrote Ad = ii x At in Chapter 1, we chose 
meters as units for d, seconds as units for t, and 
then made sure that the equation came out right 
by using meters/second as units for v. In other 
words, we let the equation define the unit for v. 
If we had already chosen some other units for v, 
say miles per hour, then we would have had to 
write instead something like 

Ad= fe X vAt 

where fe is a constant factor that matches up the 
units of d, t, and v. 

What value would k have if d were measured in 
miles, t in seconds, and v in miles per hour? 
Writing a = T„Jm before we have defined 
units of F and m is not the very best mathematical 
procedure. To be perfectly correct in expressing 
Newton's law, we would have had to write: 

- f 


where fe is a constant factor that would match up 
whatever units we choose for a, F, and m. In fact, 
we will take the easiest way out and let the 
equation define the units of F in terms of the 
units we choose for a and m, so the equation 
comes out right without using k. (Or if you prefer 
to say it that way, we choose units so that k = 1.) 

3.13 A body is being accelerated by an 
unbalanced force. If the magnitude of the net 
force is doubled and the mass of the body is 
reduced to one-third of the original value, what 

wUl be the ratio of the second acceleration to the 

3.14 What does a laboratory balance measure- 
mass or weight? What about a spring balance? 
(Hint: consider what would happen to readings on 
each if they were on the moon instead of the 
earth.) You might want to consider this question 
again after reading Sec. 3.8. 

3.1.) Describe as a thought experiment how you 
could calibrate a spring balance in force units. If 
you actually tried to do the experiments, what 
practical difficulties would you expect? 

3.16 "Hooke's law" says that the force exerted by 
a stretched or compressed spring is directly 
proportional to the amount of the compression or 
extension. As Robert Hooke put it in announcing 
his discovery: 

. . . the power of any spring is in the same 
proportion with the tension thereof: that 
is, if one power stretch or bend it one 
space, two will bend it two, three will 
bend it three, and so forward. Now as the 
theory is very short, so the way of trying 
it is very easie. 

If Hooke says it's "easie," then it might well 
be so. You can probably think immediately of how 
to test this law using springs and weights, (a) Try 
designing such an experiment; then after 
checking with your instructor, carry it out. What 
limitations do you find to Hooke's law? (b) How 
could you use Hooke's law to simplify the 
calibration procedure asked for in SG 3.15? 

3.17 Refer to the discussion in SG 3.12. Show 
that fe = 1 when we define a newton as we do on 
p. 83. 

3.18 When units for different terms in a relation 
are defined completely independently from one 
another, the numerical value of the constant 
must be found experimentally. (Later in this 
course you will see how finding the value of k in 
certain relations was very important in the 
development of physics.) Say. for example, that 
we had decided to measure force in "tugs." 
defining a tug as the force required to stretch a 
standard rubber band one inch. How could we go 
about finding k? 



3.19 Complete this table: 





a 1.0 N 



1 .0 m/sec^ 

b 24.0 












f 72.0 


g 3.6 


h 1.3 


i 30.0 


j 0.5 


k 120.0 


being applied each time. Record the 
position of the cart at equal time intervals 
by means of stroboscopic photography. 

(c) Repeat the last step in all details, but use 
two carts hooked together. Repeat again 
using all three carts hooked together. In 
all three cases it is crucial that the applied 
force be essentially the same. 

(d) Determine the value of acceleration for 
masses of m (1 cart), 2m (2 carts), and 
3m (3 carts). 

(e) Prepare a graph of a vs. m, of a vs. llm, 
and of 1/a vs. m. Comment on your 

3.22 Describe in detail the steps you would take 
in an idealized experiment to determine the 
unknown mass m of a certain object (in 
kilograms) if you were given nothing but a 
frictionless horizontal plane, a 1-kg standard, an 
uncalibrated spring balance, a meter stick, and a 

3.23 A block is dragged with constant velocity 
along a rough horizontal table top by means of a 
spring balance horizontally attached to the block. 
The balance shows a reading of 0.40 N at this 
and any other constant velocity. This means that 
the retarding frictional force between block and 
table is 0.40 N, and is not dependent on speed. 

Now the block is pulled harder and given a 
constant acceleration of 0.85 m/sec^; the balance 
is found to read 2.1 N. Compute the mass of the 

3.20 A rocket-sled has a mass of 4440 kg and is 
propelled by a solid-propellent rocket motor of 
890,000-N thrust which bums for 3.9 seconds. 

(a) What is the sled's average acceleration 
and maximum speed? 

(b) This sled has a maximum acceleration of 
30 g (= 30 Ug). How can that be, considering 
the data given? 

(c) If the sled travels a distance of 1530 m 
while attaining a top speed of 860 m/sec 
(how did it attain that high a speed?), 
what is its average acceleration? 

3.21 If you have "dynamics carts" available, 
here is one way of doing an experiment to 
demonstrate the inverse proportionality between 
acceleration and mass: 

(a) Add load blocks to one or the other of two 
carts until the carts balance when placed 
on opposite platforms of a laboratory 
balance. Balance a third cart with one of 
the first pair. Each cart now has the 
same mass m. (State two main 
assumptions involved here.) 

(b) Accelerate one cart on a level surface, 
using a rubber band; that is, pull the cart 
with the rubber band, keeping the rubber 
band stretched a known constant amount 
so that it supplies a constant force. Any 
other method can be used that will assure 
you that, within reason, the same force is 

3.24 We have claimed that any body in free fall 
is "weightless" because any weight-measuring 
device falling with it would read zero. This is not 
an entirely satisfactory explanation, because you 
feel a definite sensation during free fall that is 
exactly the same sensation you would feel if you 
were truly without weight — say deep in space far 
from any star or planet. (The sensation you feel 
on jumping off a roof or a diving board, or when 
someone pulls a chair out from under you.) Can 
you explain why your insides react in the same 
way to lack of weight and to free fall? 

3.25 Explain the statement that while the mass 
of an object is the same everywhere, its weight 
may vary from place to place. 

3.26 (a) A replica of the standard kilogram is 

constructed in Paris and then sent to the 
National Bureau of Standards near 
Washington. D.C. Assuming that this 
secondary standard is not damaged in 
transit, what is 

(i) its mass in Washington? 
(ii) its weight in Paris and in Washington? 
(In Paris, a, = 9.81 m/sec^; in 
Washington, a„= 9.80 m/sec*.) 

(b) What is the change in your own weight as 
you go from Paris to Washington? 



3.27 (a) Find your mass in kg, and your weight 

in newtons. 
(b) How much force is needed to accelerate 
you 1 m/sec^? How many kilograms can 
you lift? How many newtons of force 
must you exert to do this? 

3.28 Why is it often said that astronauts in orbit 
are weightless? 

3.29 When a runner pushes on the earth with the 
sole of his shoe, the earth pushes with an equal 
and opposite force on the sole of the shoe. This 
latter force has an accelerating effect on the 
runner, but what does the force acting on the 
earth do to the earth? From Newton's second law 
we would conclude that such an unbcdanced 
force would accelerate the earth. The mass of the 
earth is very great, however, so the acceleration 
caused by the runner is very small. A reasonable 
value for the average acceleration of a runner 
when he starts is 5 m/sec/sec, and a reasonable 
value for his mass would be 60 kg. The mass of 
the earth is approximately 60 x 10" kg. 

(a) What acceleration of the earth would the 
runner cause? 

(b) If the acceleration lasts for 2 seconds, 
what speed will the runner have reached? 

(c) What speed will the earth have reached? 

3.30 In terms of Newton's third law, assess the 
following statements: 

(a) You are standing perfectly still on the 
ground; therefore you and the earth exert 
equal and opposite forces on each other. 

(b) The reason that a propeller airplane 
cannot fly above the atmosphere is that 
there is no air to push one way while the 
plane goes the other. 

(c) Object A rests on object B. The mass of 
object A is 100 times as great as that of 
object B, but even so, the force A exerts 
on B is no greater than the force of B 
on A. 

3.31 Consider a tractor pulling a heavy log in a 
straight line. On the basis of Newton's third law, 
one might argue that the log pulls back on the 
tractor just as strongly as the tractor pulls the 
log. But why, then, does the tractor move? (Make 
a large drawing of the tractor, rope, log, and earth, 
and enter the forces.) 





ii^^^Bk-.* " '* -* 

3.32 Consider the system consisting of a 1.0-kg 
ball and the earth. The ball is dropped from a 
short distance above the ground and falls freely. 
Assuming that the mass of the earth is 
approximately 6.0 x 10" kg, 

(a) make a vector diagram illustrating the 
important forces acting on each member 
of the system. 

(b) calculate the acceleration of the earth in 
this interaction. 

(c) find the ratio of the magnitude of the 
ball's acceleration to that of the earth's 
acceleration (aja^). 

(d) make a vector diagram as in (a) but 
showing the situation when the ball has 
come to rest after hitting the ground. 

3.33 (a) A 75-kg man stands in an elevator. What 

force does the floor exert on him when the 

(i) starts moving upward with an 

acceleration of 1.5 m/sec^? 
(ii) moves upward with a constant speed 

of 2.0 m/sec? 
(iii) starts accelerating downward at 1.5 

(b) If the man were standing on a bathroom 
(spring) scale during his ride, what 
readings would the scale have under 
conditions (i), (ii), and (iii) above? 

(c) It is sometimes said that the "apparent 
weight" changes when the elevator 
accelerates. What could this mean? Does 
the weight really change? 

3.34 Useful hints for solving problems about the 
motion of an object and the forces acting on it. 

(a) make a light sketch of the physical 

(b) in heavy line, indicate the limits of 

the particular object you are interested in, 
and draw all the forces acting on that 
object. (For each force acting on it, it will 
be exerting an opposite force on something 
else — but we don't care about those.) 

(c) find the vector sum of all these forces, for 
example, by graphical construction. 

(d) using Newton's second law, set this sum, 
F„e„ equal to ma. 

(e) solve the equation for the unknown 

(f ) put in the numerical values you know and 
calculate the answer. 


A ketchup bottle whose mass is 1.0 kg rests 
on a table. If the friction force between the table 
and the bottle is a constant 6 newtons, what 
horizontal pull is required to accelerate the bottle 
from rest to a speed of 6 m/sec in 2 sec? 

First, sketch the situation: 



Second, draw in arrows to represent all the forces 
acting on the object of interest. There will be 
the horizontal pull Fp, the friction Pf, the 
gravitational pull Fg (the bottle's weight), and the 
upward force Ft exerted by the table. (There is, 
of course, also a force acting down on the table, 
but we don't care about that — we're interested 
only in the forces acting on the bottle.) 

Next, draw the arrows alone. In this sketch all 
the forces can be considered to be acting on the 
center of mass of the object. 


The mass m is given as 1.0 kg. The acceleration 
involved in going from rest to 6.0 m/sec in 2 
seconds is 


" At 

6.0 m/sec 

2 sec 

= 3.0 m/sec/sec 

So the 


force required is 

Fne, = 1.0 kg X 3.0 m/sec/sec 

= 3.0 kg m/sec/sec 

= 3.0 newtons 

If we consider toward the right to be the 
positive direction, Fnet is 3.0 newtons and Ff, which 
is directed to the left, is —3.0 newtons. 

Fnet = Fp + F/ 

3.0N = Fp + (-3.0N) 
fp = 3.0N + 3.0N 
fp = 6.0N 

If you prefer not to use + and — signs, you can 
work directly from your final diagram and use 
only the magnitudes of the forces: 

3.0>; 3. ON 


Because the bottle is not accelerating up or 
down, we know there is no net force up or down — 
so fr must just balance fg. So the net force 
acting on the bottle is just the vector sum of fp 
and ff. Using the usual tip- to- tail addition: 

from which the magnitude of Fp is obviously 6. ON. 

As the last arrow diagram shows, the horizontal 
pull must be greater than the force required for 
acceleration by an amount equal to the friction. 
We already know ff. We can find Fnet from 
Newton's second law if we know the mass and 
acceleration of the bottle, since Fnet = ^«- The net 
force required to accelerate the case is found 
from Newton's second law: 


4.1 A trip to the moon 

4.2 Projectile motion 

4.3 What is the path of a projectile? 

4.4 Moving frames of reference 

4.5 Circular motion 

4.6 Centripetal acceleration and centripetal force 

4.7 The motion of earth satellites 

4.8 What about other motions? 


". . . the greater the velocity . . . with 
which [a stone] is projected, the 
farther it goes before it falls to the 
earth. We may therefore suppose the 
velocity to be so increased, that it 
would describe an arc of 1, 2, 5, 10, 
100, 1000 miles before it arrived at the 
earth, till at last, exceeding the limits 
of the earth, it should pass into space 
without touching it." — Newton's Sys- 
tem of the World 


Understanding Motion 

4.1 A trip to the moon 

Imagine a Saturn rocket taking off from its launching pad at 
Cape Kennedy. It climbs above the earth, passing through the 
atmosphere and beyond. Successive stages of the rocket shut off 
leaving finally a capsule hurtling through the near-vacuum of space 
toward its destination 240,000 miles away. Approximately 65 hours 
after take-off, the capsule circles the moon and descends to its 
target— the center of the lunar crater Copernicus. 

The complexity of such a voyage is enormous. To direct and 
guide the flight, a great number and variety of factors must be 
taken into account. The atmospheric drag in the early part of the 
flight depends upon the rocket's speed and altitude. The engine 
thrust changes with time. The gravitational pulls of the sun, the 
earth, and the moon change as the capsule changes its position 
relative to them. The rocket's mass is changing. Moreover, it is 
launched from a spinning earth, which in turn is circling the sun, 
and the target — the moon — is moving around the earth at a speed 
of about 2,300 miles per hour. 

Yet, as for almost any complex motion, the flight can be broken 
down into small portions, each of which is relatively simple to 
describe. What we have learned in earlier chapters will be useful in 
this task. 

In simplified form, the earth-moon trip can be divided into 
these eight parts: 

Part 1. The rocket accelerates vertically upward from the 

surface of the earth. The force acting on the rocket is 
not really constant, and the mass of the rocket 
decreases as the propellent escapes. The value of the 
acceleration at any instant can be computed using 
Newton's second law; it is given by the ratio of net 
force (thrust minus weight) at that instant to the mass 
at that instant. 

Part 2. The rocket, still accelerating, follows a curved path as 
it is "injected" into an orbit about the earth. 

In his science-fiction novels of more 
than a hundred years ago, the French 
author Jules Verne (1828-1905) 
launched three spacemen to the 
moon by means of a gigantic charge 
fixed in a steel pipe deep in the earth. 




Understanding Motion 

"^ 1 ^n ■ ■■^.^^ 



Part 3. In an orbit 115 miles above the earth's surface, the 
capsule moves in a nearly circular arc at a constant 
speed of 17,380 miles/hr. 

Part 4. The rocket engines are fired again, increasing the 

capsule's speed so that it follows a much less curved 
path into space. (The minimum speed necessary to 
escape the earth completely is 24,670 miles/hr.) 

Part 5. In the flight between earth and moon, only occasional 
bursts from the capsule's rockets are required to 
keep it precisely on course. Between these 
correction thrusts, the capsule moves under the 
influence of the gravitational forces of earth, moon, 
and sun; we know from Newton's first law that the 
capsule would move with constant velocity if it were 
not for these forces. 

Part 6. On nearing the moon, the rocket engines are fired again 
to give the capsule the correct velocity to "inject" into 
a circular orbit around the moon. 

Part 7. The capsule is moving with a constant speed of about 1 
mile/sec in a nearly circular path 50 miles above the 
moon's surface. 

Part 8. After its rockets are fired in the direction of motion to 
reduce the speed, the capsule accelerates downward 
as it falls toward the surface of the moon. It follows 
an arcing path before it lands in the crater Copernicus. 
(Just before impact, the rocket engines fire a final time 
to reduce speed of fall and prevent a hard landing.) 

SG 4.2 Motion along a straight line (as in Parts 1 and 5) is easy 

enough to describe. But let us analyze in greater detail other parts 
of this trip: moving on a circular arc, as in Parts 3 and 7, and 
projectile motion, as in Part 8, are two important cases. 

How shall we go about making this analysis? Following the 
example of Galileo and Newton, we can try to learn about the 
behavior or moving objects beyond our reach, even on the moon or 
in the farthest parts of the universe, by studying the motion of 
objects near at hand. If we believe that physics is the same 
everywhere, then the path of a lunar capsule moving as in Part 8 
can be understood by studying a marble rolling off" the edge of a 
table or a bullet fired from a horizontal rifle. 

Section 4.2 

4.2 Projectile motion 


Consider this experiment: a rifle is mounted on a tower with its 
barrel parallel to the ground; the ground over which the bullet will 
travel is level for a very great distance. At the instant a bullet 
leaves the rifle, an identical bullet is dropped from the height of the 
barrel of the rifle. The second bullet has no horizontal motion 
relative to the ground; it goes only straight down. Which bullet will 
reach the ground first? 

We do not need to know anything about the speed of the bullet 
or the height of the tower in order to answer this question. 
Consider first the motion of the second bullet, the one that is dropped. 
As a freely falling object, it accelerates toward the ground with 
constant acceleration. As it falls, the time t and the corresponding 
downward displacement y are related by 

y = iagt^ 

where Ug is the acceleration due to gravity at that location. 

Now consider the bullet that is fired horizontally from the rifle. 
When the gun is fired, the bullet is driven by the force of expanding 
gases and accelerates very rapidly until it reaches the muzzle of the 
rifle. On reaching the muzzle these gases escape and no longer 
push the bullet. At that moment, however, the bullet has a large 
horizontal speed, Vj,. The air will slow the bullet slightly, but we 
shall ignore that fact and imagine an ideal case with no air 
friction. As long as air friction is ignored, there is no force acting 
on the projectile in the horizontal direction. Therefore, we expect 

the horizontal speed will remain constant. From the instant the 
bullet leaves the muzzle, we would expect its horizontal motion to 
be described by the equation 

X = Vj.t 

So much for the forward part of the motion. There is, however, 
another part that becomes more and more important as t increases. 
From the moment the bullet leaves the gun, it falls toward the 
earth while it moves forward, like any other unsupported body. 
Can we use the same equation to describe its fall that we used to 
describe the fall of the dropped bullets? And how will falling aff"ect 
the bullet's horizontal motion? These doubts raise a more 

J L 



Understanding Motion 

The two balls in this strobscopic 
photograph were released simultane- 
ously. The one on the left was simply 
dropped from rest position; the one 
on the right was given an initial veloc- 
ity in the horizontal direction. 

SG 4.3 

SG 4.4 

fundamental question that goes beyond just the behavior of the 
bullets; namely, is the vertical motion of an object affected by its 
horizontal motion? Or vice versa? 

To answer these questions, we can carry out a real experiment 
similar to our thought experiment. We can use a special laboratory 
device designed to fire a ball in a horizontal direction at the 
moment that a second ball is released to fall freely from the same 
height. We set up our apparatus so that both balls are the same 
height above a level floor. The balls are released and, although the 
motions of the balls may be too rapid for us to follow with the eye, 
we will hear that they reach the floor at the same time. This result 
suggests that the vertical motion of the projected ball is unaffected 
by its horizontal velocity. 

In the margin is a stroboscopic photograph of this experiment. 
Equally spaced horizontal lines aid our examination. Look first at 
the ball on the left, which was released without any horizontal 
motion. You see that it accelerates because it moves greater 
distances between successive flashes. Careful measurement of the 
photograph shows that the acceleration is constant, within the 
uncertainty of our measurements. 

Now compare the vertical positions of the second ball, fired to 
the right, with the vertical positions of the ball which is falling 
freely. The horizontal lines show that the distances of fall are the 
same for corresponding time intervals. The two balls obey the same 
law for motion in a vertical direction. That is, at every instant they 
both have the same constant acceleration a^,, the same downward 
velocity and the same vertical displacement. The experiment 
therefore supports the idea that the vertical motion is the same 
whether or not the ball has a horizontal motion also. The horizontal 
motion does not aff'ect the vertical motion. 

We can also use the strobe photo to see if the vertical motion of 
the projectile aff"ects its horizontal velocity, by measuring the 
horizontal distance between successive images. We find that the 
horizontal distances are practically equal. Since the time intervals 
between images are equal, we can conclude that the horizontal 
velocity v^ is constant. So we can conclude that the vertical motion 
doesn't aff'ect the horizontal motion. 

The experiment shows that the vertical and horizontal 
componerits of the motion are independent of each other. This 
experiment can be repeated from diff"erent heights, and with 
diff"erent horizontal velocities, but the results lead to the same 

The independence of motions at right angles has important 
consequences. For example, it is easy to predict the displacement 
and the velocity of a projectile at any time during its flight. We need 
merely to consider the horizontal and vertical aspects of the 
motion separately, and then add the results -vectorially. We can 
predict the magnitude of the components of displacement (x and y) 
and of the components of velocity (Vj. and v„) at any instant by 
application of the appropriate equations. For the horizontal 

Section 4.3 


component of motion, the equations are 

Vx = constant 

X= Vjct 

and for the vertical component of motion, 

Vy^ Ugt 



y = ^agt^ 

Q1 If a body falls from rest with acceleration Og, with what 
acceleration will it fall if it has an initial horizontal speed Vx? 

4.3 What is the path of a projectile? 

It is easy to see that a thrown object, such as a rock, follows a 
curved path, but it is not so easy to see just what kind of curve it 
traces. For example, arcs of circles, ellipses, parabolas, hyperbolas, 
and cycloids (to name only a few geometric figures) all provide 
likely-looking curved paths. 

Better knowledge about the path of a projectile was gained 
when mathematics was applied to the problem. This was done by 
deriving the equation that expresses the shape of the path. Only a 
few steps are involved. First let us hst equations we already know 
for a projectile launched horizontally: 

X = v„t 


y = Jttgt^ 

We would know the shape of the trajectory if we had an equation 
that gave the value of y for each value of x. We can find the fall 
distance y for any horizontal distance x by combining these two 
equations in a way that eliminates the time variable. Solving the 
equation x = v^t for t we get 

_ ^ 

Because t means the same in both equations, we can substitute 
xlvx for t in the equation for y: 



and thus 


In this last equation there are two variables of interest, x and y, 
and three constant quantities: the number y, the uniform 
acceleration of free fall a«, and the horizontal speed Vx which we 

Specialized equations such as 
these need not be memorized. 


Understanding Motion 

* t 

r c i 

b A 






Drawing of a parabolic trajectory from 
Galileo's Two New Sciences. 

SG 4.5 
SG 4.6 

"Philosophy is written in this grand 
book, the universe, which stands 
continually open to our gaze. But 
the book cannot be understood 
unless one first learns to 
comprehend the language and read 
the letters in which it is composed. 
It is written in the language of 
mathematics, and its characters 
are triangles, circles, and other 
geometric figures, without which it 
is humanly impossible to understand 
a single word of it." (Galileo, cited 
in Discoveries and Opinions of 
Galileo, translated by Stillman Drake, 
Anchor Books, pp. 237-238.) 

take to be constant for any one flight from launching to the end. 
Bringing these constants together between one set of parentheses, we 
can write the equation as 



or, letting k stand for constant (agl2v/) 

y = kx^ 

This equation shows a fairly simple relationship between x and y 
for the trajectory. We can translate it as: the distance a projectile 
falls away from a straight path is proportional to the square of the 
distance it moves sideways. For example, when the projectile goes 
twice as far horizontally from the launching point, it drops 
vertically four times as far. 

The mathematical curve represented by this relationship 
between x and y is called a parabola. Galileo deduced the parabolic 
shape of trajectories by an argument similar to the one we used. 
(Even projectiles not launched horizontally — as in the photographs 
on p. 103 and 123 — have parabolic trajectories.) With this 
discovery, the study of projectile motion became much simpler, 
because the geometric properties of the parabola had been 
established centuries earlier by Greek mathematicians. 

Here we find a clue to one of the important strategies in 
modem science. When we express the features of a phenomenon 
quantitatively and cast the relations between them into equation 
form, we can use the rules of mathematics to manipulate the 
equations, and so open the way to unexpected insights. 

Galileo insisted that ''the proper language of nature is 
mathematics," and that an understanding of natural phenomena is 
aided by translating our qualitative experiences into quantitative 
terms. If, for example, we find that trajectories have a parabolic 
shape, we can apply all we know about the mathematics of 
parabolas to describe — and predict — trajectories. Physicists have 
often drawn on the previously developed parts of pure mathematics 
to express (or to extend) their conceptions of natural 
phenomena. Sometimes, as in the case of Newton's inventing 
calculus, they have to develop new parts of mathematics. The 
physical scientist often tries to use methods from another branch of 
science, in addition to mathematics, to find a solution for his 
particular problem. For example, just as Galileo used the already- 
known mathematics of parabolas to deal with actual projectile 
motions, so the modern sound engineer solves problems in acoustics 
using ideas and mathematical techniques developed independently 
by electrical engineers. Whatever the methods of science may be, 
many ideas and concepts can often be extended from one specialty 
to another, with fruitful results. 

We can now apply our theory of projectile motion to the case 
mentioned earlier, the free motion of a space capsule toward the 

Section 4.4 


moon's surface. Let us assume that the orbit is a low one, so that 
the acceleration due to gravity is almost constant between the 
orbit and the surface. If the rocket engines are fired forward, in 
the direction of motion, the capsule's speed will be reduced and 
it will begin to fall closer to the surface. After firing, the reduced 
horizontal speed remains constant, so the capsule falls toward 
the surface on a parabolic path. Spaceflight engineers apply 
ideas like these to land a space capsule on a desired moon target. 
(See SG 4.23). 

Q2 Which of the conditions below must hold in order for the 
relationship y = kx- to describe the path of a projectile? 

(a) Ug is a constant 

(b) Ug depends on t 

(c) ttg is straight down 

(d) Vj. depends on t 

(e) air friction is negligible 

4.4 Moving frames of reference 



The critics of Galileo claimed that if 
the earth moved, a dropped stone 
would be left behind and land beyond 
the foot of the tower. 

Galileo's work on projectiles leads to thinking about reference 
frames. As you will see in Unit 2, Galileo ardently supported the 
idea that the preferred reference frame for discussing motions in 
our planetary system is one fixed to the sun, not the earth. From 
that point of view, the earth both revolves around the sun and 
rotates on its own axis. For many scientists of Galileo's time, this 
idea was impossible to accept, and they thought they could prove 
their case. If the earth rotated, they said, a stone dropped from a 
tower would not land directly at its base. For if the earth rotates 
once a day, the tower would move on for hundreds of feet for every 
second the stone is falling; hence, the stone would be left behind 
while falling through the air and consequently would land far 
behind the base of the tower. But this is not what happens. As near 
as one can tell, the stone lands directly under where it was 
released. Therefore, many of Galileo's critics believed that the tower 
and the earth could not be considered to be in motion. 

To answer these arguments, Galileo showed the same 
observation can support his view that, during the time of fall, the 
tower and the ground supporting it were moving forward together 
with the same uniform velocity. While the stone was being held at 
the top of the tower, it had the same horizontal velocity as the tower. 
Releasing the stone allows it to gain vertical speed, but by the 
principle of independence of Vj. and Vy discussed in Section 4.3, this 
does not diminish any horizontal speed it had initially on being 
released. In other words, the falling stone behaves like any other 
projectile: the horizontal and vertical components of its motion are 
independent of each other. Since the stone and tower continue to 
have the same Vj. throughout, the stone will not be left behind as it 
falls. Therefore, no matter what the speed of the earth, the stone 



Galileo argues that the falling stone 
continued to share the motion of the 
earth, so that an observer on earth 
could not tell whether or not the earth 
moved by watching the stone. 


Understanding Motion 

At high speeds, air drag will 
affect the results considerably. The 
situation is still indistinguishable 
from a car at rest— but in a high 

When relative speeds become a 
noticeable fraction of the speed 
of light (almost a billion mph), 
some deviations from this simple 
relativity principle begin to appear. 
We will consider some of them in 
Unit 5. 

SG 4.8, 4.9, 4.10 

will land at the foot of the tower. The fact that falling stones are 
not left behind is not a proof that the earth is standing stiU. 

Similarly, Galileo said, an object released from a crow's nest at 
the top of a ship's perpendicular mast will land at the foot of the 
mast, whether the ship is standing still in the harbor or moving 
with constant velocity through quiet water. This was actually tested 
by experiment in 1642 (and is also the subject of three Project 
Physics film loops). We know this to be the case from everyday 
observation: when you drop or throw a book in a bus or train or 
plane that is moving with constant velocity, you will see it moving 
just as it would if the vehicle were standing still. Or again, if an 
object is projected vertically upward from inside an open car that is 
moving at constant velocity, it will fall back into the car. A person 
in the car will see the same thing happen whether the car has 
been continuously moving at constant velocity or has been standing 

From these and other observations has come a valuable 
generalization: If there is any one laboratory in which Newton's 
laws hold, then these laws will hold equally well in any other lab 
(or "reference frame") that moves at constant velocity with 
respect to the first. This generalization is called the Galilean 
relativity principle. It holds true for all "classical" mechanical 
phenomena — that is, phenomena involving a tremendous range of 
relative velocities, up to millions of miles per hour. 

If the laws of mechanics are found to be the same for all 
reference frames moving with constant velocity with respect to 
each other, then there is no way to find the speed of one's own 
reference frame from any mechanical experiment done in the 
reference frame, nor can one pick out any one reference frame as 
the "true" frame — the one that is, say, at absolute rest. Thus 
there can be no such thing as the "absolute" velocity of a body — 
all measured velocities are only relative. 

What about observations of phenomena outside of one's own 
frame of reference? Certainly some outside phenomena can appear 
differently to observers in different reference frames — for example, 
the velocity of an airplane will have a different value when seen 
from the earth and from a moving ship. But other measurables such 
as mass, acceleration, and time interval will have the same values 
when a phenomenon is observed from different reference frames 
that move with constant velocity with respect to one another. More- 
over, certain relationships among such measurements will be 
found to be the same for these different reference frames. Newton's 
laws of motion are examples of such "invariant" relationships, and 
so are all the laws of mechanics that follow from them. 

Notice that the relativity principle, even in this restricted 
classical form, does not say "everything is relative." On the 
contrary, it asks us to look for relationships that do not change 
when reference systems are changed. 

Section 4.5 


Q3 If the laws of mechanics are found to be the same in two 
reference frames, what must be true of their motions? 

4.5 Circular motion 

A projectile launched horizontally from a tall tower strikes the 
earth at a point determined by the speed of the projectile, the 
height of the tower, and the acceleration due to the force of gravity. 
As the projectile's launch speed is increased, it strikes the earth at 
points farther and farther from the tower's base, and we would 
have to take into account that the earth is not flat but curved. If we 
suppose the launch speed to be increased even more, the projectile 
would strike the earth at points even farther from the tower, till at 
last it would rush around the earth in a nearly circular orbit. At this 
orbiting speed, the fall of the projectile away from the forward, 
straight line motion is matched by the curvature of the surface, and 
it stays at a constant distance above the surface. 

What horizontal launch speed is required to put an object into 
a circular orbit about the earth or the moon? We shall be able to 
answer this question quite easily after we have learned about 
circular motion. 

The simplest kind of circular motion is uniform circular motion, 
that is, motion in a circle at constant speed. If you are in a car or 
train that goes around a perfectly circular track so that at every 
instant the speedometer reading is forty miles per hour, you are 
executing uniform circular motion. But this is not the case if the 
track is any shape other than circular, or if your speed changes at 
any point. 

How can we find out if an object in circular motion is moving 
at constant speed? The answer is to apply the same test we used in 
deciding whether or not an object traveling in a straight line does 
so with constant speed: we measure the instantaneous speed at 
many different moments and see whether the values are the same. 
If the speed is constant, we can describe the circular motion of the 
object by means of two numbers: the radius R of the circle and 
the speed v along the path. For regularly repeated circular motion, 
we can use a quantity more easily measured than speed: either the 
time required by an object to make one complete revolution, or the 
number of revolutions the object completes in a unit of time. The 
time required for an object to complete one revolution in a circular 
path is called the period of the motion. The period is usuaUy 
denoted by the capital letter T. The number of revolutions completed 
by the same object in a unit time interval is called the frequency 
of the motion. Frequency will be denoted by the letter/. 

As an example, we will use these terms to describe a car moving 
with uniform speed on a circular track. Let us suppose the car takes 
20 seconds to make one lap around the track. Thus, T= 20 seconds. 
Alternatively, we might say that the car makes 3 laps in a minute. 

In discussing circular motion it is 
useful to keep clearly in mind a dis- 
tinction between revolution and 
rotation. We define these terms dif- 
ferently: revolution is the act of 
traveling along a circular or elliptical 
path; rotation is the act of spinning 
rather than traveling. A point on the 
rim of a phonograph turntable travels 
a long way; it is revolving about the 
axis of the turntable. But the turn- 
table as a unit does not move from 
place to place: it merely rotates. In 
some situations both processes 
occur simultaneously; for example, 
the earth rotates about its own axis, 
while it also revolves (in a nearly 
circular path) around the sun. 


Understanding Motion 

SG 4.11 

The term "revolutions" is not 
assigned any units because it is a 
pure number, a count. There is no 
need for a standard as there is for 
distance, mass, and time. So, the 
unit for frequency is usually given 
without "rev." This looks strange, 
but one gets used to it— and it is not 
very important, because it is merely 
a matter of terminology, not a fact 
of physics. 

Thus/= 3 revolutions per minute, or/= 1/20 revolution per second. 
The relationship between frequency and period (when the same 
time unit is used) is/= 1/T. If the period of the car is 20 sec/rev, 
then the frequency is 

1 1 rev 

20 sec 



All units are a matter of convenience. Radius may be expressed 
in terms of centimeters, kilometers, miles, or any other distance 
unit. Period may be expressed in seconds, minutes, years, or any 
other time unit. Correspondingly, the frequency may be expressed 
as "per second," "per minute," or "per year." The most widely used 
units of radius, period, and frequency in scientific work are meter, 
second, and per second. 

Table 4.1 Comparison of the frequency and period for various kinds of 
circular motion. Note the differences between units. 




Electron in circular accelerator 




per sec 





per sec 

Hoover Dam turbine 




per sec 

Rotation of earth 




per min 

Moon around the earth 




per hour 

Earth about the sun 




per day 

If an object is in uniform circular motion, and if we know the 
frequency of revolution / and the radius R of the path, we can 
compute the speed v of the object without difficulty. The distance 
traveled in one revolution is simply the perimeter of the circular 
path, that is, 2ttR. The time for one revolution is by definition the 
period T. Since for uniform motion it is always true that 

speed = 
by substitution we get 

distance traveled 
time elapsed 


To express this equation for circular motion in terms of the 
frequency/, wejrewrite it as 

V = 2ttR X j^ 

now, since by definition 

J i-p 

we can write 

V = 2ttR X / 
If the body is in uniform, circular motion, the speed computed 

Section 4.6 


with the aid of this equation is both its instantaneous speed and its 
average speed. If the motion is not uniform, the formula gives only 
the average speed; the instantaneous speed for any point on the 
circle can be determined if we find Ad/At from measurements of 
very small segments of the path. 

Let us now see how the last equation can be used. We can, for 
example, calculate the speed of the tip of a helicopter rotor blade in 
its motion around the central shaft. On one model, the main rotor 
has a diameter of 7.50 m and a frequency of 480 revolutions/minute 
under standard conditions. Thus/= 480 per minute = 8.00 per 
second and R = 3.75 m, and 

V = 27rRf 

v = 2 (3.14)(3.75)(8.00) meters/second 

V = 189 m/sec 

or about 420 miles/hr. 

Q4 If a phonograph turntable is running at 45 rev.olutions per 

(a) What is its period (in minutes)? 

(b) What is its period (in seconds)? 

(c) What is its frequency in cycles per second? 

Q5 What is the period of the minute hand of an ordinary clock? 
If the hand is 3.0 cm long, what is the linear speed of the tip of the 
minute hand? 

Q6 The terms frequency and period can also be used for any 
other periodic, repetitive phenomenon. For example, if your heart 
beats 80 times per minute, what are the frequency and period for 
your pulse? 

4.6 Centripetal acceleration and centripetal force 

Let us assume that a stone on a string is moving with uniform 
circular motion, for example in a horizontal plane as the stone is 
whirled overhead. The speed of the stone is constant. The velocity, 
however, is always changing. Velocity is a vector quantity, which 
includes both speed and direction. Up to this point we have dealt 
with accelerations in which only the speed was changing. In 
uniform circular motion the speed of the revolving object remains 
the same, while the direction of motion changes continually. The 
figure shows the whirling stone at three successive moments in its 
revolution. At any instant, the direction of the velocity vector is 
tangent to the curving path. Notice that its speed, represented by 
the length of the velocity arrow, does not vary; but its direction 
changes from moment to moment. Since acceleration is defined 
as a change in velocity, the stone is in fact accelerating. 

But to produce an acceleration a net force is needed. In the case 
of the whirling stone, a force is exerted on the stone by the string, 
and if we neglect the weight of the stone or air resistance, that 

SG 4.12 a to f 

a'r and K are parallel, but iTis perpen- 
dicular to a^. and T,.. Note that usually 
one should not draw different kinds of 
vector quantities on the same drawing. 


Understanding Motion 

The adjective centripetal means 
literally "moving, or directed, 
toward the center." 

In uniform circular motion, the 
instantaneous velocity and the 
centripetal force at any instant of 
time are perpendicular, one being 
along the tangent, the other along 
the radius. So instantaneous velocity 
and the acceleration are also 
always at right angles. 

will be the net force. If the string were suddenly cut, the stone 
would go flying off on a tangent with the velocity it had at the 
instant the string was cut — on a tangent to the circular path. As 
long as the string holds, the stone is forced into a circular path. 

The direction of this force acting on the stone is along the 
string. Thus the force vector is always pointing toward the center of 
rotation. This kind of force — always directed toward the center of 
rotation — is called centripetal force. 

From Newton's second law we know that force and 
acceleration are in the same direction, so the acceleration vector is 
also directed toward the center. We shall call this acceleration 
centripetal acceleration, and give it the symbol ap. Any object 
moving along a circular path has a centripetal acceleration. 

We know now the direction of centripetal acceleration. What is 
its magnitude? An expression for Op can be derived from the 
definition of acceleration Op = Ax;/At. The details of such a derivation 
are given on the next page. The result shows that a<, depends on v 
and R, and in fact the magnitude of ac is given by 

Let us verify this relationship with a numerical example. If, as 
sketched in the diagram, a car goes around a circular curve of 
radius R = 100 m at a uniform speed of i; = 20 m/sec, what is its 
centripetal acceleration a^ toward the center of curvature? By the 
equation derived on the gray page: 


_ V sec/ 
100 m 

400 sec' 
~ 100 m 

This is about 4/10 of a„, and could 
be called an acceleration of "0.4g." 

= 4.0 


Derivation of the equation Sc = -^ 

Assume the stone is moving uniformly in a circle of radius R. 
We can find what the relationship between ac, v, and R is by treating 
a small part of the circular path as the combination of a tangential 
motion and an acceleration toward the center. To follow the circular 
path, the stone must accelerate toward the center through a 
distance h in the same time that it would move through a tangential 
distance d. The stone, with speed v, would travel a tangential 
distance d given by d = vM. In the same time At, the stone, with 
acceleration a^ would travel toward the center through a distance h 
given hy h = ^UcM^. (We can use this last equation because at 
t = 0, the stone's velocity toward the center is zero.) 

We can now apply the Pythagorean Theorem to the triangle in 
the figure at the right. 

R2 + ^2 = (R + h7 

= R^ + 2Rh + h^ 

When we subtract R^ from each side of the equation we are left 

d2 = 2Rh + h^ 

We can simphfy this expression by making an approximation: since 
h is very small compared to R, h^ will be very small compared to Rh. 
If we choose At to be vanishingly small (as we must to get the 
instantaneous acceleration), h^ will become vanishingly small 
compared to Rh; so we shall neglect h^ and write 

d^ = 2Rh 

Also, we know d = t;Af and h = ia^At^, so we can substitute for d^ 
and for h accordingly. Thus 

(vMy = 2R • jUciMy 
vKMy = RadMY 


ar = -7r 

The approximation becomes better and better as At becomes 
smaller and smaller. In other words. v^lR is the magnitude of the 
instantaneous centripetal acceleration for a body moving on a 
circular arc of radius R. For uniform circular motion, V'^IR is the 
magnitude of the centripetal acceleration at every point of the path. 
(Of course it does not have to be a stone on a string. It can be a 
point particle on the rim of a rotating wheel, or a house on the 
rotating earth, or a coin sitting on a rotating phonograph disk, or a 
car in a curve on the road.) 

112 Understanding Motion 

^^- -^-^^ Does this make sense? We can check the result by going back 

'^ ^■^<\2.0m to the basic vector definition of acceleration: a„,. = AvIAt. We will 

/ V make a scale drawing of the car's velocity vector at two instants a 

\ short time At apart, measure the change in velocity Av between 

.,-,i„, i.,, o,.n 1^ '' \ them, and divide the magnitude of Av by At to get a„,. over the 

1fnm = Imiy'hr ' ^ interval. 

! Consider a time interval of At = 1 second. Since the car is 

moving at 20 m/sec, its position will change 20 m during At. Two 
positions P and P', separated by 20 m, are marked in diagram B. 
/ Now draw arrows representing velocity vectors. If we choose a 

/ scale of 1 cm = 10 m/sec, the velocity vector for the car will be 

represented by an arrow 2 cm long. These are drawn at P and P' in 
diagram C. 

If we put these two arrows together tail to tail as in diagram 
D, it is easy to see what the change in the velocity vector has been 
during At. Notice that if Ai; were drawn halfway between P and P', 
it would point directly toward the center of the curve; so the 
average acceleration between P and P' is indeed directed 
centripetally. Measurement of the Az; arrow in the diagram shows 
that it has a magnitude of 0.40 cm; so it represents a velocity 
change of 4.0 m/sec. This change occurred during At = 1 second, so 
the rate of change is 4.0 m/sec/sec — the same value we found using 
the relation Uc = v^/Rl 
/ The best way of showing that a^ = z/^/R is entirely consistent 

with the mechanics we have developed in Unit 1 is to do some 
experiments to measure the centripetal force required to keep an 
object moving in a circle. If, for example, the mass of the car were 
1000 kg, there would have to be a centripetal force acting on the 

^ m 

\ = 1000 kg X 4.0 -^ 

\ se& 


' m 

' = 4000 kg = 4000 N (or about 1800 pounds). 

I sec^ 


/ This force would be directed toward the center of curvature of 

the road — that is, it would always be sideways to the direction the 

car is moving. This force is exerted on the tires by the road. If the 

road is wet or icy, and can not exert a force of 4000 N sideways on 

SG 4.13 the tires, the centripetal acceleration will be less than 4.0 m/sec — so 

SG 4.14 the car will follow a less curved path as sketched in the margin on 

the next page. In situations where the car's path is less curved 

than the road, we would say the car "left the road" -although it 

might be just as appropriate to say the road left the car. 

The sideways force exerted on tires by a road is not easy to 

measure. But in Project Physics Handbook 1 there are a number of 

ways suggested for you to check experimentally whether Fc = muc 

or Fr^m vVR. 

For uniform motion in repeated cycles, it is often easier to 

measure the frequency / or period T than it is to measure v directly. 

Section 4.7 


We can substitute the relations v = 27rR/or v = 2itRIT into the 
equation equation for a^ to get alternative and equivalent ways of 
calculating a^: 



\ T / 

_ 47r2R2/2 




= 4rrmp 



Q7 In which of the following cases can a body have an 

(a) moves with constant speed 

(b) moves in a circle with constant radius 

(c) moves with constant velocity 

Q8 In what direction would a piece from a rapidly spinning 
fly-wheel go if it suddenly shattered? 

Q9 If a car of mass m going at speed v enters a curve of radius 
R, what is the force required to keep the car curving with the road? 

Q10 If a rock of mass m is being whirled overhead at 1 
revolution/second on a string of length R, what is the force which 
the string must be exerting? 

4.7 The motion of earth satellites 

Nature and technology provide many examples of the type of 
motion where an object is in uniform circular motion. The wheel 
has been a main characteristic of our civilization, first as it 
appeared on crude carts and then later as an essential part of 
complex machines. The historical importance of rotary motion in 
the development of modern technology has been described by the 
historian V. Gordon Childe: 

Rotating machines for performing repetitive 
operations, driven by water, by thermal power, or by 
electrical energy, were the most decisive factors of the 
industrial revolution, and, from the first steamship till the 
invention of the jet plane, it is the application of rotary 
motion to transport that has revolutionized communica- 
tions. The use of rotary machines, as of any other human 
tools, has been cumulative and progressive. The inventors 
of the eighteenth and nineteenth centuries were merely 
extending the applications of rotary motion that had 
been devised in previous generations, reaching back 
thousands of years into the prehistoric past. . . . 
[The History of Technology.] 

SG 4.12 g, h 
SG 4.16 
SG 4.17 
SG 4.18 

Chariot. Alberto Giacometti, 1950. 


Understanding Motion 

SG 4.19 

As you will see in Unit 2, there is another rotational motion that 
has also been one of the central concerns of man throughout 
recorded history: the orbiting of planets around the sun and of the 
moon around the earth. 

Since the kinematics and dynamics for any uniform circular 
motion are the same, we can apply what you have learned so far to 
the motion of artificial earth satellites in circular (or nearly 
circular) paths. As an illustration, we will select the satellite 
Alouette I, Canada's first satellite, which was launched into a 
nearly circular orbit on September 29, 1962. 

Tracking stations located in many places around the world 
maintain a record of any satellite's position in the sky. From the 
position data, the satellite's distance above the earth at any time 
and its period of revolution are found. By means of such tracking, 
we know that Alouette I moves at an average height of 630 miles 
above sea level, and takes 105.4 minutes to complete one revolution. 

We can now quickly calculate the orbital speed and the 
centripetal acceleration of Alouette I. The relationship v = IttRIT 
allows us to find the speed of any object moving uniformly in a 
circle if we know its period T and its distance R from the center 
of its path (in this case, the center of the earth). Adding 630 miles 
to the earth's radius of 3963 miles, we get R = 4594 miles, and 


277 X 4593 mi 
105.4 min 

28, 860 mi 
105.4 min 

= 274 mi/min 

or roughly 16,400 mi/hr. 

To calculate the centripetal acceleration of Alouette I, we can 
use this value of v along with the relationship a^ = v'^IR. Thus 

SG 4.20 
SG 4.21 

_ (274 mi/min)^ 
4,594 mi 

= 16.3 mi/min^ 

which is equivalent to 7.3 m/secl (To get the same result, we could 
just as well have used the values of R and T directly in the 
relationship a^ = ^tt'^RIT^) 

What is the origin of the force that gives rise to this 
acceleration? Although we will not make a good case for it until 
Chapter 8, you surely know already that it is due to the earth's 
attraction. Evidently the centripetal acceleration a<. of the 
satellite is just the gravitational acceleration a„ at that height, 
which has a value 25% less than Qg very near the earth's surface. 

Section 4.7 


Earlier we asked the question, "What speed is required for an 
object to stay in a circular orbit about the earth?" You can answer 
this question now for an orbit 630 miles above the surface of the 
earth. To get a general answer, you need to know how the 
acceleration due to gravity changes with distance. In Chapter 8 we 
will come back to the problem of injection speeds for orbits. 

The same kind of analysis applies to an orbit around the moon. 
For example, on the first manned orbit of the moon (Apollo 8, in 
1968), the mission control group wanted to put the capsule into a 
circular orbit 70 miles above the lunar surface. They believed that 
the acceleration due to the moon's gravity at that height would be 
ttg = 1.43 m/sec^. What direction and speed would they give the 
capsule to "inject" it into lunar orbit? 

The direction problem is fairly easy — to stay at a constant 
height above the surface, the capsule would have to be moving 
horizontally at the instant the orbit correction was completed. So 
injection would have to occur just when the capsule was moving on 
a tangent, 70 miles up, as shown in the sketch in the margin. What 
speed (relative to the moon, of course) would the capsule have to 
be given? The circular orbit has a radius 70 miles greater than the 
radius of the moon, which is 1080 miles; so R = 1080 mi + 70 mi 
= 1150 mi; this is equal to 1.85 x 10^ meters. The centripetal 
acceleration is just the acceleration caused by gravity, which was 
supposed to be 1.43 m/sec^, so 

= x/(1.85x 10«m)x 1.43 


= A/2.65mX 10« 


= 1.63x 103 


The necessary speed for an orbit at 70 miles above the surface is 
therefore 1630 m/sec (about 3600 mi/hr). Knowing the capsule's 
speed, ground control could calculate the necessary speed changes 
to reach 1630 m/sec. Knowing the thrust force of the engines and 
the mass of the capsule, they could calculate the time of thrust 
required to make this speed change. 

SG 4.22 
SG 4.23 
SG 4.24 

Q1 1 What information was necessary to calculate the speed for 
an orbit 70 miles above the moon's surface? 


Understanding Motion 

Table 4.2 Some information on selected artificial satellites. 


Sputnik 1 
1957 (USSR) 

Oct. 4, 1957 


Explorer 7 
1958 (USA) 

Jan. 31, 1958 


Lunik 3 
1959 (USSR) 

Oct. 4, 1959 


Vostok 1 
1961 (USSR) 

Apr. 12, 1961 


Midas 3 
1961 (USA) 

July 12, 1961 


Telestar 1 
1962 (USA) 

July 10, 1962 


Alouette 1 
1962 (USA- 

Sept. 29, 1962 


Luna 4 

Apr. 2, 1963 


1963-08 (USSR) 

Vostok 6 June 16, 1963 "about 5 

1963-23 (USSR) tons" 

Syncom 2 July 26, 1963 

1963-31 (USA) 


HEIGHT (miles) 

RIOD (min) 

Perigee- Apogee 













REMARKS ('"eliding 









First earth satellite. Internal 
temperature, pressure inside satellite. 

Cosmic rays, micrometeorites, 
internal and shell temperatures, 
discovery of first Van Allen belts. 

Transmitted photographs of far side of 

First manned orbital flight (Major Yuri 
Gagarin; one orbit) 

Almost circular orbit. 

Successful transmission across the 
Atlantic: telephony, phototelegraphy, 
and television. 

Joint project between NASA and 
Canadian Defense Research Board; 
measurement in ionosphere. 

Passed 5,300 miles from moon; very 
large orbit. 

First orbital flight by a woman; 
(Valentina Terishkive; 48 orbits) 

Successfully placed in near- 
Synchronous orbit (stays above same 
spot on earth). 

4.8 What about other motions? 

So far we have described straight-line motion, projectile motion, 
and uniform circular motion. In all these cases we considered only 
examples where the acceleration was constant -at least in 
magnitude if not in direction -or very nearly constant. There is 
another basic kind of motion that is equally common and important 
in physics, where the acceleration is always changing. A common 
example of this type of motion is that seen in playground swings, or 
in vibrating guitar strings. Such back and forth motion, or 
oscillation, about a center position occurs when there is a force 
always directed toward the center position. When a guitar string is 
pulled aside, for example, a force arises which tends to restore the 
string to its undisturbed center position. If it is pulled to the other 
side, a similar restoring force arises in the opposite direction. 

A very common type of such motion is one for which the 
restoring force is proportional, or nearly proportional, to how far the 
object is displaced. This is true for the guitar string, if the 
displacements are not too large; pulling the string aside 2 mm will 
produce twice the restoring force that pulling it aside 1 mm will. 
Oscillation with a restoring force proportional to the displacement 

Section 4.8 


is called simple harmonic motion. The mathematics for describing 
simple harmonic motion is relatively simple, and many phenomena, 
from pendulum motion to the vibration of atoms, have aspects that 
are very close to simple harmonic motion. Consequently, the 
analysis of simple harmonic motion is used very widely in physics. 
The Project Physics Handbook 1 describes a variety of activities you 
can do to become familiar with oscillations and their description. 

Either simply or in combination, the dynamics discussed in this 
chapter will cover most of the motions that will interest us, and is a 
good start toward understanding apparently very complicated 
motions, whether those of water ripples on a pond, a person running, 
the swaying of a tall building or bridge in the wind, a small 
particle zig-zagging through still air, an amoeba seen under a 
microscope, or a high-speed nuclear particle moving in the field of a 
magnet. The methods we have developed in this and the preceding 
chapters give us means for dealing with any kind of motion 
whatsoever, on earth or anywhere in the universe. 

When we considered the forces needed to produce motion, 
Newton's laws supplied us with the answers. Later, when we shall 
discuss other motions ranging from the elliptical motion of planets 
to the hyperbolic motion of an alpha particle passing near a nucleus, 
we shall continue to find in Newton's laws the tool for inferring the 
magnitude and direction of the forces acting in each case. 

Conversely, if we know the magnitude and direction of the 
forces acting on an object, we can determine what its change in 
motion will be. If in addition we know also the present position, 
velocity and mass of an object, we can reconstruct how it moved 
in the past, and we can predict how it will move in the future under 
these forces. Thus Newton's laws provide a comprehensive view of 
forces and motion. It is not surprising that Newtonian mechanics 
became a model for many other sciences: here seemed to be a 
method for understanding all motions, no matter how mysterious 
they previously may have appeared to be. 

SG 4.25 
SG 4.26 

118 Understanding Motion 

EPILOGUE The purpose of this Unit was to deal with the 
fundamental concepts of nnotion. We decided to start by analyzing 
particularly simple kinds of motion in the expectation that they are 
indeed the "ABC's" of physics. These ideas would allow us to turn our 
attention back to some of the more complex features of the world. To 
what extent were these expectations fulfilled? 

We did find that a relatively few basic concepts allowed us to gain 
a considerable understanding of motion. First of all, we found that 
useful descriptions of the motion of objects can be given using the 
concepts of distance, displacement, time, speed, velocity, and 
acceleration. If to these we add force and mass and the relationships 
expressed in Newton's three laws of motion, it becomes possible to 
account for observed motion in an effective way. The surprising thing is 
that these concepts of motion, which were developed in extraordinarily 
restricted circumstances, can in fact be so widely applied. For example, 
our work in the laboratory centered around the use of sliding dry ice 
pucks and steel balls rolling down inclined planes. These are not 
objects found moving around ordinarily in the everyday "natural" world. 
Even so, we found that the ideas obtained from those specialized 
experiments could lead us to an understanding of objects falling near 
the earth's surface, of projectiles, and of objects moving in circular 
paths. We started by analyzing the motion of a disk of dry ice moving 
across a smooth surface and ended up analyzing the motion of a space 
capsule as it circles the moon and descends to its surface. 

Thus, we have made substantial progress in analyzing complex 
motions. On the other hand, we cannot be satisfied that we have here 
all the intellectual tools needed to understand all of the phenomena that 
interest us. In Unit 3 we shall add to our stock of fundamental concepts 
a few additional ones, particularly those of momentum, work, and 
energy. They will help us when we turn our attention away from 
interactions involving a relatively few objects of easily discernible size, 
and to interactions involving countless numbers of submicroscopic 
objects-molecules and atoms. 

In this Unit we have dealt primarily with concepts that owe their 
greatest debts to Galileo, Newton, and their followers. If space had 
permitted, we should also have included the contributions of Ren^ 
Descartes and the Dutch scientist Christian Huyghens. The 
mathematician and philosopher, A. N. Whitehead has summarized the 
role of these four men and the significance of the concepts we have 
been dealing with in the following words: 

This subject of the formation of the three laws of motion 
and of the law of gravitation [which we shall take up in Unit 2] 
deserves critical attention. The whole development of 
thought occupied exactly two generations. It commenced 
with Galileo and ended with Newton's Principia: and 
Newton was born in the year that Galileo died. Also the 
lives of Descartes and Huyghens fall within the period 
occupied by these great terminal figures. The issue of the 
combined labours of these four men has some right to be 



considered as the greatest single intellectual success which 
mankind has achieved. (Science and the Modern World) 

The laws of motion Whitehead speaks of, the subject of this Unit, 
were important most of all because they suddenly allowed a new 
understanding of celestial motion. For at least twenty centuries man 
had been trying to reduce the complex motions of the stars, sun, moon, 
and planets to an orderly system. The genius of Galileo and Newton 
was in studying the nature of motion of objects as it occurs on earth, 
and then to assume the same laws would apply to objects in the 
heavens beyond man's reach. 

Unit 2 is an account of the immense success of this idea. We shall 
trace the line of thought, starting with the formulation of the problem of 
planetary motion by the ancient Greeks, through the work of 
Copernicus, Tycho Brahe, Kepler, and Galileo to provide a planetary 
model and the laws for planetary motion, and finally to Newton's 
magnificent synthesis of terrestrial and celestial physics in his Law of 
Universal Gravitation. 


4.1 The Project Physics learning materials 
particularly appropriate for Chapter 4 include 
the following: 

Curves of Trajectories 

Prediction of Trajectories 

Centripetal Force 

Centripetal Force on a Turntable 


Projectile Motion Demonstration 

Speed of a Stream of Water 

Photographing a Waterdrop Parabola 

Ballistic Cart Projectiles 

Motion in a Rotating Reference Frame 

Penny and Coat Hanger 

Measuring Unknown Frequencies 

Reader Articles 

Galileo's Discussion of Projectile Motion 
Newton's Laws of Dynamics 
Rigid Body 
Fun in Space 

Film Loops 

A Matter of Relative Motion 

Galilean Relativity- Ball Dropped from Mast 

of Ship 

Galilean Relativity -Object Dropped 

from Aircraft 

Galilean Relativity — Projectile Fired 


Analysis of Hurdle Race I 

Analysis of Hurdle Race II 

4.2 The thrust developed by a Saturn Apollo 
rocket is 7,370,000 newtons (approximately 
1,650,000 lbs.) and its mass is 540,000 kg. What is 
the acceleration of the vehicle relative to the 
earth's surface at lift off"? How long would it take 
for the vehicle to rise 50 meters? 

The acceleration of the vehicle increases 
greatly with time (it is 47 m/sec^ at first stage 
burnout) even though the thrust force does not 
increase appreciably. Explain why the acceleration 

4.H A hunter points his gun barrel directly at a 
monkey in a distant palm tree. Will the bullet 
follow the line of sight along the barrel? If the 
animal, startled by the flash, drops out of the 
branches at the very instant of firing, will it then 
be hit by the bullet? Explain. 

4.4 The displacement d'of an object is a vector 
giving the straightline distance from the 
beginning to the end of an actual path; ?can 
be thought of as made up of a horizontal (x) and 
a vesical (y) component of displacement; that is, 
d = x + y (added vectorially). 

In a trajectory, x, y, and the total 
displacement d can be thought of as the 
magnitudes of the sides of right triangles. So can 
Vx, v^ and the magnitude of the velocity v. 

(a) Find an expression for d in terms of jc 
and y. 


(b) Find an expression for v in terms of Vj 
and Vu 

(c) Rewrite the expression for d and i^in 
terms of v^, a„, and t. 

4.5 If you like algebra, try this general proof. 
If a body is launched with speed v at some 

angle other than 0°. it will initially have both a 
horizontal speed v^. and a vertical speed v^. The 
equation for its horizontal displacement is x = v^t, 
as before. But the equation for its vertical 
displacement has an additional term: y = Vyt + 
jayt'\ Show that the trajectory is still parabolic 
in shape. 

4.6 A lunch pail is accidently kicked off a steel 
beam on a skyscraper under construction. Suppose 
the initial horizontal speed v^ is 1.0 m/sec. Where 
is the pail (displacement), and what is its speed 
and direction (velocity) 0.5 sec after launching? 

4.7 In Galileo's drawing on page 104. the 
distances be, cd. de, etc. are equal. What is the 
relationship among the distances ho. oq. ql, and 

4.cS You are inside a van that is moving with a 
constant velocity. You drop a ball. 

(a) What would be the ball's path relative to 
the van? 

(b) Sketch its path relative to a person driving 
past the van at a high uniform speed. 

(c) Sketch its path relative to a person 
standing on the road. 

You are inside a moving van that is 
accelerating uniformly in a straight line. When 
the van is traveling at lOmph (and still 
accelerating) you drop a ball from near the roof of 
the van onto the floor. 

(d) What would be the ball's path relative to 
the van? 

(e) Sketch its path relative to a person driving 
past the van at a high uniform speed. 

(f ) Sketch its path relative to a person 
standing on the road. 

4.9 Two persons watch the same object move. 
One says it accelerates straight downward, but 
the other claims it falls along a curved path. 
Describe conditions under which each would be 
reporting correctly what he sees. 

4.10 An airplane has a gun that fires bullets 
straight ahead at the speed of 600 mph when 
tested on the ground while the plane is stationary. 



The plane takes off and flies due east at 600 mph. 
Which of the following describes what the pilot 
of the plane will see? In defending your answers, 
refer to the Galilean relativity principle: 

(a) When fired directly ahead the bullets 
move eastward at a speed of 1200 mph. 

(b) When fired in the opposite direction, the 
bullets dropped vertically downward. 

(c) If fired vertically downward, the bullets 
move eastward at 600 mph. while they 

Specify the frames of reference from which (a), 
(b), and (c) are the correct observations. 

1.11 Many commercial record turntables are 
designed to rotate at frequencies of 16 2/3 rpm 
(called transcription speed). 33 1/3 rpm (long 
playing). 45 rpm (pop singles), and 78 rpm (old 
fashioned). What is the period corresponding to 
each of these frequencies? 

4.12 Two blinkies are resting on a rotating 
turntable and are photographed in a setup as 
shown in the figure below. The outer blinky has a 
frequency of 9.4 flashes/sec and is located 15.0 
cm from the center. For the inner blinky, the 
values are 9.1 flashes/sec and 10.6 cm. 

(a) What is the period of the turntable? 

(b) What is the frequency of rotation of the 
turntable? Is this a standard phonograph 

(c) What is the speed of the turntable at the 
position of the outer blinky? 

(d) What is the speed of the turntable at the 
position of the inner blinky? 

(e) What is the speed of the turntable at the 
very center? 

(f ) What is the angular speed of each 
blinky — that is. the rate of rotation 
measured in degrees/sec? Are they equal? 

(g) What is the centripetal acceleration 
experienced by the inner blinky? 

(h) What is the centripetal acceleration 
experienced by the outer blinky? 

(i) If the turntable went faster and faster, 
which would leave the turntable first, and 

4.1.3 Passengers on the right side of the car in 
a left turn have the sensation of being "thrown 

against the door." Explain what actually happens 
to the passengers in terms of force and 

4.14 The tires of the turning car in the example 
on page 112 were being pushed sideways by the 
road with a total force of 1800 lb. Of course the 
tires would be pushing on the road with a total 
force of 1800 lb also, (a) What happens if the road 
is covered with loose sand or gravel? (b) How 
would softer (lower pressure) tires help? (c) How 
would banking the road (that is. tilting the 
surface toward the center of the curve) help? 
(Hint: consider the extreme case of banking in 
the bob-sled photo on p. 110.) 

4.15 Using a full sheet of paper, make and 
complete a table like the one below. 





Length of a 
path between 
any two points, 
as measured 
along the path. 

Straight line 
distance and 
direction from 
Detroit to 



An airplane 
flying west 
at 400 mph at 

Time rate of 
change of 



The drive shaft 
of some 
turns 600 rpm 
in low gear. 

The time it 
takes to make 
one complete 



4.16 Our sun is located at a point in our galaxy 
about 30,000 light years (1 light year= 9.46 x 
10'^ km) from the galactic center. It is thought 
to be revolving around the center at a linear 
speed of approximately 250 km/sec. 

(a) What is the sun's centripetal acceleration 
with respect to the center of the galaxy? 

(b) The sun's mass can be taken to be 1.98 
X 10*" kg; what centripetal force is 
required to keep the sun moving in a 
circular orbit about the galactic center? 

(c) Compare the centripetal force in (b) with 
that necessary to keep the earth in orbit 
about the sun. (The earth's mass is 

5.98 X 10^^ kg and its average distance 
from the sun is 1.495 x 10* km.) 

4.17 The hammer thrower in the photograph 
below is exerting a large centripetal force to keep 
the hammer moving fast in a circle, and applies 
it to the hammer through a connecting wire. The 
mass of the 16-pound hammer is 7.27 kg. (a) 
Estimate the radius of the circle and the period, 
and calculate a rough value for the amount of 
force required just to keep it moving in a circle. 
(b) What other components are there to the total 
force he exerts on the hammer? 

4.18 Contrast rectilinear motion, projectile 
motion, and uniform circular motion by 

(a) defining each 

(b) giving examples. 

(c) describing the relation between velocity 
and acceleration in each case. 


4.19 These questions are asked with reference to 
Table 4.2 on page 116. 

(a) Which satellite has the most nearly 
circular orbit? 

(b) Which satellite has the most eccentric 
orbit? How did you arrive at your answer? 

(c) Which has the longest period? 

(d) How does the position of Syncom 2 
relative to a point on earth change over 
one day? 

4.20 If the earth had no atmosphere, what would 
be the period of a satellite skimming just above 
the earth's surface? What would its speed be? 

4.21 Explain why it is impossible to have an earth 
satelhte orbit the earth in 80 minutes. Does this 
mean that it is impossible for any object to go 
around the earth in less than 80 minutes? 

4.22 What was the period of the "70 mi" Apollo 8 
lunar orbit? 

4.23 Knowing Ug near the moon's surface, and 
the orbital speed in an orbit near the moon's 
surface, we can now work an example of Part 8 
of the earth-moon trip described in Sec 4.1. The 
Apollo 8 capsule was orbiting about 100 kilometers 
above the surface. The value of a^ near the 
moon's surface is about 1.5 m/sec^. 

If the capsule's rocket engines are fired in the 
direction of its motion, it will slow down. 
Consider the situation in which the rockets fire 
long enough to reduce the capsule's horizontal 
speed to 100 m/secl 

(a) About how long will the fall to the moon's 
surface take? 

(b) About how far will it have moved 
horizontally during the fall? 

(c) About how far in advance of the landing 
target might the "braking" maneuver be 

4.24 Assume that a capsule is approaching the 
moon along the right trajectory, so that it will be 
moving tangent to the desired orbit. Given the 
speed v„ necessary for orbit and the current speed 
V, how long should the engine with thrust F fire to 
give the capsule of mass m the right speed? 

4.25 The intention of the first four chapters has 
been to describe "simple" motions and to progress 
to the description of more "complex" motions. Put 
each of the following examples under the heading 
"simplest motion." "more complex." or "very 
complex." Be prepared to say why you place any 
one example as you did and state any assumptions 
you made. 

(a) helicopter shown on p. 109 

(b) "human cannon ball" in flight 

(c) car going from 40 mph to a complete stop 

(d) tree growing 

(e) child riding a Ferris wheel 

(f ) rock dropped 3 mi. 

(g) person standing on a moving escalator 
(h) climber ascending Mt. Everest 

(i) person walking 

( j ) leaf falling from a tree 

4.26 Write a short essay on the physics involved 
in the motions shown in one of the four pictures 
on the opposite page, using the ideas on motion 
from Unit 1. 





Pp. 1-4 Fermi, Laura, Atoms in the Family, 
U. of Chicago Press, pp. 83-100 not inclusive. 
Chapter Two 

P. 6 Aristotle, De Caelo, trans. J. L. Stokes, Book 
I, Chapter 6, Oxford University Press, p. 273b. 

Pp. 44-60 Galilei, Galileo, Two New Sciences, 
trans. Crew and DeSalvio, Dover Publications, pp. 
62-243 not inclusive. 
Chapter Three 

P. 86 Newton, Sir Isaac, The Principia, Vol. I, 
Mott's translation revised by Florian Cajori, U. of 
Calif. Press, pp. 13-14. 

Pp. 86-87 Ibid., pp. XIII-XV. 

P. 88 Magie, W. P., A Source Book in Physics, 
McGraw-Hill, p. 94. 
Chapter Four 

P. 92 Newton, Sir Isaac, op. cit.. Vol. II, p. 251. 

P. Ill Childe, V. Gordon, "Rotary Motion," A 
History of Technology, Vol. I, Oxford University 
Press, p. 187. 

P. 117 Pope, Alexander, Epitaph Intended for 
Sir Isaac Newton (1732). 

P. 127 Whitehead, A. N., Science and the 
Modern World, a Mentor Book published by The 
New American Library, pp. 46-47. 

Picture Credits 


P. 4 U.S. Atomic Energy Commission. 

P. 6 (left) Mt. Wilson and Palomar Observa- 
tories: (right) Professor Erwin W. Mueller, The 
Pennsylvania State University. 

P. 7 (left) Museum of Comparative Zoology, 
Harvard University; (right) Brookhaven National 
Chapter 1 

P. 8 Yale University Art Gallery, Collection 
Societe Anonyme. 

P. 10 United Press International, LIFE 
Magazine, © Time Inc. 

P. 21 (solar flare) reproduced from Sydney 
Chapman's IGY: Year of Discovery, by courtesy of 
The University of Michigan Press; (glacier) from 
the film strip "Investigating a Glacier" © 1966, 
Encyclopaedia Britannica Educational Corpora- 
tion, Chicago; (plants) Dr. Leland Earnest, Dept. 
of Biology, Eastern Nazarene College. 

P. 26 (1) Bayerisches Nationalmuseum, Munich; 

(2) (4) George Eastman House, Rochester, N.Y.; 

(3) Bill Eppridge, LIFE MAGAZINE, © Time Inc. 

P. 27 (5) (6) (7) Dr. Harold E. Edgerton. 
Massachusetts Institute of Technology, 
Cambridge, Mass. 

P. 30 George Silk, LIFE MAGAZINE, © Time 

P. 35 George Eastman House, Rochester, N.Y. 
Chapter 2 

P. 36 Cabinet des Dessins, Louvre Museum. 

P. 38 Vatican Museum. Rome. 

P. 43 (signature) Smith Collection, Columbia 
University Libraries. 

P. 44 Houghton Library, Harvard University. 

P. 45 Courtesy of Educational Development 
Center, Newton, Mass. 

P. 53 Alinari-Art Reference Bureau. 
Chapter 3 

P. 66 A. G. Mill, © Time Inc. 

P. 72 G. Kew, © Time Inc. 

P. 74 C. T. Polumbaum, © Time Inc. 

P. 85 (balance) Collection of Historical 
Scientific Instruments, Harvard University. 

P. 87 Dr. Harold E. Edgerton, MIT. 

P. 89 National Aeronautics and Space 

P. 92 The Museum of Modem Art, New York. 

P. 95 U.S. Air Force. 
Chapter 4 

P. 99 National Aeronautics and Space 
Administration; Verne, Jules, De la terre a la lune, 
Paris, 1866. 

P. 102 from PSSC Physics, D.C. Heath & Co., 
Boston, 1965. 

P. 103 (skater) National Film Board of Canada; 
(fireworks) Stan Wayman, LIFE MAGAZINE, © 
Time Inc. 

P. 108 (carousel) Ernst Haas, Magnum Photos, 

P. 109 (helicopter) Andreas Feininger, 

P. 117 (guitar) Photo by Albert B. Gregory, Jr. 

P. 117 (runner) Associated Newspapers, 
Pictorial Parade, Inc., New York City. 
Facing p. 1 (Fermi at the blackboard) University 
of Chicago 

P. 96 Caterpillar Tractor. 

P. 110 (train) P. Stackpole, LIFE MAGAZINE, © 
Time Inc. 

P. 123 (Bouncing ball) Dr. Harold E. Edgerton. 
MIT.; (Bicyclists) Walker Art Center, Minneapolis: 
(acrobats) A. E. Clar|c, LIFE MAGAZINE, © 
Time Inc. 
(Cathedral spires) 

All photographs not credited above were made 
by the staff' of Harvard Project Physics. 


The Projects Physics Course 

Concepts of Motion 

Picture Credits 

Cover: (top left) Cartoon by Charles Gary Solin and 
reproduced by his permission only; (top right) from 
the film loop Galilean Relativity l-Ball Dropped from 
Mast of Ship. 

P. 135 Isogonic chart through the courtesy of the En- 
vironmental Sciences Services Administration, Coast 
and Geodetic Survey. 

Pp. 135, 138, 143, 165, 170, 171, 183. 187 (cartoons). 
By permission of Johnny Hart and Field Enterprises 

P. 152 Photography unlimited by Ron Church from 
Rapho Guillumette Pictures. New York. 
P.184 (water drop parabola) Courtesy of Mr. Harold M. 
Waage, Palmer Physical Laboratory, Princeton Uni- 

P. 185 (water drop parabola -train) Courtesy of Edu- 
cational Development Center, Newton. Mass. 
All photographs used with film loops courtesy of 
National Film Board of Canada. 
Photographs of laboratory equipment and of stu- 
dents using laboratory equipment were supplied with 
the cooperation of the Project Physics staff and 
Damon Corporation. 




Keeping Records 1 29 

Using the Polaroid Camera 132 

The Physics Reader 133 


1. Naked Eye Astronomy 134 

2. Regularity and Time 142 

3. Variations in Data 144 

Cliapter 1 The Language of IVIotion 


4. Measuring Uniform Motion 145 

Using the Electronic Stroboscope 151 
Making Frictionless Pucks 151 

Ciiapter 2 Free Fall— Galileo Describes IVIotion 


5. A Seventeenth-Century Experiment 153 

6. Twentieth-Century Version of Galileo's 
Experiment 157 

7. Measuring the Acceleration of Gravity a, 158 

(A) a, by Direct Fall 158 

(B) Ug from a Pendulum 159 

(C) a, with Slow-Motion Photography (FUm Loop) 

(D) a, from Falling Water Drops 161 

(E) a, with Falling Ball and Turntable 162 

(F) Ug with Strobe Photography 162 

When is Air Resistance Important? 163 
Measuring Your Reaction Time 163 
Falling Weights 163 
Extrapolation 1 63 
Fihn Loops 164 

1. Acceleration Due to Gravity-I 164 

2. Acceleration Due to Gravity-II 165 

Chapter 4 Understanding Motion 


10. Curves of Trajectories 176 

11. Prediction of Trajectories 179 

12. Centripetal Force 181 

13. Centripetal Force on a Turntable 182 

Projectile Motion Demonstration 184 
Speed of a Stream of Water 184 
Photographing a Waterdrop Parabola 184 
Ballistic Cart Projectiles 185 
Motion in a Rotating Reference Frame 185 
Penny and Coat Hanger 186 
Measuring Unknown Frequencies 186 
Film Loops 

4. A Matter of Relative Motion 187 

5. Galilean Relativity— Ball Dropped from 
Mast of Ship 188 

6. Galilean Relativity— Object Dropped 
from Aircraft 189 

7. Galilean Relativity— Projectile Fired 
Vertically 190 

8. Analysis of a Hurdle Race I 190 

9. Analysis of a Hurdle Race II 191 


Chapter 3 The Birth of Dynamics-Newton Explains Motion 


8. Newton's Second Law 166 

9. Mass and weight 169 

Checker Snapping 170 

Beaker and Hammer 1 70 

Pulls and Jerks 170 

Experiencing Newton's Second Law 

Make One of These Accelerometers 

(A) The Liquid-Surface Accelerometer-I 170 

(B) Automobile Accelerometer-I 172 

(C) Automobile Accelerometer-II 173 

(D) Damped-Pendulum Accelerometer 173 
Film Loop 

3. Vector Addition-Velocity of a Boat 174 


This Handbook is your guide to observa- 
tions, experiments, activities, and explorations, 
far and wide, in the realms of physics. 

Prepare for challenging work, fun and 
some surprises. One of the best ways to learn 
physics is by doing physics, in the laboratory 
and out. Do not rely on reading alone. Also, 
this Handbook is different from laboratory 
manuals you may have worked with before. 
Far more projects are described here than you 
alone can possibly do, so you will need to pick 
and choose. 

Although only a few of the experiments 
and activities will be assigned, do any addi- 
tional ones that interest you. Also, if an activity 
occurs to you that is not described here, dis- 
cuss with your teacher the possibility of doing 
it. Some of the most interesting science you 
will experience in this course will be the result 
of the activities which you choose to pursue 
beyond the regular assignments of the school 

This Handbook contains a section corre- 
sponding to each chapter of the Text. Usually 
each section is divided further in the following 

The Experiments contain full in- 
structions for the investigations you will 
be doing with your class. 

The Activities contain many sugges- 
tions for construction projects, demon- 
strations, and other activities you can do 
by yourself. 

The Film Loop notes give instruc- 
tions for the use of the variety of film 
loops that have been specially prepared 
for the course. 

In each section, do as many of these things 
as you can. With each, you will gain a better 
grasp of the physical principles and relation- 
ships involved. 

Introduction 129 

Keeping Records 

Your records of observations made in the lab- 
oratory or at home can be kept in many ways. 
Your teacher will show you how to write up 
your records of observations. But regardless of 
the procedure followed, the key question for 
deciding what kind of record you need is this: 
"Do I have a clear enough record so that I 
could pick up my lab notebook a few months 
from now and explain to myself or others what 
I did?" 

Here are some general rules to be followed 
in every laboratory exercise. Your records 
should be neatly written without being fussy. 
You should organize all numerical readings in 
tables, if possible, as in the sample lab write up 
on pages 130 and 131. You should always iden- 
tify the units (centimeters, kilograms, seconds, 
etc.) for each set of data you record. Also, iden- 
tify the equipment you are using, so that you 
can find it again later if you need to recheck 
your work. 

In general, it is better to record more rather 
than less data. Even details that may seem to 
have little bearing on the experiment you are 
doing — such as the temperature and whether 
it varied during the observations, and the time 
when the data were taken — may turn out to be 
information that has a bearing on your analy- 
sis of the results. 

If you have some reason to suspect that a 
particular datum may be less reliable than 

other data— perhaps you had to make the read- 
ing very hurriedly, or a line on a photograph 
was very faint— make a note of the fact. But 
don't erase a reading. When you think an entry 
in your notes is in error, draw a single line 
through it— don't scratch it out completely or 
erase it. You may find it was significant after 

There is no "wrong" result in an experi- 
ment, although results may be in considerable 
error. If your observations and measurements 
were carefully made, then your result will be 
correct. What ever happens in nature, includ- 
ing the laboratory, cannot be "wrong." It may 
have nothing to do with your investigation. Or 
it may be mixed up with so many other events 
you did not expect that your report is not use- 
ful. Therefore, you must think carefully about 
the interpretation of your results. 

Finally, the cardinal rule in a laboratory is 
to choose in favor of "getting your hands dirty" 
instead of "dry-labbing." In 380 B.C., the Greek 
scientist, Archytas, summed this up pretty 


In subjects of which one has no l<nowledge, 
one must obtain knowledge either by learning from 
someone else, or by discovering it for oneself. That 
which is learnt, therefore, comes from another and 
by outside help; that which is discovered comes by 
one's own efforts and independently. To discover 
without seeking is difficult and rare, but if one 
seeks, it is frequent and easy; if, however, one does 
not know how to seek, discovery is impossible. 



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On these two pages is shown an example of a student's lab notebook report. The table 
is used to record both observed quantities (mass, scale position) and calculated quan- 
tities (force, extension of rubber band). The graph shows at a glance how the extension 
of the rubber band changes as the force acting on it is increased. 









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Points THAT 



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Using the Polaroid Camera 

You will find the Polaroid camera is a very use- 
ful device for recording many of your labora- 
tory observations. Section 1.3 of your textbook 
shows how the camera is used to study moving 
objects. In the experiments and activities 
described in this Handbook, many suggestions 
are made for photographing moving objects, 
both with an electronic stroboscope (a rapidly 
flashing xenon light) and with a mechanical 
disk stroboscope (a slotted disk rotating in front 
of the camera lens). Thi setup of the rotating 
disk stroboscope with a Polaroid camera is 
shown below. 

Camera Cable Release 

a LaoieKeiease -^ 
,ft'** X — ^ Camera 

Rotating Disc 


Electric Motor 

Strobe Mounting 


In the opposite column is a check list of 
operations to help you use the modified Polaroid 
Land camera model 210. For other models, 
your teacher will provide instructions. Check 
list of operations for Polaroid Land camera 
model 002 

1. Make sure that there is film in the camera. 
If no white tab shows in front of the door 
marked "4" you must put in new film. 

2. Fasten camera to tripod or disk strobe base. 
If you are using the disk strobe technique, fix 
the clip-on slit in front of the lens. 

3. Check film (speed) selector. Set to suggested 
position (75 for disk strobe or blinky; 3000 for 
xenon strobe). 

4. If you are taking a "bulb" exposure, cover 
the electric eye. 

5. Check distance from lens to plane of object 
to be photographed. Adjust focus if necessary'. 
Work at the distance that gives an image just 
one-tenth the size of the object, if possible. 
This distance is about 120 cm. 

6. Look through viewer to be sure that what- 
ever part of the event you are interested in will 
be recorded. (At a distance of 120 cm the field 
of view is just under 100 cm long.) 

7. Make sure the shutter is cocked (by depress- 
ing the number 3 button). 

8. Run through the experiment a couple of 
times without taking a photograph, to accus- 
tom yourself to the timing needed to photo- 
graph the event. 

9. Take the picture: keep the cable release 
depressed only as long as necessary to record 
the event itself. Don't keep the shutter open 
longer than necessary. 

10. Pull the white tab all the way out of the 
camera. Don't block the door (marked "4" on 
the camera). 

11. Pull the large yellow tab straight out— all 
the way out of the camera. Begin timing de- 

12. Wait 10 to 15 seconds (for 3000-speed 
black-and-white film). 

13. Ten to 15 seconds after removing film from 
the camera, strip the white print from the 

14. Take measurements immediately. (The 
magnifier may be helpful.) 

15. After initial measurements have been 
taken, coat your picture with the preservative 
supplied with each pack of film. Let this dry 
thoroughly, label it on the back for identifica- 
tion and mount the picture in your (or a part- 
ner's) lab report. 

16. The negative can be used, too. Wash it 
carefully with a wet sponge, and coat with 

17. Recock the shutter so it will be set for next 

18. Always be careful when moving around 
the camera that you do not inadvertently kick 
the tripod. 

19. Always keep the electric eye covered when 
the camera is not in use. Otherwise the batter- 
ies inside the camera will run down quickly. 

Introduction 133 

The Physics Readers 

Your teacher probably will not often assign 
reading in the Project Physics Reader, but you 
are encouraged to look through it for articles 
of interest to you. In the Unit 1 Reader most 
students enjoy the chapter from Fred Hoyle's 
science fiction novel, The Black Cloud. This 
chapter, "Close Reasoning," is fictional, but 
nevertheless accurately reflects the real ex- 
citement of scientists at work on a new and 
important problem. 

Since different people have very different 
interests, nobody can tell you which articles 
you will most enjoy. Those with interests in art 
or the humanities will probably like Gyorgy 
Kepes' article "Representation of Movement." 
If you are interested in history and in the role 
science plays in historical development, you 
are urged to read the Butterfield and Willey 

The Reader provides several alternative 
treatments of mechanics which either supple- 
ment or go beyond the Unit 1 Text. Thus Sawyer 
gives a discussion of the concept of speed dif- 
ferent from that used in the Text. Clifford's 
approach is interesting because it uses geom- 
etry rather than algebra in explaining funda- 
mental ideas. For those seeking a deeper un- 
derstanding of mechanics, we particularly 
recommend the article from the Feynman 
Lectures on Physics. For articles that deal 
with applications of physics, you can turn to 
Strong on "The Dynamics of the Golf Club," 

Kirkpatrick on "Bad Physics in Athletic Mea- 
surements," and DuBridge on "Fun in Space." 
Practice the art of browsing! Don't decide 
from the titles alone whether you are inter- 
ested, but read portions of articles here and 
there, and you may well discover something 
new and interesting. 

Project Physics Reader 

An Introduction to Physics 

Concepts of Motion 

134 Experiment 1 




The purpose of this first experiment is to fa- 
miliarize you with the continually changing 
appearance of the sky. By watching the heav- 
enly bodies closely day and night over a period 
of time, you wOl begin to understand what is 
going on up there and gain the experience you 
will need in working with Unit 2, Motion in 
the Heavens. 

Do you know how the sun and the stars, 
the moon and the planets, appear to move 
through the sky? Do you know how to tell a 
planet from a star? Do you know when you can 
expect to see the moon during the day? How do 
the sun and planets move in relation to the 

The Babylonians and Egyptians knew the 
answers to these questions over 5000 years 
ago. They found them simply by watching the 
everchanging sky. Thus, astronomy began 
with simple observations of the sort you can 
make with your unaided eye. 

You know that the earth appears to be at 
rest while the sun, stars, moon, and planets are 
seen to move in various paths through the sky. 
Our problem, as it was for the Babylonians, is 
to describe what these paths are and how they 
change from day to day, from week to week, 
and from season to season. 

Some of these changes occur very slowly. 
In fact, this is why you may not have noticed 
them. You will need to watch the motions in 
the sky carefully, measuring them against 
fixed points of reference that you establish. 
You will need to keep a record of your obser- 
vations for at least four to six weeks. 

Choosing References 

To locate objects in the sky accurately, you 
first need some fixed lines or planes to which 
your measurements can be referred, just as 
a map maker uses lines of latitude and longi- 
tude to locate places on the earth. 

For example, you can establish a north- 
south line along the ground for your first refer- 

ence. Then with a protractor held horizontally, 
you can measure the position of an object in 
the sky around the horizon from this north- 
south line. The angle of an object around the 
horizon from a north-south line is called the 
object's azimuth. Azimuths are measured 
from the north point (0°) through east (90°) 
to south (180°) and west (270°) and around 
to north again (360°or 0°). 

To measure the height of an object in the 
sky, you can measure the angle between the 
object and a horizontal plane, such as the 
horizon, for your second reference. This plane 
can be used even when the true horizon is 
hidden by trees or other obstructions. The 
angle between the horizontal plane and the 
line to an object in the sky is called the altitude 
of the object. 


\ Altitude30° 







/fVJ70\^ ^~- 




Establishing References 

You can establish your north-south line in 
several different ways. The easiest is to use a 
compass to establish magnetic north but this 
may not be the same as true north. A magnetic 
compass responds to the total magnetic effect 
of all parts of the earth, and in most localities 
the compass does not point true north. The 
angle between magnetic north and true north 
is called the angle of magnetic declination. At 
some places the magnetic declination is zero, 
and the compass points toward true north. 

At places east of the line where the de- 
clination is zero, the compass points west of 
true north; at places west of the line, the com- 
pass points east of true north. You can find the 

angle of decimation and its rate of change per 
year for your area from the map below. 

At night you can use the North Star (Po- 
laris) to establish the north-south line. Polaris 
is the one fairly bright star in the sky that 
moves least from hour to hour or with the 
seasons. It is almost due north of an observer 
anywhere in the Northern Hemisphere. 

To locate Polaris, first find the "Big Dip- 
per" which on a September evening is low in 
the sky and a little west of north. (See the star 
map, Fig. 1-1 page 136.) The two stars forming 
the end of the dipper opposite the handle are 
known as the "pointers," because they point to 
the North Star. A line passing through them 
and extended upward passes very close to a 
bright star, the last star in the handle of the 
"Little Dipper." This bright star is the Pole 
Star, Polaris. On September 15 at 8:30 P.M. 
these constellations are arranged about as 
shown in the diagram below. 

Experiment 1 135 

B.C. by John Hart 




^^y^^^^ *5c: 


cti— 1 

By pennission of John Hart and Field Enterprises, Inc. 

IZO- 115' 


136 Experiment 1 

5 November 20 a 

Fig. 1-1. 

02 ^^H 

This chart of the stars will help you locate some of the bright stars and the constel- 
lations. To use the map, face north and turn the chart until today's date is at the top. 
Then move the map up nearly over your head. The stars will be in these positions at 
8 P.M. For each hour ear//er than 8 p.m., rotate the chart 15 degrees (one sector) clock- 
wise. For each hour later than 8 p.m., rotate the chart counter-clockwise. If you are 
observing the sky outdoors with the map, cover the glass of a flashlight with fairly 
transparent red paper to look at the map. This will prevent your eyes from losing their 
adaptation to the dark when you look at the map. 

Experiment 1 137 

Imagine a line from Polaris straight down 
to the horizon. The point where this line meets 
the horizon is nearly due north of you. 

Now that you have established a north- 
south line, either with a compass or from the 
North Star, note its position with respect to 
fixed landmarks, so that you can use it day 
or night. 

You can establish the second reference, 
the plane of the horizon, and measure the 
altitude of objects in the sky from the horizon, 
with an astrolabe, a simple instrument you can 
obtain easily or make yourself, very similar to 
those used by ancient viewers of the heavens. 
Use the astrolabe in your hand or on a flat 
table mounted on a tripod or on a permanent 
post. A simple hand astrolabe you can make is 
described in the Unit 2 Handbook, in the ex- 
periment dealing with the size of the earth. 

Sight along the surface of the flat table to 
be sure it is horizontal, in line with the horizon 
in all directions. If there are obstructions on 
your horizon, a carpenter's level turned in all 
directions on the table will tell you when the 
table is level. 

Turn the base of the astrolabe on the table 

until the north-south line on the base points 
along your north-south line. Or you can obtain 
the north-south line by sighting on Polaris 
through the astrolabe tube. Sight through the 
tube of the astrolabe at objects in the sky you 
wish to locate and obtain their altitude above 
the horizon in degrees from the protractor on 
the astrolabe. With some astrolabes, you can 
also obtain the azimuth of the objects from the 
base of the astrolabe. 

To follow the position of the sun with the 
astrolabe, slip a large piece of cardboard with 
a hole in the middle over the sky-pointing end 
of the tube. (Caution: Never look directly at 
the sun. It can cause permanent eye damage!) 
Standing beside the astrolabe, hold a small 
piece of white paper in the shadow of the large 
cardboard, several inches from the sighting 
end of the tube. Move the tube about until the 
bright image of the sun appears through the 
tube on the paper. Then read the altitude of 
the sun from the astrolabe, and the sun's azi- 
muth, if your instrument permits. 


Now that you know how to establish your ref- 
erences for locating objects in the sky, here are 
suggestions for observations you can make on 
the sun, the moon, the stars, and the planets. 
Choose at least one of these objects to observe. 
Record the date and time of all your observa- 


Experiment 1 

tions. Later compare notes with classmates 
who concentrated on other objects. 

A. Sun 

CAUTION: NEVER look directly at the sun; 
it can cause permanent eye damage. Do not 
depend on sun glasses or fogged photographic 
film for protection. It is safest to make sun 
observations on shadows. 

1. Observe the direction in which the sun sets. 
Always make your observation from the same 
observing position. If you don't have an un- 
obstructed view of the horizon, note where the 
sun disappears behind the buildings or trees in 
the evening. 

2. Observe the time the sun sets or disappears 
below your horizon. 

3. Try to make these observations about once 
a week. The first time, draw a simple sketch 
on the horizon and the position of the setting 

4. Repeat the observation a week later. Note 
if the position or time of sunset has changed. 
Note if they change during a month. Try to 
continue these observations for at least two 

5. If you are up at sunrise, you can record the 
time and position of the sun's rising. (Check 
the weather forecast the night before to be 
reasonably sure that the sky will be clear.) 

6. Determine how the length of the day, from 
sunrise to sunset, changes during a week; 
during a month; or for the entire year. You 
might like to check your own observations of 
the times of sunrise and sunset against the 
times as they are often reported in newspapers. 
Also if the weather does not permit you to 
observe the sun, the newspaper reports may 
help you to complete your observations. 

7. During a single day, observe the sun's azi- 
muth at various times. Keep a record — of the 
azimuth and the time of observation. Deter- 
mine whether the azimuth changes at a con- 
stant rate during the day, or whether the sun's 
apparent motion is more rapid at some times 
than at others. Find how fast the sun moves 
in degrees per hour. See if you can make a 
graph of the speed of the sun's change in azi- 

Similarly, find out how the sun's angular 
altitude changes during the day, and at what 
time its altitude is greatest. Compare a graph 
of the speed of the sun's change in altitude 
with a graph of its speed of change in azimuth. 
8. Over a period of several months — or even an 
entire year — observe the altitude of the sun at 
noon — or some other convenient hour. (Don't 
worry if you miss some observations.) Deter- 
mine the date on which the noon altitude of 
the sun is a minimum. On what date would the 
sun's altitude be a maximum? 

B.C. by John Hart 

By permission of John Hart and Field Enterprises, Inc. 

B. Moon 

1. Observe and record the altitude and azi- 
muth of the moon and draw its shape on suc- 
cessive evenings at the same hour. Carry your 
observations through at least one cycle of 
phases, or shapes, of the moon, recording in 
your data the dates of any nights that you 

For at least one night each week, make a 
sketch showing the appearance of the moon 
and another "overhead" sketch of the relative 
positions of the earth, moon, and sun. If the 
sun is below the horizon when you observe the 
moon, you will have to estimate the sun's po- 

2. Locate the moon against the background of 

Experiment 1 


the stars, and plot its position and phase on a 
sky map suppHed by your teacher. 

3. Find the full moon's maximum altitude. 
Find how this compares with the sun's maxi- 
mum altitude on the same day. Determine how 
the moon's maximum altitude varies from 
month to month. 

4. There may be a total eclipse of the moon 
this year. Consult Table 1 on page 140, or the 
Celestial Calendar and Handbook, for the 
dates of lunar eclipses. Observe one if you 
possibly can. 

C. Stars 

1. On the first evening of star observation, 
locate some bright stars that will be easy to 
find on successive nights. Later you will iden- 
tify some of these groups with constellations 
that are named on the star map in Fig. 1-1, 
which shows the constellations around the 
North Star, or on another star map furnished 
by your teacher. Record how much the stars 
have changed their positions compared to your 
horizon after an hour; after two hours. 

2. Take a time exposure photograph of several 
minutes of the night sky to show the motion 
of the stars. Try to work well away from bright 
street lights and on a moonless night. Include 
some of the horizon in the picture for refer- 
ence. Prop up your camera so it won't move 
during the time exposures of an hour or more. 
Use a small camera lens opening (large f- 
number) to reduce fogging of your film by 
stray light. 

3. Viewing at the same time each night, find 

This multiple exposure picture of the moon was taken 
with a Polaroid Land camera by Rick Pearce, a twelfth- 
grader in Wheat Ridge, Colorado. The time intervals 
between successive exposures were 15 min, 30 min, 
30 min, and 30 min. Each exposure was for 30 sec using 
2000-speed film. Which way was the moon moving in 
the sky? 

A time exposure photograph of Ursa Major (The Big 
Dipper) taken with a Polaroid Land camera on an au- 
tumn evening in Cambridge, Massachusetts. 

whether the positions of the star groups are 
constant in the sky from month to month. Find 
if any new constellations appear after one 
month; after 3 or 6 months. Over the same 
periods, find out if some constellations are no 
longer visible. Determine in what direction and 
how much the positions of the stars shift per 
week and per month. 

D. Planets and meteors 

1. The planets are located within a rather 
narrow band across the sky (called the ecliptic) 
along which the sun and the moon also move. 
For details on the location of planets, consult 
Table 1 on page 140, or the Celestial Calendar 
and Handbook, or the magazine Sky and Tele- 
scope. Identify a planet and record its position 
in the sky relative to the stars at two-week 
intervals for several months. 

2. On almost any clear, moonless night, go 
outdoors away from bright lights and scan as 
much of the sky as you can see for meteors. 
Probably you will glimpse a number of fairly 
bright streaks of meteors in an hour's time. 
Note how many meteors you see. Try to locate 
on a star map like Fig. 1-1 where you see them 
in the sky. 

Look for meteor showers each year around 
November 5 and November 16, beginning 
around midnight. Dates of other meteor show- 
ers are given in Table 2 on page 141. Remem- 
ber that bright moonlight will interfere with 
meteor observation. 

Additional sky observations you may wish 
to make are described in the Unit 2 Handbook. 


Experiment 1 



Check your local newspaper for eclipse times and extent of eclipse in your locality. 








Visible for about one 

Visible for several months 

Very bright for 





week around 


around stated time. 

one month on 

bright for sev- 

bright for two 


each side of 
given lime. 

en months be- 
yond stated 

months on 
each side of 

Mercury and Venus arc 

best viewed the hour 

Observable for 


given time. 

before dawn v 

when indicated as a.m. and the 

16 months sur. 

Visible for 13 

hour after sunset when 

indicated as p.m. 

rounding given 



mid Feb.: 


Mar. 7: total 

1 late Apr.: 


late May: 

early Dec: 

in Fla., par- 

9 early June 

: a.m. 

early Nov.: p.m. 



Feb. 21 

tial in east- 

7 mid Aug.: 


mid Dec: a.m. 



Aug. 17 

em and 

late Sept.: 





early Dec. 


U. S. 

mid Jan.: 


1 late Mar.: 


early Sept.: 

late June: 

late Dec: 

9 mid May: 





Feb. 10 

7 late July: 





1 mid Sept.: 





late Nov.: 


early Jan.: 


1 mid Mar.: 


late July: 

9 early May: 



July 10 

7 mid July: 


mid May: p.m. 


Jan. 30 

partial in 

2 late Aug.: 


early Aug.: a.m. 


July 26 

northern U.S. 

early Nov. 


mid Dec: 


late Feb.: 


1 late Apr.: 


late Nov.: 

early Sept.: 

early Jan.: 

9 late June : 


late Dec: p.m. 




Dec. 10 

7 early Aug. 





3 mid Oct.: 





early Dec. 


mid Feb.: 


1 late Mar.: 


mid Oct.: 

late Jan.: 

9 early June 


early Mar.: a.m. 



June 4 

7 mid July: 




Nov. 29 

4 late Sept.: 




early Nov. 


late Jan.: 


1 early Mar. 


early Nov.: 

early Feb.: 

9 mid May: 


mid-late July: p.m. 



May 25 

7 early July: 


early Oct.: a.m. 



Nov. 18 

5 mid Sept.: 




late Oct.: 


mid Jan.: 


1 late Feb.: 


late Jan.: 

early Dec: 

late Feb.: 

9 early May: 






7 mid June: 





6 late Aug.: 





early Oct.: 


mid Dec: 


early Feb.: 


1 early Apr.: 


early Mar.: p.m. 

9 late May: 


mid Apr.: a.m. 

Apr. 4 

7 mid Aug.: 


7 late Sept.: 


Experiment 1 







Jan. 3-5 



Apr. 19-23 



July 27-Aug. 17 


Oct. 15-25 


Nov. 14-18 



Dec. 9-14 


Rises in the east 
around 2 a.m., 
upper eastern sky 
at 5 a.m. 

Rises in the east 
around 10 p.m., 
western sky at 
5 a.m. 

Rises in the east 
at 10 p.m., 
towards the west 
at 5 a.m. 

Rises in the east 
at midnight, di- 
rectly overhead 
at 5 a.m. 

Rises in the east 
at 2 a.m., upper 
eastern sky at 
5 a.m. 

Rises in the east 
at 8 p.m., 
towards the far 
west at 5 a.m. 



Through early 
August, after 
Aug. 10 



Good 1 



Aug. 3-17 

Oct. 20 


Good 1 




July 27-Aug. 11 

Oct. 18-25 


Poor 1 



July 27-Aug. 2 
Aug. 7-17 



Good 1 




Aug. 2-17 

Oct. 15-20 

Nov. 14-16 

Good 1 



Apr. 21-23 

July 27-Aug. 9 



Poor 1 




Aug. 7-17 



Good 1 





Oct. 21-25 


Dec. 9-12 1 




July 27-Aug. 5 
Aug. 12-17 



Good 1 



Aug. 3-17 

Oct. 15-21 


Good 1 



Experiment 2 


You will often encounter regularity in your 
study of science. Many natural events occur 
regularly— that is, over and over again at equal 
time intervals. But if you had no clock, how 
would you decide how regularly an event re- 
curs? In fact, how can you decide how regular 
a clock is? 

The first part of this exercise is intended 
merely to show you the regularity of a few 
natural events. In the second part, you will try 
to measure the regularity of an event against 
a standard and to decide what is really meant 
by the word "regularity." 

Part A 

You work with a partner in this part. Find 
several recurring events that you can time in 
the laboratory. You might use such events as a 
dripping faucet, a human pulse, or the beat of 
recorded music. Choose one of these events as 
a "standard event." All the others are to be 
compared to the standard by means of the 
strip chart recorder. 

One lab partner marks each "tick" of the 
standard on one side of the strip chart recorder 
tape while the other lab partner marks each 
"tick" of the event being tested. After a long 

run has been taken, inspect the tape to see how 
the regularities of the two events compare. 
Run for about 300 ticks of the standard. For 
each 50 ticks of the standard, find on the tape 
the number of ticks of the other phenomenon, 
estimating to ^ of a tick. Record your results 
in a table something like this: 



First 50 ticks 


Second 50 ticks 


Third 50 ticks 


Fourth 50 ticks 


The test event's frequency is almost certain to 
be different from test to test. The difference 
could be a real difference in regularity, or it 
could come from your error in measuring. 
Ql If you think that the difference is larger 
than you would expect from human error, then 
which of the two events is not regular? 

Part B 

In this part of the lab, you will compare the 
regularity of some devices specifically designed 
to be regular. The standard here will be the 
time recording provided by the telephone com- 
pany or Western Union. To measure two peri- 

ods of time, you will have to make three calls 
to the time station, for example, 7 p.m., 7 a.m., 
and 7 p.m. again. Agreement should be reached 
in class the day before on who will check wall 
clocks, who will check wristwatches, and so 
on. Watch your clock and wait for the record- 
ing to announce the exact hour. Tabulate your 
results something like this: 



"7 P.M. exactly"! 
"7 A.M. exactly"^ 
"7 P.M. exactly" 

12:00:00 hr 
12:00:00 hr 



In Part I, you found that to test regularity you 
need a standard that is consistent, varying as 
little as possible. The standard is understood, 
by definition, to be regular. 
Q2 What is the standard against which the 
time station signal is compared? Call to find 
out what this standard is. Try to find the final 
standard that is used to define regularity— the 
time standard against which all other recur- 
ring events are tested. How can we be sure of 
the regularity of this standard? 


Experiment 2 143 

By Johnny Hart 






Experiment 3 


If you count the number of chairs or people in 
an ordinary sized room, you will probably get 
exactly the right answer. But if you measure 
the length of this page with a ruler, your an- 
swer will have a small margin of uncertainty. 
That is, numbers read from measuring in- 
struments do not give the exact measure- 
ments in the sense that one or two is exact 
when you count objects. Every measurement 
is to some extent uncertain. 

Moreover, if your lab partner measures the 
length of this page, he will probably get a dif- 
ferent answer from yours. Does this mean that 
the length of the page has changed? Hardly! 
Then can you possibly find the length of the 
page without any uncertainty in your measure- 
ment? This lab exercise is intended to show 
you why the answer is "no." 

Various stations have been set up around 
the room, and at each one you are to make 
some measurement. Record each measure- 
ment in a table like the one shown here. When 
you have completed the series, write your 
measurements on the board along with those 
of your classmates. Some interesting patterns 
should emerge if your measurements have not 

been influenced by anyone else. Therefore, do 
not talk about your results or how you got them 
until everyone has finished. 



Experiment 4 



Chapter I The Language of Motion 


If you roll a ball along a level floor or table, 
eventually it stops. Wasn't it slowing down all 
the time, from the moment you gave it a push? 
Can you think of any things that have uniform 
motion in which their speed remains constant 
and unchanging? Could the dry-ice disk pic- 
tured in Sec. 1-3 of the Text really be in uni- 
form motion, even if the disk is called "friction- 
less"? Would the disk just move on forever? 
Doesn't everything eventually come to a stop? 

In this experiment you check the answers 
to these questions for yourself. You observe 
very simple motion, like that pictured below, 
and make a photo record of it, or work with 
similar photos. You measure the speed of the 
object as precisely as you can and record your 
data in tables and draw graphs from these 
data. From the graphs you can decide whether 
the motion was uniform or not. 

Your decision may be harder to make than 
you would expect, since your experimental 
measurements can never be exact. There are 
likely to be ups and downs in your final results. 
Your problem will be to decide whether the ups 
and downs are due partly to real changes in 
speed or due entirely to uncertainty in your 

If the speed of your object turns out to be 
constant, does this mean that you have pro- 
duced an example of uniform motion? Do you 
think it is possible to do so? 

Doing the Experiment 

Various setups for the experiment are shown 
on pages 145 and 146. It takes two people to 
photograph a disk sliding on a table, or a glider 


Experiment 4 

Fig. 1-1. Stroboscopic photograph of a moving CO2 disk. 

on an air track, or a steadily flashing light 
(called a blinky) mounted on a small box which 
is pushed by a toy tractor. Your teacher will 
explain how to work with the set up you are 
using. Excellent photographs can be made of 
any of them. 

If you do not use a camera at all, or if you 
work alone, then you may measure a trans- 
parency or a movie film projected on the chalk 
board or a large piece of paper. 

Or you may simply work from a previously 
prepared photograph such as Fig. 1-1, above. 
If there is time, you might try several of these 

One setup uses for the moving object a disk 
made of metal or plastic. A few plastic beads 
sprinkled on a smooth, dust- free table top (or 
a sheet of glass) provide a surface for the disk 
to slide with almost no friction. Make sure the 
surface is quite level, too, so that the disk will 
not start to move once it is at rest. 

Set up the Polaroid camera and the strobo- 
scope equipment according to your teacher's 
instructions. Instructions for operating the 
Polaroid model 210, and a diagram for mount- 
ing this camera with a rotating disk strobo- 
scope is shown on page 132. A ruler need not be 
included in your photograph as in the photo- 
graph above. Instead, you can use a magnifier 
with a scale that is more accurate than a ruler 
for measuring the photograph. 

Either your teacher or a few trials will 
give you an idea of the camera settings and of 
the speed at which to launch the disk, so that 
the images of your disk are clear and well- 
spaced in the photograph. One student launch- 
es the disk while his companion operates the 
camera. A "dry run" or two without taking a 

picture will probably be needed for practice 
before you get a good picture. A good picture 
is one in which there are at least five sharp and 
clear images of your disk far enough apart for 
easy measuring on the photograph. 



"T~T"r ! ' ' ' 


Fig, 1-2. Estimating to tenths of a scale division. 

Making Measurements 

Whatever method you have used, your next 
step is to measure the spaces between succes- 
sive images of your moving object. For this, 
use a ruler with millimeter divisions and esti- 
mate the distances to the nearest tenth of a 
millimeter, as shown in Fig. 1-2 above. If 
you use a magnifier with a scale, rather than 
a ruler, you may be able to estimate these quite 
precisely. List each measurement in a table 
like Table 1. 

Since the intervals of time between one 
image and the next are equal, you can use that 
interval as a unit of time for analyzing the 
event. If the speed is constant, the distances 
of travel would turn out to be all the same, and 
the motion would be uniform. 
Ql How would you recognize motion that is 
not uniform? 

Q2 Why is it unnecessary for you to know the 
time interval in seconds? 

Experiment 4 147 










0.48 cm 






Table 1 has data that indicate uniform 
motion. Since the object traveled 0.48 cm dur- 
ing each time interval, the speed is 0.48 cm 
per unit time. 

It is more likely that your measurements 
go up and down as in Table 2, particularly if 
you measure with a ruler. 







0.48 cm 











Q3 Is the speed constant in this case? Since 
the distances are not all the same, you might 
well say, "No, it isn't." Or perhaps you looked 
again at a couple of the more extreme data in 
Table 2, such as 0.46 and 0.50 cm, checked 
these measurements, and found them doubt- 
ful. Then you might say, "The ups and downs 
are because it is difficult to measure to 0.01 cm 
with the ruler. The speed really is constant as 
nearly as I can tell." Which statement is right? 

Look carefully at the divisions or marks on 
your ruler. Can you read your ruler accurately 
to the nearest 0.01 cm? If you are like most 
people, you read it to the nearest mark of 0.1 
cm (the nearest whole millimeter) and esti- 
mate the next digit between the marks for the 
nearest tenth of a millimeter (0.01 cm), as 
illustrated in Fig. 1-2 at the left. 

In the same way, whenever you read the 
divisions of any measuring device you should 

read accurately to the nearest division or mark 
and then estimate the next digit in the mea- 
surement. Then probably your measurement, 
including your estimate of a digit between 
divisions, is not more than half of a division 
in error. It is not likely, for example, that in 
Fig. 1-2 on page 146 you would read more 
than half a millimeter away from where the 
edge being measured comes between the 
divisions. In this case, in which the divisions 
on the ruler are millimeters, you are at most no 
more than 0.5 mm (0.05 cm) in error. 

Suppose you assume that the motion really 
is uniform and that the slight differences be- 
tween distance measurements are due only to 
the uncertainty in reading the ruler. What is 
then the best estimate of the constant dis- 
tance the object traveled between flashes? 

Usually, to find the "best" value of dis- 
tance you must average the values. The aver- 
age for Table 2 is 0.48 cm, but the 8 is an un- 
certain measurement. 

If the motion recorded in Table 2 really is 
uniform, the measurement of the distance 
traveled in each time interval is 0.48 cm plus 
or minus 0.05 cm, written as 0.48±0.05 cm. 
The ±0.05 is called the uncertainty of your 
measurement. The uncertainty for a single 
measurement is commonly taken to be half a 
scale division. With many measurements, this 
uncertainty may be less, but you can use it to 
be on the safe side. 

Now you can return to the big question: Is 
the speed constant or not? Because the num- 
bers go up and down you might suppose that 
the speed is constantly changing. Notice 
though that in Table 2 the changes of data 
above and below the average value of 0.48 cm 
are always smaller than the uncertainty, 0.05 
cm. Therefore, the ups and down may all be 
due to the difficulty in reading the ruler to 
better than 0.05 cm — and the speed may, in 
fact, be constant. 

Our conclusion from the data given here 
is that the speed is constant to within the 
uncertainty of measurement, which is 0.05 
cm per unit time. If the speed goes up or down 
by less than this amount, we simply cannot 
reliably detect it with our ruler. 

148 Experiment 4 

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Experiment 4 149 


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Experiment 4 

Study your own data in the same way. 
Q4 Do they lead you to the same conclusion? 
If your data vary as in Table 2, can you think 
of anything in your setup that could have been 
making the speed actually change? Even if 
you used a magnifier with a scale, do you still 
come to the same conclusion? 

Measuring More Precisely 

A more precise measuring instrument than a 
ruler or magnifier with a scale might show 
that the speed in our example was not con- 
stant. For example, if we used a measuring 
microscope whose divisions are 0.001 cm apart 
to measure the same picture again more pre- 
cisely, we might arrive at the data in Table 3. 
Such precise measurement reduces the uncer- 
tainty greatly from ±0.05 cm to ±0.0005 cm. 







0.4826 cm 











Just as in the example on Text page 19, lay 
off time intervals along the horizontal axis of 
the graph. Your units are probably not seconds; 
they are "blinks" if you used a stroboscope or 
simply "arbitrary time units" which mean here 
the equal time intervals between positions of 
the moving object. 

Then lay off the total distances traveled 
along the vertical axis. The beginning of each 
scale is in the lower left-hand comer of the 

Choose the spacing of your scale division 
so that your data will, if possible, spread across 
most of the graph paper. 

The data of Table 2 on page 147 are plotted 
as an example on the graph of the sample 
write up of Experiment 4 on pages 148 and 149. 
Q6 In what way does the graph on page 149 
show uniform motion? Does your own graph 
show uniform motion too? 

If the motion in your experiment was not 
uniform, review Sec. 1.9of the Text. Then from 
your graph find the average speed of your 
object over the whole trip. 
Q7 Is the average speed for the whole trip 
the same as the average of the speeds between 
successive measurements? 

Q5 Is the speed constant when we measure to 
such high precision as this? 

The average of these numbers is 0.4804, 
and they are all presumably correct within 
half a division which is 0.0005 cm. Thus our 
best estimate of the true value is 0.4804 ± 
0.0005 cm. 

Drawing a Graph 

If you have read Sec. 1.5 in the Text, you have 
seen how speed data can be graphed. Your data 
provide an easy example to use in drawing a 

Additional Questions 

Q8 Could you use the same methods you used 
in this experiment to measure the speed of a 
bicycle? a car? a person running? (Assume 
they are moving uniformly.) 
Q9 The divisions on the speedometer scale 
of many cars are 5 mi/hr in size. You can 
estimate the reading to the nearest 1 mi/hr. 

(a) What is the uncertainty in a speed 
measurement by such a speedometer? 

(b) Could you measure reliably speed 
changes as small as 2 mi/hr? 1 mi/hr? 
0.5 mi/hr? 0.3 mi/hr? 



Examine some moving objects illuminated by 
an electronic stroboscope. Put a piece of tape 
on a fan blade or mark it with chalk and watch 
the pattern as you turn the fan on and off. 
How can you tell when there is exactly one 
flash of light for each rotation of the fan blade? 
Observe a stream of water from a faucet, 
objects tossed into the air, or the needle of a 
running sewing machine. If you can darken 
the room completely, try catching a thrown 
ball lighted only by a stroboscope. How many 
flashes do you need during the flight of the ball 
to be able to catch it reliably? 

Method 1. Use a flat piece of dry ice on a very 
smooth surface, like glass or Formica. When 
you push the piece of dry ice (frozen carbon 
dioxide), it moves in a frictionless manner 
because as the carbon dioxide changes to a 
vapor it creates a layer of CO2 gas between 
the solid and the glass. (CAUTION: Don't 
touch dry ice with your bare hands; it can 
give you a severe frost bite!) 

Method 2. Make a balloon puck if your lab does 
not have a supply. First cut a 4-inch diameter 
disk of 1-inch-thick Masonite. Drill a y" diam- 
eter hole part way through the center of the 
disk so it will hold a rubber stopper. Then 
drill a ■^" diameter hole on the same center 
the rest of the way through the disk. Drill a 
iV" hole through the center of a stopper in the 
hole in the masonite disk. Place the disk on 
glass or Formica. 

Method 3. Make a pressure pump puck. Make 
a disk as described in Method 2. Instead of 
using a balloon, attach a piece of flexible tub- 
ing, attached at the other end to the exhaust 
of a vacuum pump as shown in the diagram. 
Run the tubing over an overhead support so 

*ilth ViG* hole 

Masonite I* thicJc 

it does not interfere with the motion of the 

Method 4. Drill a ■^" hole in the bottom of a 
smooth-bottomed cylindrical can, such as one 
for a typewriter ribbon. Break up dry ice 
(DON'T touch it with bare hands) and place 
the pieces inside the can. Seal the can with 
tape, and place it on a very smooth surface. 

152 Experiments 

Chapter £m Free Fall— Galileo Describes Motion 

Accelerated motion goes on all around you 
every day. You experience many accelerations 
yourself, although not always as exciting as 
those shown in the photographs. What accel- 
erations have you experienced today? 

When you get up from a chair, or start to 

walk from a standstill, hundreds of sensations 
are gathered from all over your body in your 
brain, and you are aware of these normal ac- 
celerations. Taking off in a jet or riding on an 
express elevator, you experience much sharper 
accelerations. Often this feeling is in the pit 
of your stomach. These are very complex 

Note how stripped down and simple the 
accelerations are in the following experiments, 
film loops, activities. As you do these, you will 
learn to measure accelerations in a variety of 
ways, both old and new, and become more 
familiar with the fundamentals of accelera- 

If you do either of the first two experiments 
of this chapter, that is, numbers 5 and 6. you 
will try to find, as Galileo did, whether dlt' 
is a constant for motion down an inclined 
plane. The remaining experiments are mea- 
surements of the value of the acceleration due 
to gravity which is represented by the sym- 
bol a„. 

Experiment 5 



This experiment is similar to the one discussed 
by Galileo in the Two New Sciences. It will 
give you firsthand experience in working with 
tools similar to those of a seventeenth-century 
scientist. You will make quantitative measure- 
ments of the motion of a ball rolling down an 
incline, as described by Galileo. 

From these measurements you should be 
able to decide for yourself whether Galileo's 
definition of acceleration was appropriate 
or not. Then you should be able to tell whether 
it was Aristotle or Galileo who was correct 
about his thinking concerning the acceleration 
of objects of different sizes. 

Reasoning Behind the Experiment 

You have read in Sec. 2.6 of the Text how Gali- 
leo expressed his belief that the speed of free- 
falling objects increases in proportion to the 
time of fall— in other words, that they accel- 
erate uniformly. But since free fall was much 
too rapid to measure, he assumed that the 
speed of a ball rolling down an incline in- 
creased in the same way as an object in free 
fall did, only more slowly. 

But even a ball rolling down a low incline 
still moved too fast to measure the speed for 
different parts of the descent accurately. So 
Galileo worked out the relationship d oo t^ (or 
dlt^ = constant), an expression in which speed 
differences have been replaced by the total 
time t and total distance d rolled by the ball. 
Both these quantities can be measured. 

Be sure to study Text Sec. 2.7 in which the 
derivation of this relationship is described. 
If Galileo's original assumptions were true, 
this relationship would hold for both freely 
falling objects and rolling balls. Since total 
distance and total time are not difficult to mea- 
sure, seventeenth-century scientists now had 
a secondary hypothesis they could test by 
experiment. And so have you. Sec. 2.8 of the 
Text discusses much of this. 


The apparatus that you will use is shown in 
Fig. 2-1 below. It is similar to that described 
by Galileo. 

You will let a ball roll various distances 
down a channel about six feet long and time 
the motion with a water clock. 

You use a water clock to time this experi- 

Water clock, operat-ed by 
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Fig. 2-1 

154 Experiments 

Experiment 5 155 

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Experiment 5 

ment because that was the best timing device 
available in Galileo's time. The way your own 
clock works is very simple. Since the volume 
of water is proportional to the time of flow, you 
can measure time in milliliters of water. Start 
and stop the flow with your fingers over the 
upper end of the tube inside the funnel. When- 
ever you refill the clock, let a little water run 
through the tube to clear out the bubbles. 

Compare your water clock with a stop 
watch when the clock is full and when it is 
nearly empty to determine how accurate it is. 
Ql Does the clock's timing change? If so, by 
how much? 

It is almost impossible to release the ball 
with your fingers without giving it a slight 
push or pufl. Therefore, dam the ball up, with 
a ruler or pencil, and release it by quickly 
moving this dam away from it down the in- 
clined plane. The end of the run is best marked 
by the sound of the ball hitting the stopping 

Brief Comment on Recording Data 

A good example of a way to record your data 
appears on page 154. We should emphasize 
again the need for neat, orderly work. Orderly 
work looks better and is more pleasing to you 
and everyone else. It may also save you from 
extra work and confusion. If you have an or- 
ganized table of data, you can easily record 
and find your data. This will leave you free to 
think about your experiment or calculations 
rather than having to worry about which of 
two numbers on a scrap of paper is the one you 
want, or whether you made a certain measure- 
ment or not. A few minutes' preparation before 
you start work will often save you an hour or 
two of checking in books and with friends. 

Operating Suggestions 

You should measure times of descent for 
several diff"erent distances, keeping the in- 
clination of the plane constant and using 
the same bah. Repeat each descent about four 
times, and average your results. Best results 
are found for very small angles of inclination 
(the top of the channel raised less than 30 cm). 
At greater inclinations, the ball tends to slide 
as well as to roll. 

From Data to Calculations 

Galileo's definition of uniform acceleration 
(Text, page 49) was "equal increases in speed 
in equal times." Galileo showed that if an 
object actually moved in this way, the total 
distance of travel should be directly propor- 
tional to the square of the total time of fall, 
or d » t^. 

Q2 Show how this follows from Galileo's defi- 
nition. (See Sec. 2.7 in the Text if you cannot 
answer this.) 

If two quantities are proportional, a graph 
of one plotted against the other will be a 
straight line. Thus, making a graph is a good 
way to check whether two quantities are pro- 
portional. Make a graph of d plotted against 

Q3 Does your graph support the hypothesis? 
How accurate is the water clock you have been 
using to time this experiment? 

If you have not already done so, check your 
water clock against a stopwatch or, better 
yet, repeat several trials of your experiment 
using a stopwatch for timing. 
Q4 How many seconds is one milliliter of time 
for your water clock? Can the inaccuracy of 
your water clock explain the conclusion you 
arrived at in Q2 above? 

Going Further 

1. In Sec. 2.7 of the Text you learned that 
a = 2dlt'\ Use this relation to calculate the 
actual acceleration of the ball in one of your 

2. If you have time, go on to see whether Gali- 
leo or Aristotle was right about the accelera- 
tion of objects of various sizes. Measure dlt'^ 
for several diff"erent sizes of balls, all rolling 
the same distance down a plane of the same 

Q5 Does the acceleration depend on the size 
of the ball? In what way does your answer 
refute or support Aristotle's ideas on falling 

Q6 Galileo claimed his results were accurate 
to Jo of a pulse beat. Do you believe his results 
were that accurate? Did you do that well? How 
could you improve the design of the water 
clock to increase its accuracy? 

Experiment 6 157 


Galileo's seventeenth-century experiment had 
its limitations, as you read in the Text, Sec. 2.9. 
The measurement of time with a water clock 
was imprecise and the extrapolation from ac- 
celeration at a small angle of inclination to 
that at a verticle angle (90°) was extreme. 

With more modern equipment you can 
verify Galileo's conclusions; further, you can 
get an actual value for acceleration in free 
fall (near the earth's surface). But remember 
that the idea behind the improved experiment 
is still Galileo's. More precise measurements 
do not always lead to more significant con- 

Determine Ug as carefully as you can. This 
is a fundamental measured value in modern 
science. It is used in many ways — from the 
determination of the shape of the earth and 
the location of oil fields deep in the earth's 
crust to the calculation of the orbits of earth 
satellites and spacecrafts in today's impor- 
tant space research programs. 

Apparatus and Procedure 

For an inclined plane use the air track. For 
timing the air track glider use a stopwatch 
instead of the water clock. Otherwise the pro- 
cedure is the same as that used in Experiment 
5. As you go to higher inclinations you should 
stop the glider by hand before it is damaged 
by hitting the stopping block. 

Instead of a stopwatch, you may wish to 
use the Polaroid camera to make a strobe 
photo of the glider as it descends. A piece of 
white tape on the glider will show up well in 
the photograph. Or you can attach a small 
light source to the glider. You can use a mag- 
nifier with a scale attached to measure the 
glider's motion recorded on the photograph. 

Here the values of d will be millimeters on 
the photograph and t will be measured in an 
arbitrary unit, the "blink" of the stroboscope, 
or the "slot" of the strobe disk. 

Plot your data as before on a graph of d vs. 
t^. Compare your plotted lines with graphs 
of the preceding cruder seventeenth-century 
experiment, if they are available. Explain 
the differences between them. 
Ql Is d/t^ constant for an air track glider? 
Q2 What is the significance of your answer to 
the question above? 

As further challenge, if time permits, try 
to predict the value of Ug, which the glider 
approaches as the air track becomes vertical. 
To do this, of course, you must express d and 
t in famOiar units such as meters or feet, and 
seconds. The accepted value of Ug is 9.8 m/sec^ 
or 32 ft/sec^ near the earth's surface. 
Q3 What is the percentage error in your cal- 
culated value? That is, what percent is your 
error of the accepted value? 
Percentage error 

accepted value — calculated value 
= X 100 

accepted value 
so that if your value of Ug is 30 ft/sec^ your 
percentage error 

32 ft/sec^ - 30 ft/sec^ 

32 ft/sec^ 
= -^x 100 =6% 

X 100 

Notice that you cannot carry this 6% out 
to 6.25% because you only know the 2 in the 


fraction 32" to one digit. Hence, you can only 
know one digit in the answer, 6%. A calculated 
value like this is said to have one significant 
digit. You cannot know the second digit in the 
answer untO you know the digit following the 
2. To be significant, this digit would require 
a third digit in the calculated values of 30 
and 32. 
Q4 What are some of the sources of your error? 


Experiment 7 


Aristotle's idea that falling bodies on earth 
are seeking out their natural places sounds 
strange to us today. After all, we know the 
answer: It's gravity that makes things fall. 

But just what is gravity? Newton tried to 
give operational meaning to the idea of gravity 
by seeking out the laws according to which 
it acts. Bodies near the earth fall toward it 
with a certain acceleration due to the gravita- 
tional "attraction" of the earth. But how can 
the earth make a body at a distance fall toward 
it? How is the gravitational force transmitted? 
Has the acceleration due to gravity always 
remained the same? These and many other 
questions about gravity have yet to be an- 
swered satisfactorily. 

Whether you do one or several parts of 
this experiment, you will become more famil- 
iar with the effects of gravity— you find the 
acceleration of bodies in free fall yourself— 
and you will learn more about gravity in later 

Part A: a^ by Direct Fall* 

In this experiment you measure the accel- 
eration of a falling object. Since the distance 
and hence the speed of fall is too small for air 
resistance to become important, and since 
other sources of friction are very small, the 
acceleration of the falling weight is very 
nearly Ug. 

Doing the Experiment 

The falling object is an ordinary laboratory 
hooked weight of at least 200 g mass. (The 
drag on the paper strip has too great an effect 
on the fall of lighter weights.) The weight 
is suspended from about a meter of paper 
tape as shown in the photograph. Reinforce 
the tape by doubling a strip of masking tape 
over one end and punch a hole in the rein- 
forcement one centimeter from the end. With 
careful handling, this can support at least 
a kilogram weight. 

♦Adapted from R. F. Brinckerhoff and D. S. Taft, Modern 
Laboratory Experiments in Physics, by permission of 
Science Electronics, Inc., Nashua, New Hampshire. 

When the suspended weight is allowed to 
fall, a vibrating tuning fork will mark equal 
time intervals on the tape pulled down after 
the weight. 

The tuning fork must have a frequency 
between about 100 vibrations/sec and about 
400 vibrations/sec. In order to mark the tape, 
the fork must have a tiny felt cone (cut from a 
marking pen tip) glued to the side of one of its 
prongs close to the end. Such a small mass 
affects the fork frequency by much less than 
1 vibration/sec. Saturate this felt tip with a 
drop or two of marking pen ink, set the fork 
in vibration, and hold the tip very gently 
against the tape. The falling tape is most con- 
veniently guided in its fall by two thumbtacks 
in the edge of the table. The easiest procedure 
is to have an assistant hold the weighted tape 
straight up until you have touched the vi- 
brating tip against it and said "Go." After a 
few practice runs, you will become expert 
enough to mark several feet of tape with a 
wavy line as the tape is accelerated past the 
stationary vibrating fork. 

Instead of using the inked cone, you may 
press a corner of the vibrating tuning fork 

gently against a 1-inch square of carbon paper 
which the thumbtacks hold ink surface in- 
wards over the falling tape. With some prac- 
tice, this method can be made to yield a series 
of dots on the tape without seriously retarding 
its fall. 

Analyzing Your Tapes 

Label with an A one of the first wave crests 
(or dots) that is clearly formed near the begin- 
ning of the pattern. Count 10 intervals be- 
tween wave crests (or dots), and mark the end 
of the tenth space with a B. Continue marking 
every tenth crest with a letter throughout 
the length of the record, which ought to be at 
least 40 waves long. 

At A, the tape already had a speed of v^. 
From this point to B, the tape moved in a time 
t, a distance we shall call dj. The distance 
di is described by the equation of free fall: 

di = -yot + 2 

In covering the distance from A to C, the tape 
took a time exactly twice as long, 2t, and fell 
a distance d^ described (on substituting 2t for 
t and simplifying) by the equation: 

dz = 2vot + 


In the same way the distances AB, AE, etc., are 
described by the equations: 

da — 3Vot H — 

d^ = 4vot + 


and so on. 

All of these distances are measured from 
A, the arbitrary starting point. To find the dis- 
tances fallen in each 10-crest interval, you 
must subtract each equation from the one 
before it, getting: 





= Vot + 



^Vot + 



= v.t. + 



AB = Vot + 


From these equations you can see that the 
weight falls farther during each time interval. 
Moreover, when you subtract each of these 
distances, AB, BC, CD, . . . from the subsequent 
distance, you find that the increase in dis- 
tance fallen is a constant. That is, each dif- 
ference BC - AB = CD - BC = DE - CD = agt\ 
This quantity is the increase in the distance 
fallen in each successive 10-wave interval 
and hence is an acceleration. Our formula 
shows that a body falls with a constant ac- 

From your measurements of AB, AC, AD, 
etc., make a column of AB, BC, CD, ED, etc., 
and in the next column record the resulting 
values of Ugt^. The values of Ugt^ should aU be 
equal (within the accuracy of your measure- 
ments). Why? Make all your measurements as 
precisely as you can with the equipment you 
are using. 

Find the average of all your values of 
Ugt^, the acceleration in centimeters/( 10-wave 
interval) ^ You want to find the acceleration 
in cm/sec^ If you call the frequency of the 
tuning fork n per second, then the length of 
the time interval t is 10/n seconds. Replacing 
t of 10 waves by 10/n seconds gives you the 
acceleration, Ug in cm/sec^. 

The ideal value of Ug is close to 9.8 m/sec^ 
but a small force of friction impeding a falling 
object is sufficient to reduce the observed value 
by several percent. 

Ql What errors would be introduced by using 
a tuning fork whose vibrations are slower than 
about 100 vibrations per second? higher than 
about 400 vibrations per second? 

Part B: a^, from a Pendulum 

You can easily measure the acceleration due to 
gravity by timing the swinging of a pendulum. 


Experiment 7 

Of course the pendulum is not falling straight 
down, but the time it takes for a round-trip 
swing still depends on Ug. The time T it takes 
for a round-trip swing is 

T = 27rxff 

In this formula I is the length of the pendulum. 
If you measure I with a ruler and T with a 
clock, you should be able to solve for a^. 

You may learn in a later physics course 
how to derive the formula. Scientists often use 
formulas they have not derived themselves, 
as long as they are confident of their validity. 

Making the Measurements 

The formula is derived for a pendulum with 
all the mass concentrated in the weight at the 
bottom, called the bob. Hence the best pen- 
dulum to use is one whose bob is a metal 
sphere hung by a fine thread. In this case you 
can be sure that almost all the mass is in the 
bob. The pendulum's length, I, is the distance 
from the point of suspension to the center of 
the bob. 

Your suspension thread can have any con- 
venient length. Measure / as accurately as 
possible, either in feet or meters. 

Set the pendulum swinging with small 
swings. The formula doesn't work well for 
large swings, as you can test for yourself later. 

Time at least 20 complete round trips, 
preferably more. By timing many round trips 
instead of just one you make the error in start- 
ing and stopping the clock a smaller fraction 
of the total time being measured. (When you 
divide by 20 to get the time for a single round 
trip, the error in the calculated value for one 
will be only jo as large as if you had measured 
only one.) 

Divide the total time by the number of 
swings to find the time T of one swing. 

Repeat the measurement at least once as 
a check. 

Finally, substitute your measured quan- 
tities into the formula and solve it for a^. 

If you measured I in meters, the accepted 
value of ttg is 9.80 meters/sec^. 

If you measured I in feet, the accepted 
value of ttg is 32.1 ft/sec^. 

Finding Errors 

You probably did not get the accepted value. 
Find your percentage error by dividing your 
error by the accepted value and multiplying by 

Percentage error 

_ accepted value — your value 

accepted value 

X 100 

your error 
accepted value 

X 100 

With care, your value of Op should agree within 
about 1%. 

Which of your measurements do you think 
was the least accurate? 

If you believe it was your measurement of 
length and you think you might be off by as 
much as 0.5 cm. change your value of / by 0.5 
cm and calculate once more the value of ag. 
Has Ug changed enough to account for your 
error? (If a^ went up and your value of a, was 
already too high, then you should have altered 
your measured I in the opposite direction. Try 

If your possible error in measuring is not 
enough to explain your difference in Oy try 
changing your total time by a few tenths of a 
second— a possible error in timing. Then you 
must recalculate T and hence a,. 

If neither of these attempts work (nor 
both taken together in the appropriate direc- 
tion) then you almost certainly have made an 
error in arithmetic or in reading your measur- 
ing instruments. It is most unlikely that a„ in 
your school differs from the accepted value by 
more than one unit in the third digit. 
Q2 How does the length of the pendulum af- 
fect your value of T? of ag? 

Q3 How long is a pendulum for which T = 2 
seconds? This is a useful timekeeper. 

Part C: a,, with Slow-Motion 
Photography (Film Loop) 

With a high speed movie camera you could 
photograph an object falling along the edge of 
a vertical measuring stick. Then you could 

Experiment 7 


determine a^ by projecting the film at standard 
speed and measuring the time for the object 
to fall specified distance intervals. 

A somewhat similar method is used in 
Film Loops 1 and 2. Detailed directions are 
given for their use in the Film Loop notes on 
pages 164-165. 

Part D: a^ from Falling Water 

You can measure the acceleration due to grav- 
ity Ug simply with drops of water falling on a 
pie plate. 

Put the pie plate or a metal dish or tray on 
the floor and set up a glass tube with a stop- 
cock, valve, or spigot so that drops of water 
from the valve will fall at least a meter to the 
plate. Support the plate on three or four pen- 
cils to hear each drop distinctly, like a drum 

Adjust the valve carefully until one drop 

strikes the plate at the same instant the next 
drop from the valve begins to fall. You can do 
this most easily by watching the drops on the 
valve while listening for the drops hitting the 
plate. When you have exactly set the valve, 
the time it takes a drop to fall to the plate is 
equal to the time interval between one drop 
and the next. 

With the drip rate adjusted, now find the 
time interval t between drops. For greater 
accuracy, you m.ay want to count the number 
of drops that fall in half a minute or a minute, 
or to time the number of seconds for 50 to 100 
drops to fall. 

Your results are likely to be more accurate 
if you run a number of trials, adjusting drip 
rate each time, and average your counts of 
drops or seconds. The average of several trials 
should be closer to actual drip rate, drop count, 
and time intervals than one trial would be. 

Now you have all the data you need. You 
know the time t it takes a drop to fall a dis- 
tance d from rest. From these you can calcu- 
late Ug, since you know that d = 2^gt'^ for ob- 
jects falling from rest. Rewrite this relationship 
in the form ag=. . . 

Q4 When you have calculated Ug by this meth- 
od, what is your percentage error? How does 
this compare with your percentage error by any 
other methods you have used? What do you 
think led to your error? Could it be leaking 
connections, allowing more water to escape 
sometimes? How would this affect your an- 

Distance of fall lessened by a puddle form- 
ing in the plate: How would this change your 

Less pressure of water in the tube after a 
period of dripping: Would this increase or 
decrease the rate of dripping? Do you get the 
same counts when you refill the tube after 
each trial? 

Would the starting and stopping of your 
counting against the watch or clock affect your 
answer? What other things may have shown 
up in your error? 

Can you adapt this method of measuring 
the acceleration of gravity so that you can do it 
at home? Would it work in the kitchen sink? 

162 Experiment 7 

or if the water fell a greater distance, such as 
down a stairwell? 

Part E: a^ with Falling Ball and 

You can measure a^ with a record-player turn- 
table, a ring stand and clamp, carbon paper, 
two balls with holes in them, and thin thread. 

Ball X and ball Y are draped across the 
prongs of the clamp. Line up the balls along a 
radius of the turntable, and make the lower 
ball hang just above the paper. 

With the table turning, the thread is burned 
and each ball, as it hits the carbon paper, will 
leave a mark on the paper under it. 

Measure the vertical distance between the 
balls and the angular distance between the 
marks. With these measurements and the 
speed of the turntable, determine the free- 
fall time. Calculate your percentage error and 
suggest its probable source. 

White l=^per 


can be graphed and analyzed to give an aver- 
age value of ttg. The 12-slot strobe disk gives 
a very accurate 60 slots per second. (Or, a 
neon bulb can be connected to the ac line out- 
let in such a way that it will flash a precise 60 
times per second, as determined by the line 
frequency. Your teacher has a description of 
the approximate circuit for doing this.) 

Part F: a,, with Strobe Photography 

Photographing a falling light source with the 
Polaroid Land camera provides a record that 



By taking strobe photos of various falling ob- 
jects, you can find when air resistance begins 
to play an important role. You can find the 
actual value of the terminal speed for less 
dense objects such as a Ping-Pong or styro- 
foam ball by dropping them from greater and 
greater heights until the measured speeds do 
not change with further increases in height. 
(A Ping-Pong ball achieves terminal speed 
within 2 m.) Similarly, ball bearings and mar- 
bles can be dropped in containers of liquid 
shampoo or cooking oil to determine factors 
affecting terminal speed in a liquid as shown 
in the adjoining photograph. 


A magnet is a 
handy aid in rais- 
ing the steel ball 
to the top of the 


Your knowledge of physics can help you cal- 
culate your reaction time. Have someone hold 
the top of a wooden ruler while you space your 
thumb and forefinger around the bottom (zero) 
end of the ruler. As soon as the other person 
releases the ruler, you catch it. You can com- 
pute your reaction time from the relation 

d = Ugt^ 
by solving for t. Compare your reaction time 
with that of other people, both older and young- 
er than yourself. Also try it under different con- 

ditions—lighting, state of fatigue, distracting 
noise, etc. Time can be saved by computing d 
for Jo sec or shorter intervals, and then taping 
reaction-time marks on the ruler. 

A challenge is to try this with a one-dollar 
bill, telling the other person that he can have 
the dollar if he can catch it. 


This demonstration shows that the time it 
takes a body to fall is proportional to the square 
root of the vertical distance (d °^ t^). Suspend 
a string, down a stairwell or out of a window, 
on which metal weights are attached at the 
following heights above the ground: 3", 1', 
2'3", 4', 6'3", 9', 12'3", 16'. Place a metal tray 
or ashcan cover under the string and then drop 
or cut the string at the point of suspension. The 
weights will strike the tray at equal intervals 
of time— about ,8 second. 

Compare this result with that obtained 
using a string on which the weights are sus- 
pended at equal distance intervals. 


Many arguments regarding private and public 
policies depend on how people choose to ex- 
trapolate from data they have gathered. From 
magazines, make a report on the problems of 
extrapolating in various cases. For example: 

1. The population explosion 

2. The number of students in your high 
school ten years from now 

3. The number of people who will die in 
traffic accidents over next holiday 

4. The number of lung cancer cases that 
will occur next year among cigarette 

5. How many gallons of punch you 
should order for your school's Junior 

To become more proficient in making statis- 
tics support your pet theory— and more cautious 
about common mistakes— read How to Lie 
with Statistics by Darrell Huff, published by 
W. W. Norton and Company. 



A bowling ball in free fall was filmed in real 
time and in slow motion. Using the slow- 
motion sequence, you can measure the ac- 
celeration of the ball due to gravity. This film 
was exposed at 3900 frames/sec and is pro- 
jected at about 18 frames/sec; therefore, the 
slow-motion factor is 3900/18, or about 217. 
However, your projector may not run at ex- 
actly 18 frames/sec. To calibrate your pro- 
jector, time the length of the entire film which 
contains 3331 frames. (Use the yellow circle 
as the zero frame.) 

To find the acceleration of the falling body 
using the definition 

acceleration = 

change in speed 
time interval 

you need to know the instantaneous speed at 
two different times. You cannot directly mea- 
sure instantaneous speed from the film, but 
you can determine the average speed during 
small intervals. Suppose the speed increases 
steadily, as it does for freely falling bodies. 
During the first half of any time interval, the 
instantaneous speed is less than the average 
speed; during the second half of the interval, 
the speed is greater than average. Therefore, 
for uniformly accelerated motion, the average 
speed Vf,,. for the interval is the same as the 
instantaneous speed at the mid-time of the 

If you find the instantaneous speed at the 
midtimes of each of two intervals, you can 
calculate the acceleration a from 

a = 


tj ti 
where v, and v.2 are the average speeds during 

the two intervals, and where t, and ^2 are the 
midtimes of these intervals. 

Two intervals 0.5 meter in length are 
shown in the film. The ball falls 1 meter be- 
fore reaching the first marked interval, so it 
has some initial speed when it crosses the first 
line. Using a watch with a sweep second hand, 
time the ball's motion and record the times at 
which the ball crosses each of the four lines. 
You can make measurements using either the 
bottom edge of the ball or the top edge. With 
this information, you can determine the time 
(in apparent seconds) between the midtimes 
of the two intervals and the time required for 
the ball to move through each j-meter inter- 
val. Repeat these measurements at least once 
and then find the average times. Use the slow- 
motion factor to convert these times to real 
seconds; then, calculate the two values of 
Va,,. Finally, calculate the acceleration a. 

This film was made in Montreal, Canada, 
where the acceleration due to gravity, rounded 
off to ± 1%, is 9.8 m/sec^. Try to decide from 
the internal consistency of your data (the re- 
peatability of your time measurements) how 
precisely you should write your result. 

Film Loops 165 


A bowling ball in free fall was filmed in slow 
motion. The film was exposed at 3415 frames/ 
sec and it is projected at about 18 frames/ 
sec. You can calibrate your projector by timing 
the length of the entire film, 3753 frames. 
(Use the yellow circle as a reference mark.) 

If the ball starts from rest and steadily 
acquires a speed v after falling through a 
distance d, the change in speed A t; is x; — 0, or 

V, and the average speed is Vgv = 

O + i; 

= -9V. The 

time required to fall this distance is given by 

The acceleration a is given by 

a = 

change of speed _ At; 
time interval At 



Thus, if you know the instantaneous speed v 
of the falling body at a distance d below the 
starting point, you can find the acceleration. 
Of course you cannot directly measure the 

instantaneous speed but only average speed 
over the interval. For a small interval, how- 
ever, you can make the approximation that 
the average speed is the instantaneous speed 
at the midpoint of the interval. (The average 
speed is the instantaneous speed at the mid- 
time, not the midpoint; but the error is small 
if you use a short enough interval.) 

In the film, small intervals of 20 cm are 
centered on positions Im, 2m, 3m, and 4m 
below the starting point. Determine four aver- 
age speeds by timing the ball's motion across 
the 20 cm intervals. Repeat the measurements 
several times and average out errors of mea- 
surement. Convert your measured times into 
real times using the slow-motion factor. Com- 
pute the speeds, in m/sec, and then compute 
the value of v^l2d for each value of d. 

Make a table of calculated values of a, in 
order of increasing values of d. Is there any 
evidence for a systematic trend in the values? 
Would you expect any such trend? State the 
results by giving an average value of the 
acceleration and an estimate of the possible 
error. This error estimate is a matter of judg- 
ment based on the consistency of your four 
measured values of the acceleration. 

B.C. by John Hart 

By permission of John Hart and Field Enterprises, Inc. 

166 Experiments 

Chapter O 

The Birth of Dynamics — Newton Explains Motion 


Newton's second law of motion is one of the 
most important and useful laws of physics. 
Review Text Sec. 3.7 on Newton's second law 
to make sure you are familiar with it. 

Newton's second law is part of a much 
larger body of theory than can be proved by any 
simple set of laboratory experiments. Our 
experiment on the second law has two pur- 

First, because the law is so important, it 
is useful to get a feeling for the behavior of 
objects in terms of force (F), mass (m), and 
acceleration (a). You do this in the first part 
of the experiment. 

Second, the experiment permits you to 
consider the uncertainties of your measure- 
ments. This is the purpose of the latter part of 
the experiment. 

You will apply different forces to carts of 
different masses and measure the accelera- 

Fig. 3-1 

How the Apparatus Works 

You are about to find the mass of a loaded cart 
on which you then exert a measurable force. 
From Newton's second law you can predict 
the resulting acceleration of the loaded cart. 

Arrange the apparatus as shown in Fig. 
3-1. A spring scale is firmly taped to a dynam- 
ics cart. The cart, carrying a blinky, is pulled 
along by a cord attached to the hook of the 
spring scale. The scale therefore measures 
the force exerted on the cart. 

The cord runs over a pulley at the edge of 
the lab table and from its end hangs a weight. 

Fig. 3-2 

Experiment 8 167 

(Fig. 3-2.) The hanging weight can be changed 
so as to produce various tensions in the cord 
and hence various accelerating forces on the 

Now You Are Ready to Go 

Measure the total mass of the cart, the blinky, 
the spring scale, and any other weights you 
may want to include with it to vary the mass. 
This is the mass m being accelerated. 

Release the cart and allow it to accelerate. 
Repeat the motion several times while watch- 
ing the spring-scale pointer. You may notice 
that the pointer has a range of positions. The 
midpoint of this range is a fairly good measure- 
ment of the average force Fav producing the 

Record Fav in newtons. 

Our faith in Newton's law is such that we 
assume the acceleration is the same and is 
constant every time this particular Fav acts on 
the mass m. 

Use Newton's law to predict what the aver- 
age acceleration Uav was during the run. 

Then find a directly to see how accurate 
your prediction was. 

To measure the average acceleration Uav 
take a Polaroid photograph through a rotating 
disk stroboscope of a light source mounted on 
the cart. As alternatives you might use a liquid 
surface accelerometer described in detail on 
page 170, or a blinky. Analyze your results just 
as in the experiments on uniform and accel- 
erated motion 4, 5, and 6 to find Uav 

This time, however, you must know the 
distance traveled in meters and the time in- 
terval in seconds, not just in blinks, flashes or 
other arbitrary time units. 
Ql Does Fav (as measured) equal maav (as com- 
puted from measured values)? 

You may wish to observe the following 
effects without actually making numerical 

1. Keep the mass of the cart constant and ob- 
serve how various forces affect the accelera- 

2. Keep the force constant and observe how 
various masses of the cart affect the accelera- 

Q2 Do your observations support Newton's 
second law? Explain. 

Experimental Errors 

It is unlikely that your values of Fav and maav 
were equal. 

Does this mean that you have done a poor 
job of taking data? Not necessarily. As you 
think about it, you will see that there are at 
least two other possible reasons for the in- 
equality. One may be that you have not yet 
measured everything necessary in order to 
get an accurate value for each of your three 

In particular, the force used in the calcu- 
lation ought to be the net, or resultant, force 
on the cart— not just the towing force that 
you measured. Friction force also acts on your 
cart, opposing the accelerating force. You can 
measure it by reading the spring scale as you 
tow the cart by hand at constant speed. Do it 
several times and take an average, Ff. Since 
Ff acts in a direction opposite to the towing 
force Ft, 

Piiel — Ft — Ff 

If Ff is too small to measure, then F„e, = Ft. 
which is simply the towing force that you 
wrote as Far in the beginning of the experi- 

Another reason for the inequality of Fav 
and niav may be that your value for each of 
these quantities is based on measurements 
and every measurement is uncertain to some 

You need to estimate the uncertainty of 
each of your measurements. 

Uncertainty in average force F„,. Your uncer- 
tainty in the measurement of F„,. is the amount 
by which your reading of your spring scale 
varied above and below the average force, 
Fav Thus if your scale reading ranged from 1.0 
to 1.4N the average is 1.2N, and the range of 
uncertainty is 0.2N. The value of F„,. would be 
reported as 1.2 ± 0.2N. 

Q3 What is your value of F„,. and its uncer- 

Uncertainty in mass m Your uncertainty in m 
is roughly half the smallest scale reading of 


Experiment 8 

the balance with which you measured it. The 
mass consisted of a cart, a bhnky, and a spring 
scale (and possibly an additional mass). If the 
smallest scale reading is 0.1 kg, your record of 
the mass of each of these in kilograms might 
be as follows: 

m cart = 0.90 ± 0.05 kg 
m blinky = 0.30 ± 0.05 kg 
w scale =0.10 ±0.05 kg 

The total mass being accelerated is the sum 
of these masses. The uncertainty in the total 
mass is the sum of the three uncertainties. 
Thus, in our example, m — 1.30 ± 0.15 kg. 
Q4 What is your value of m and its uncer- 

Uncertainty in average acceleration a„,. Finally, 
consider a„^,. You found this by measuring 
Ad/At for each of the intervals between the 
points on your blinky photograph. 

Adj — »|< Ld^ 

|^Ad.-4^ Ad- 
Fig. 3-3 

Suppose the points in Fig. 3-3 represent 
images of a light source photographed through 
a single slot— giving 5 images per second. 
Calculate Ad/At for several intervals. 

If you assume the time between blinks to 
have been measured very accurately, the 
uncertainty in each value of Ad/At is due pri- 
marily to the fact that the photographic images 
are a bit fuzzy. Suppose that the uncertainty 
in locating the distance between the centers 
of the dots is 0.1 cm as shown in the first 
column of the Table below. 

Average speeds Average accelerations 

Ad,/At = 2.5 ± 0.1 cm/sec 
AdJM = 3.4 ± 0.1 cm/sec 
AdJAt = 4.0 ± 0.1 cm/sec 
AdJAt = 4.8 ± 0.1 cm/sec 

Aj/,/At = 0.9 ± 0.2 cm/sec^ 
AyJM = 0.6 ± 0.2 cm/sec^ 
Ay,/ At = 0.8 ± 0.2 cm/sec^ 
Average = 0.8 ± 0.2 cm/sec^ 

When you take the differences between 
successive values of the speeds, Ad/At, you 
get the accelerations, At;/At, which are re- 
corded in the second column. When a differ- 
ence in two measurements is involved, you 

find the uncertainty of the differences (in this 
case, AvIM) by adding the uncertainties of the 
two measurements. This results in an uncer- 
tainty in acceleration of (±0.1) + (±0.1) or ±0.2 
cm/sec^ as recorded in the table. 
Q5 What is your value of aav and its uncer- 

Comparing Your Results 

You now have values of Far, Tn and aav, their 
uncertainties, and you consider the uncer- 
tainty of ma„,. When you have a value for the 
uncertainty of this product of two quantities, 
you will then compare the value of ma„,. with 
the value of Fav and draw your final conclu- 
sions. For convenience, we have dropped the 
"av" from the symbols in the equations in the 
following discussion. When two quantities are 
multiplied, the percentage uncertainty in 
the product never exceeds the sum of the 
percentage uncertainties in each of the fac- 
tors. In our example, m x a = 1.30 kg x 0.8 
cm/sec^ = 1.04 newtons. The uncertainty in 
a (0.8 ± 0.2 cm/sec") is 25% (since 0.2 is 25% 
of 0.8). The uncertainty in m is 11%. Thus the 
uncertainty in ma is 25% + 11% = 36% and we 
can write our product as ma = 1.04 N +0.36% 
which is, to two significant figures, 

ma = 1.04 ±0.36 N 
(The error is so large here that it really isn't 
appropriate to use the two decimal places; 
we could round off to 1.0 ± 0.4 N.) In our ex- 
ample we found from direct measurement 
that Fnet = 1.2 ± 0.2 N. Are these the same 

Although 1.0 does not equal 1.2, the range 
of 1.0 ± 0.4 overlaps the range of 1.2 + 0.2, so 
we can say that "the two numbers agree within 
the range of uncertainty of measurement. ' 

An example of lack of agreement would 
be 1.0 ± 0.2 and 1.4 ±0.1. These are presum- 
ably not the same quantity since there is no 
overlap of expected uncertainties. 

In a similar way, work out your own values 
of F„e, and maa,- 

Q6 Do your own values agree within the range 
of uncertainty of your measurement? 
Q7 Is the relationship F„p, ^maav consistent 
with your observations? 

Experiment 9 169 


You know from your own experience that an 
object that is pulled strongly toward the earth 
(like, say, an automobile) is difficult to ac- 
celerate by pushing. In other words, objects 
with great weight also have great inertia. But 
is there some simple, exact relationship be- 
tween the masses of objects and the gravita- 
tional forces acting on them? For example, 
if one object has twice the mass of another, 
does it also weigh twice as much? 

Measuring Mass 

The masses of two objects can be compared 
by observing the accelerations they each ex- 
perience when acted on by the same force. 
Accelerating an object in one direction with a 
constant force for long enough to take mea- 
surements is often not practical in the labora- 
tory. Fortunately there is an easier way. If 
you rig up a puck and springs between two 
rigid supports as shown in the diagram, you 





can attach objects to the puck and have the 
springs accelerate the object back and forth. 
The greater the mass of the object, the less 
the magnitude of acceleration will be, and the 
longer it will take to oscillate back and forth. 
To "calibrate" your oscillator, first time 
the oscillations. The time required for 5 com- 
plete round trips is a convenient measure. 
Tape pucks on top of the first one, and time 
the period for each new mass. (The units of 
mass are not essential here, for we will be 
interested only in the ratio of masses.) Then 
plot a graph of mass against the oscillation 
period, drawing a smooth curve through your 
experimental plot points. Do not leave the 
pucks stuck together. 

Ql Does there seem to be a simple relation- 
ship between mass and period? Could you 
write an algebraic expression for the relation- 


To compare the gravitational forces on two 
objects, they can be hung on a spring scale. 
In this investigation the units on the scale 
are not important, because we are interested 
only in the ratio of the weights. 

Comparing Mass and Weight 

Use the puck and spring oscillator and calibra- 
tion graph to find the masses of two objects 
(say, a dry cell and a stapler). Find the gravi- 
tational pulls on these two objects by hanging 
each from a spring scale. 

Q2 How does the ratio of the gravitational 
forces compare to the ratio of the masses? 
Q3 Describe a similar experiment that would 
compare the masses of two iron objects to the 
magnetic forces exerted on them by a large 

You probably will not be surprised to find 
that, to within your uncertainty of measure- 
ment, the ratio of gravitational forces is the 
same as the ratio of masses. Is this really worth 
doing an experiment to find out, or is the an- 
swer obvious to begin with? Newton didn't 
think it was obvious. He did a series of very 
precise experiments using many different 
substances to find out whether gravitational 
force was always proportional to inertial mass. 
To the limits of his precision, he found the 
proportionality to hold exactly. (Newton's 
results have been confirmed to a precision of 
±0.000000001%, and extended to gravitational 
attraction to bodies other than the earth). 

Newton could offer no explanation from 
his physics as to why the attraction of the 
earth for an object should increase in exact 
proportion to the object's inertia. No other 
forces bear such a simple relation to inertia, 
and this remained a complete puzzle for two 
centuries until Einstein related inertia and 
gravitation theoretically. (See "Outside and 
Inside the Elevator" in the Unit 5 Reader.) 
Even before Einstein, Ernst Mach made the 
ingenious suggestion that inertia is not the 
property of an object by itself, but is the re- 
sult of the gravitational forces exerted on an 
object by everything else in the universe. 



Stack several checkers. Put another checker 
on the table and snap it into the stack. On the 
basis of Newton's first law, can you explain 
what happened? 


One teacher suggests placing a glass beaker 
half full of water on top of a pile of three wooden 
blocks. Three quick back-and-forth swipes 
(NOT FOUR!) of a hammer leave the beaker 
sitting on the table. 


Hang a weight (such as a 
heavy wooden block that 
just barely supports it, 
and tie another identical 
string below the weight. A 
slow, steady pull on the 
string below the weight 
breaks the string above 
the weight. A quick jerk 
breaks it below the weight. 


One way for you to get the feel of Newton's 
second law is actually to pull an object with a 
constant force. Load a cart with a mass of 
several kilograms. Attach one end of a long 
rubber band to the cart and, pulling on the 
other end, move along at such a speed that the 

rubber band is maintained at a constant length 
— say 70 cm. Holding a meter stick above the 
band with its 0-cm end in your hand will help 
you to keep the length constant. 

The acceleration will be very apparent to 
the person applying the force. Vary the mass 
on the cart and the number of rubber bands 
(in parallel) to investigate the relationship 
between F, m, and a. 


An accelerometer is a device that measures 
acceleration. Actually, anything that has mass 
could be used for an accelerometer. Because 
you have mass, you were acting as an accelero- 
meter the last time you lurched forward in the 
seat of your car as the brakes were applied. 
With a knowledge of Newton's laws and cer- 
tain information about you, anybody who 
measured how far you leaned forward and how 
tense your muscles were would get a good 
idea of the magnitude and direction of the 
acceleration that you were undergoing. But 
it would be complicated. 

Here are two accelerometers of a much 
simpler kind. With a little practice, you can 
learn to read accelerations from them directly, 
without making any difficult calculations. 

A. The Liquid-Surface 

This device is a hollow, fiat plastic container 


By John Hart 

By permission of John Hart and Fteld Enterprises, Inc. 



partly filled with a colored liquid. When it is 
not being accelerated, the liquid surface is 
horizontal, as shown by the dotted line in Fig. 
3-4. But when it is accelerated toward the left 
(as shown) with a uniform acceleration a, the 
surface becomes tilted, with the level of the 
liquid rising a distance h above its normal 
position at one end of the accelerometer and 
falling the same distance at the other end. 
The greater the acceleration, the more steeply 
the surface of the liquid is slanted. This means 
that the slope of the surface is a measure of 
the magnitude of the acceleration a. 


Fig. 3-4 

The length of the accelerometer is 11, as 
shown in Fig. 3-4 above. So the slope of the 
surface may be found by 

_ vertical distance 
horizontal distance 



Theory gives you a very simple relation- 

ship between this slope and the acceleration 

h a 
slope ^ 7 = — 

Notice what this equation tells you. It says 
that if the instrument is accelerating in the 
direction shown with just a^ (one common way 
to say this is that it has a "one-G acceleration"), 
the acceleration of gravity, then the slope of 
the surface is just 1 ; that is, h = I and the sur- 
face makes a 45° angle with its normal, hori- 
zontal direction. If it is accelerating with j 
flg, then the slope will be 2"; that is fi = |- /. In 
the same way, if h = j I, then a'=\ Uy^ and so 
on with any acceleration you care to measure. 

To measure h, stick a piece of centimeter 
tape on the front surface of the accelerometer 
as shown in Fig. 3-5 below. Then stick a piece 
of white paper or tape to the back of the in- 
strument to make it easier to read the level 
of the liquid. Solving the equation above for 
a gives 


a^ UaX 

wWit^ p&pe.r on 
back, of c^eJI y 
\ 'M^ 

Onacc-eieratcei liquid Level 

Accelerate licjoid sur-face 
Fig. 3-5 

cm soale on 
front of eel 


By John Hart 

By permission of John Hart and Field Enterprises, Inc. 



This shows that if you place a scale 10 scale 
units away from the center you can read accel- 
arations directly in joth's of "G's." Since ay 
is very close to 9.8m/sec^ at the earth's surface 
if you place the scale 9.8 scale units from the 
center you can read accelerations directly in 
m/sec^. For example, if you stick a centimeter 
tape just 9.8 cm from the center of the liquid 
surface, one cm on the scale is equivalent to 
an acceleration of one m/sec^. 

Calibration of the Accelerometer 

You do not have to trust blindly the theory 
mentioned above. You can test it for yourself. 
Does the accelerometer really measure accel- 
erations directly in m/sec^? Stroboscopic meth- 
ods give you an independent check on the 
correctness of the theoretical prediction. 

Set the accelerometer on a dynamics cart 
and arrange strings, pulleys, and masses as 
you did in Experiment 9 to give the cart a uni- 
form acceleration on a long tabletop. Don't 
forget to put a block of wood at the end of the 
cart's path to stop it. Make sure that the ac- 
celerometer is fastened firmly enough so that 
it will not fly off" the cart when it stops sud- 
denly. Make the string as long as you can, so 
that you use the entire length of the table. 

Give the cart a wide range of accelerations 
by hanging different weights from the string. 
Use a stroboscope to record each motion. To 
measure the accelerations from your strobe 
records, plot t'^ against d, as you did in Experi- 
ment 5. (What relationship did Galileo dis- 
cover between dlt^ and the acceleration?) Or 
use the method of analysis you need in Experi- 
ment 9. 

Compare your stroboscopic measurements 
with the readings on the accelerometer during 
each motion. It takes some cleverness to read 
the accelerometer accurately, particularly 
near the end of a high-acceleration run. One 
way is to have several students along the table 
observe the reading as the cart goes by; use 
the average of their reports. If you are using 
a xenon strobe, of course, the readings on the 
accelerometer will be visible in the photograph; 
this is probably the most accurate method. 

Plot the accelerometer readings against 

the stroboscopically measured accelerations. 
This graph is called a "calibration curve." If 
the two methods agree perfectly, the graph 
will be a straight line through the origin at a 
45° angle to each axis. If your curve turns out 
to have some other shape, you can use it to 
convert "accelerometer readings" to "accel- 
erations"— if you are willing to assume that 
your strobe measurements are more accurate 
than the accelerometer. (If you are not willing, 
what can you do?) 

B. Automobile Accelerometer-I 

With a liquid-surface accelerometer mounted 
on the front-back line of a car, you can measure 
the magnitude of acceleration along its path. 
Here is a modification of the liquid-surface 
design that you can build for yourself. Bend a 
small glass tube (about 30 cm long) into a 
U-shape, as shown in Fig. 3-6 below. 


0-4- i 



1 0-2 

j o- \ 





-O- 1 , 

j — 

-o-2>_:;: 1 



Iw -■ 



Fig. 3-6 

Calibration is easiest if you make the long 
horizontal section of the tube just 10 cm long; 
then each 5 mm on a vertical arm represents 
an acceleration of jq g = (about) 1 m/sec^. 
by the same reasoning as before. The two 
vertical arms should be at least three-fourths 
as long as the horizontal arm (to avoid splash- 
ing out the liquid during a quick stop). Attach 
a scale to one of the vertical arms, as shown. 
Holding the long arm horizontal, pour colored 
water into the tube until the water level in 
the arm comes up to the zero mark. How can 
you be sure the long arm is horizontal? 

To mount your accelerometer in a car. 
fasten the tube with staples (carefully) to a 
piece of plywood or cardboard a little bigger 
than the U-tube. To reduce the hazard from 
broken glass while you do this, cover all but 



the scale (and the arm by it) with cloth or card- 
board, but leave both ends open. It is essential 
that the accelerometer be horizontal if its 
readings are to be accurate. When you are 
measuring acceleration in a car, be sure the 
road is level. Otherwise, you will be reading 
the tilt of the car as well as its acceleration. 
When a car accelerates — in any direction — it 
tends to tilt on the suspension. This will in- 
troduce error in the accelerometer readings. 
Can you think of a way to avoid this kind of 

C. Automobile Accelerometer-ll 

An accelerometer that is more directly related 
to F = ma can be made from a 1-kg cart and a 
spring scale marked in newtons. The spring 
scale is attached between a wood frame and 
the cart as in the sketch below. If the frame is 
kept level, the acceleration of the system can 


U)ood J*; 

Ikg cart \0^^ 


be read directly from the spring scale, since 
one newton of force on the 1-kg mass indicates 
an acceleration of one m/sec'. (Instead of a 
cart, any 1-kg object can be used on a layer 
of low-friction plastic beads.) 

A damped-pendulum accelerometer, on 
the other hand, indicates the direction of any 
horizontal acceleration; it also gives the mag- 
nitude, although less directly than the pre- 
vious instruments do. 

Hang a smaU metal pendulum bob by a 
short string fastened to the middle of the lid 
of a one-quart mason jar as shown on the left 
hand side of the sketch at the bottom of the 
page. Fill the jar with water and screw the 
lid on tight. For any position of the pendulum, 
the angle that it makes with the vertical de- 
pends upon your position. What would you 
see, for example, if the bottle were accelerating 
straight toward you? Away from you? Along 
a table with you standing at the side? (Careful: 
this last question is trickier than it looks. 

To make a fascinating variation on the 
damped-pendulum accelerometer, simply re- 
place the pendulum bob with a cork and turn 
the bottle upside down as shown on the right 
hand side of the sketch at the bottom of the 
page. If you have punched a hole in the bottle 
lid to fasten the string, you can prevent leakage 
with the use of sealing wax, parafin, or tape. 

This accelerometer will do just the opposite 
from what you would expect. The explanation 
of this odd behavior is a little beyond the scope 
of this course: it is thoroughly explained in 
The Physics Teacher, vol. 2, no. 4 (April 1964) 
page 176. 

D. Damped-Pendulum Accelerometer 

One advantage of liquid-surface acceler- 
ometers is that it is easy to put a scale on them 
and read accelerations directly from the instru- 
ment. They have a drawback, though; they 
give only the component of acceleration that 
is parallel to their horizontal side. If you ac- 
celerate one at right angles to its axis, it doesn't 
register any acceleration at all. And if you don't 
know the direction of the acceleration, you 
have to use trial-and-error methods to find it 
with the accelerometers we have discussed up 
to this point. 



A motorboat was photographed from a bridge 
in this film. The boat heads upstream, then 
downstream, then directly across stream, and 
at an angle across the stream. The operator 
of the boat tried to keep the throttle at a con- 
stant setting to maintain a steady speed rela- 
tive to the water. The task before you is to 
find out if he succeeded. 

This photograph was taken from one bank of the stream. 
It shows the motorboat heading across the stream and 
the camera filming this loop fixed on the scaffolding on 
the bridge. 

First project the film on graph paper and 
mark the lines along which the boat's image 
moves. You may need to use the reference 
crosses on the markers. Then measure speeds 
by timing the motion through some predeter- 
mined number of squares. Repeat each mea- 
surement several times, and use the average 
times to calculate speeds. Express all speeds 
in the same unit, such as "squares per second" 
(or "squares per cm" where cm refers to mea- 
sured separations between marks on the mov- 
ing paper of a dragstrip recorder). Why is there 
no need to convert the speeds to meters per 

second? Why is it a good idea to use a large 
distance between the timing marks on the 
graph paper? 



Fig. 3-7 

The head-to-tail method of adding vectors. For a review 
of vector addition see Project Physics Programmed 
instruction Booklet entitled Vectors II. 

The head-to-tail method of adding vectors 
is illustrated in Fig. 3-7. Since velocity is a 
vector with both magnitude and direction, you 
can study vector addition by using velocity 
vectors. An easy way of keeping track of the 
velocity vectors is by using subscripts: 

v^E velocity of boat relative to earth 

Vb» velocity of boat relative to water 

VnE velocity of water relative to earth 
Then^ _^ _^ 

For each heading of the boat, a vector 
diagram can be drawn by laying off the ve- 
locities to scale. A suggested procedure is to 
record data (direction and speed) for each of 
the five scenes in the film, and then draw the 
vector diagram for each. 

Scene 1 : Two blocks of wood are dropped over- 
board. Time the blocks. Find the speed of the 
river, the magnitude of v,, £• 

Film Loops 175 

Scene 2: The boat heads upstream. Measure 
Vbe^ then find Vbw using a vector diagram simi- 
lar to Fig. 3-8. 




Fig. 3-8 

Scene 3: The boat heads downstream. Measure 
Vbe^ then find Vb» using a vector diagram. 



Scene 4: The boat heads across stream and 
drifts downstream. Measure the speed of the 
boat and the direction of its path to find v^be- 
Also measure the direction of Vbw, the direc- 
tion the boat points. One way to record data is 
to use a set of axes with the 0° - 180° axis pass- 
ing through the markers anchored in the 
river. A diagram, such as Fig. 3-9, will help 
you record and analyze your measurements. 
(Note that the numbers in the diagram are 
deliberately not correct.) Your vector diagram 
should be something like Fig. 3-10. 

Fig. 3-10 

Scene 5: The boat heads upstream at an angle, 
but moves directly across stream. Again find 
a value for Vbw- 

Checking your work: (a) How well do the four 
values of the magnitude of Vbh agree with 
each other? Can you suggest reasons for any 
discrepancies? (b) From scene 4, you can cal- 
culate the heading of the boat. How well does 
this angle agree with the observed boat head- 
ing? (c) In scene 5. you determine a direction 
for 7^1,. Does this angle agree with the ob- 
served boat heading? 

Fig. 3-9 


Experiment 10 

Chapter 4 Understanding Motion 


Imagine you are a ski-jumper. You lean for- 
ward at the top of the sHde, grasp the raihng 
on each side, and yank yourself out into the 
track. You streak down the trestle, crouch and 
give a mighty leap at the takeoff lip, and soar 
up and out, looking down at tiny fields far 
below. The hill flashes into view and you thump 
on its steep incline, bobbing to absorb the 

This exciting experience involves a more 
complex set of forces and motions than you 
can deal with in the laboratory at one time. 
Let's concentrate therefore on just one aspect: 
your flight through the air. What kind of a path, 
or trajectory, would your flight follow? 

At the moment of projection into the air a 
skier has a certain velocity (that is, a certain 
speed in a given direction), and throughout 
his flight he must experience the downward 
acceleration due to gravity. These are circum- 
stances that we can duplicate in the laboratory. 
To be sure, the flight path of an actual ski- 
jumper is probably aff"ected by other factors, 
such as air, velocity and friction; but we now 
know that it usually pays to begin experiments 
with a simplified approximation that allows 
us to study the effects of a few factors at a 
time. Thus, in this experiment you will launch 
a steel ball from a ramp into the air and try to 
determine the path it follows. 

How to Use the Equipment 

If you are assembling the equipment for this 
experiment for the first time, follow the manu- 
facturer's instructions. 

The apparatus being used by the students 
in the photograph on page 177 consists of two 
ramps down which you can roll a steel ball. 
Adjust one of the ramps (perhaps with the help 
of a level) so that the ball leaves it horizontally. 

Tape a piece of squared graph paper to the 
plotting board with its left-hand edge behind 
the end of the launching ramp. 

To find a path that extends all across the 
graph paper, release the ball from various 
points up the ramp until you find one from 
which the ball falls close to the bottom right- 
hand corner of the plotting board. Mark the 
point of release on the ramp and release the 
ball each time from this point. 

Attach a piece of carbon paper to the im- 
pact board, with the carbon side facing the 
ramp. Then tape a piece of thin onionskin 
paper over the carbon paper. 

Now when you put the impact board in its 
way, the ball hits it and leaves a mark that you 
can see through the onionskin paper, auto- 
matically recording the point of impact be- 
tween ball and board. (Make sure that the 
impact board doesn't move when the ball hits 
it; steady the board with your hand if neces- 
sary.) Transfer the point to the plotting board 
by making a mark on it just next to the point 
on the impact board. 

Do not hold the ball in your fingers to re- 
lease it — it is impossible to let go of it in the 
same way every time. Instead, dam it up with 

Experiment 10 177 

a ruler held at a mark on the ramp and release 
the ball by moving the ruler quickly away from 
it down the ramp. 

Try releasing the ball several times (always 
from the same point) for the same setting of 
the impact board. Do all the impact points 
exactly coincide? 

Repeat this for several positions of the 
impact board to record a number of points on 
the ball's path. Move the board equal dis- 
tances every time and always release the ball 
from the same spot on the ramp. Continue 
until the ball does not hit the impact board 
any longer. 

Now remove the impact board, release the 
ball once more, and watch carefully to see that 
the ball moves along the points marked on the 
plotting board. 

The curve traced out by your plotted points 
represents the trajectory of the ball. By ob- 
serving the path the ball follows, you have 
completed the first phase of the experiment. 

If you have time, you will find it worth 
while to go further and explore some of the 
properties of your trajectory. 

Analyzing Your Data 

To help you analyze the trajectory, draw a 
horizontal line on the paper at the level of the 

end of the launching ramp. Then remove the 
paper from the plotting board and draw a 
smooth continuous curve through the points 
as shown in the figure at the bottom of the 

You already know that a moving object 
on which there is no net force acting will move 
at constant speed. There is no appreciable 
horizontal force acting on the ball during its 
fall, so we can make an assumption that its 
horizontal progress is at a constant speed. 
Then equally spaced hnes will indicate equal 
time intervals. 

Draw vertical hnes through the points on 
your graph. Make the first line coincide with 
the end of the launching ramp. Because of 
your plotting procedure these lines should 
be equally spaced. If the horizontal speed of 
the ball is uniform, these vertical lines are 
drawn through positions of the ball separated 
by equal time intervals. 

Now consider the vertical distances fallen 
in each time interval. Measure down from your 
horizontal line the vertical fall to each of your 

178 Experiment 10 

plotted points. Record your measurements in a 
column. Alongside them record the corre- 
sponding horizontal distances measured from 
the first vertical line. A sample of results as 
recorded in a student notebook is shown on 
the right. 

Ql What would a graph look like on which you 
plot horizontal distance against time? 

Earlier in your work with accelerated 
motion you learned how to recognize uniform 
acceleration (see Sees. 2.5-2.8 in the Text and 
Experiment 5). Use the data you have just 
collected to decide whether the vertical motion 
of the ball was uniformly accelerated motion. 
Q2 What do you find? 

Q3 Do the horizontal and the vertical motions 
affect each other in any way? 
Q4 Write an equation that describes the hori- 
zontal motion in terms of horizontal speed v, 
the horizontal distance, Ax, and the time of 
travel, At. 

Q5 What is the equation that describes the 
vertical motion in terms of the distance fallen 
vertically, At/, the vertical acceleration, Uy, 
and the time of travel, At? 

7 Vtxluti . 








Try These Yourself 

There are many other things you can do with 

this apparatus. Some of them are suggested 

by the following questions. 

Q6 What do you expect would happen if you 

repeated the experiment with a glass marble 

of the same size instead of a steel ball? 

Q7 What will happen if you next try to repeat 

the experiment starting the ball from a dif- 
ferent point on the ramp? 
Q8 What do you expect if you use a smaller or 
larger ball starting always from the same 
reference point on the ramp? 
Q9 Plot the trajectory that results when you 
use a ramp that launches the ball at an angle 
to the horizontal. In what way is this curve 
similar to your first trajectory? 

Experiment 11 



You can predict the landing point of a ball 
launched horizontally from a tabletop at any 
speed. If you know the speed, Vq, of the ball as 
it leaves the table, the height of the table above 
the floor and a^, you can then use the equation 
for projectile motion to predict where on the 
floor the ball will land. 

You know an equation for horizontal mo- 

^x = v At 

and you know an equation for free-fall from 

The time interval is difficult to measure. Be- 
sides, in talking about the shape of the path, 
all we really need to know is how Ay relates to 
Ax. Since, as you found in the previous experi- 
ment, these two equations still work when an 
object is moving horizontally and falling at 
the same time, we can combine them to get an 
equation relating Ay and Ax, without At ap- 
pearing at all. We can rewrite the equation for 
horizontal motion as: 

Then we can substitute this expression for t 
into the equation for fall: 

At = 


_ 1 

Ay = ^a 


Thus the equation we have derived should 
describe how Ay changes with Ax— that is, 
it should give us the shape of the trajectory. 
If we want to know how far out from the edge 
of the table the ball will land (Ax), we can 
calculate if from the height of the table (Ay), 
Ug, and the ball's speed v along the table. 

Doing the Experiment 

Find V by measuring with a stopwatch the 
time t that the ball takes to roll a distance d 
along the tabletop. (See Fig. 4-1 below.) Be sure 
to have the ball caught as it comes off the end 
of the table. Repeat the measurement a few 
times, always releasing the ball from the same 
place on the ramp, and take the average value 
of V. 

Measure Ay and then use equation for Ai/ 
to calculate Ax. Place a target, a paper cup, 
perhaps, on the floor at your predicted landing 
spot as shown below. How confident are you of 
your prediction? Since it is based on measure- 
ment, some uncertainty is involved. Mark an 
area around the spot to indicate your uncer- 

bal I mu^'t' be 
e)tiTl in air 


Fig. 4-1 

180 Experiment 11 

^• — -^ 


-^ - -_ - 

:---- ,t-- -— TT.^^ 


1-".""^ - .^-- 



/ Thread. 




V7 -p ^* 



Now release the ball once more. This time, 
let it roll off the table and land, hopefully, on 
the target as shown in the figure above. 

If the ball actually does fall within the 
range of values of x you have estimated, then 
you have supported the assumption on which 
your calculation was based, that vertical and 
horizontal motion are not affected by each 

QJ How could you determine the range of a 
ball launched horizontally by a slingshot? 
Q2 Assume you can throw a baseball 40 meters 
on the earth's surface. How far could you 
throw that same ball on the surface of the 
moon, where the acceleration of gravity is 
one-sixth what it is at the surface of the earth? 
Q^ Will the assumptions made in the equations 
l^=vbA and Ai/ = jagC At)^ hold for a Ping-Pong 
ball? If the table were 1000 meters above the 
floor, could you still use these equations? Why 
or why not? 

(Tyial.y C^ .j fy- 

^iH^ "'^^ 

The path taken by a cannon ball according to a drawing by Ufano (1621). He shows 
that the same horizontal distance can be obtained by two different firing angles. Gun- 
ners had previously found this by experience. What angles give the maximum range? 

Experiment 12 



The motion of an earth sateUite and of a weight 
swung around your head on the end of a string 
are described by the same laws of motion. 
Both are accelerating toward the center of 
their orbit due to the action of an unbalanced 

In the following experiment you can dis- 
cover for yourself how this centripetal force 
depends on the mass of the satellite and on its 
speed and distance from the center. 

How the Apparatus Works 

Your "satellite" is one or more rubber stop- 
pers. When you hold the apparatus in both 
hands, as shown in the photo above, and swing 
the stopper around your head, you can measure 
the centripetal force on it with a spring scale 
at the base of the stick. The scale should read 
in newtons or else its readings should be con- 
verted to newtons. 

You can change the length of the string 
so as to vary the radius R of the circular orbit, 
and you can tie on more stoppers to vary the 
satellite mass m. 

The best way to set the frequency / is to 
swing the apparatus in time with some peri- 
odic sound from a metronome or an earphone 
attachment to a blinky. You keep the rate con- 
stant by adjusting the swinging until you see 
the stopper cross the same point in the room at 
every tick. 

Hold the stick vertically and have as little 
motion at the top as possible, since this would 
change the radius. Because the stretch of the 
spring scale also alters the radius, it is helpful 
to have a marker (knot or piece of tape) on the 
string. You can move the spring scale up or 
down slightly to keep the marker in the same 

Doing the Experiment 

The object of the experiment is to find out how 
the force F read on the spring scale varies with 
m, with /, and with R. 

You should only change one of these three 
quantities at a time so that you can investigate 
the effect of each quantity independently of 
the others. It's easiest to either double or triple 
m, f, and R (or halve them, and so on, if you 
started with large values). 

Two or three different values should be 
enough in each case. Make a table and clearly 
record your numbers in it. 

Ql How do changes in m affect F when R and 
/ are kept constant? Write a formula that 
states this relationship. 

Q2 How do changes in / affect F when m and 
R are kept constant? Write a formula to ex- 
press this too. 

Q3 What is the effect of R on F? 
Q4 Can you put m, f, and R all together in a 
single formula for centripetal force, R? 

How does your formula compare with the 
expression derived in Sec. 4.7 of the Text. 

182 Experiment 13 


You may have had the experience of spinning 
around on an amusement park contraption 
known as the Whirhng Platter. The riders seat 
themselves at various places on a large flat 
polished wooden turntable about 40 feet in 
diameter. The turntable gradually rotates 
faster and faster until everyone (except for the 
person at the center of the table) has slid off. 
The people at the edge are the first to go. Why 
do the people slide off? 

Unfortunately you probably do not have a 
Whirling Platter in your classroom, but you do 
have a Masonite disk that fits on a turntable. 
The object of this experiment is to predict the 
maximum radius at which an object can be 
placed on the rotating turntable without slid- 
ing off. 

If you do this under a variety of conditions, 
you will see for yourself how forces act in cir- 
cular motion. 

Before you begin, be sure you have studied 
Sec. 4.6 in your Text where you learned that 
the centripetal force needed to hold a rider in a 
circular path is given by F = mv-jR. 

Studying Centripetal Force 

For these experiments it is more convenient 
to write the formula F = mvVR in terms of the 
frequency/. This is because/can be measured 
more easily than v. We can rewrite the form- 
ula as follows: 

^ ^ distance traveled ^ number of revolu- 
in one revolution tions per sec 

= 2ttR xf 

Substituting this expression for v in the form- 
ula gives: 

P _ 7nx {2Tr Rfy 

Friction on a Rotating Disk 

For objects on a rotating disk, the centripetal 
force is provided by friction. On a frictionless 
disk there could be no such centripetal force. 
As you can see from the equation we have 
just derived, the centripetal acceleration is 
proportional to R and to/"-^. Since the frequency 
/ is the same for any object moving around 
with a turntable, the centripetal acceleration 
is directly proportional to R, the distance from 
the center. The further an object is from the 
center of the turntable, therefore, the greater 
the centripetal force must be to keep it in a 
circular path. 

You can measure the maximum force 
F,„„j, that friction can provide on the object, 
measure the mass of the object, and then cal- 
culate the maximum distance from the center 
R,„ax that the object can be without sliding off. 
Solving the centripetal force equation for R 

_ 4ir^mR^P 

= AttZ 


You can measure all the quantities in this 


Use a spring scale to measure the force needed 
to make some object (of mass m from 0.2 to 
1.0 kg) start to slide across the motionless 

Experiment 13 


disk. This will be a measure of the maximum 
friction force that the disk can exert on the 

Then make a chalk mark on the turntable 
and time it (say, for 100 sec)— or accept the 
marked value of rpm— and calculate the fre- 
quency in rev/sec. 

Make your predictions of R,„ax for turn- 
table frequencies of 33 revolutions per minute 
(rpm), 45 rpm, and 78 rpm. 

Then try it! 
Ql How great is the percentage difference 

between prediction and experiment for each 
turntable frequency? Is this reasonable agree- 

Q2 What efTect would decreasing the mass 
have on the predicted value of R? Careful! 
Decreasing the mass has an effect on F also. 
Check your answer by doing an experiment. 
Q3 What is the smallest radius in which you 
can turn a car if you are moving 60 miles an 
hour and the friction force between tires and 
road is one-third the weight of the car? (Care- 
ful! Remember that weight is equal to a, x m.) 

B.C. by John Hart 

NO,MO, srURP... THE 

By permission of John Hart and Field Enterprises, Inc. 



Here is a simple way to demonstrate projectile 
motion. Place one coin near the edge of a table. 
Place an identical coin on the table and snap 
it with your finger so that it flies off" the table, 
just ticking the first coin enough that it falls 
almost straight down from the edge of the 
table. The fact that you hear only a single 
ring as both coins hit shows that both coins 
took the same time to fall to the floor from the 
table. Incidentally, do the coins have to be 
identical? Try different ones. 


You can use the principles of projectile motion 
to calculate the speed of a stream of water 
issuing from a horizontal nozzle. Measure the 
vertical distance Ay from the nozzle to the 
ground, and the horizontal distance Ax from 
the nozzle to where the water hits the ground. 

Use the equation relating Ax and Ay that 
was derived in Experiment 11, solving it for 




V = Ax 


The quantities on the right can all be measured 
and used to compute v. 


Using an electronic strobe light, a doorbell 
timer, and water from a faucet, you can photo- 
graph a water drop parabola. The principle of 
independence of vertical and horizontal mo- 
tions will be clearly evident in your picture. 

Remove the wooden block from the timer. 
Fit an "eye dropper" barrel in one end of some 
tubing and fit the other end of the tubing onto 
a water faucet. (Instead of the timer you can 
use a doorbell without the bell.) Place the tube 
through which the water runs under the clap- 
per so that the tube is given a steady series of 
sharp taps. This has the effect of breaking the 
stream of water into separate, equally spaced 
drops (see photo on previous page). 

To get more striking power, run the vibra- 
tor from a variable transformer (Variac) con- 
nected to the 110 volt a.c, gradually increasing 
the Variac from zero just to the place where 
the striker vibrates against the tubing. Adjust 
the water flow through the tube and eye drop- 
per nozzle. By viewing the drops with the 
xenon strobe set at the same frequency as the 
timer, a parabola of motionless drops is seen. 
A spot-light and disk strobe can be used in- 
stead of the electronic strobe light, but it is 
more difficult to match the frequencies of 
vibrator and strobe. The best photos are made 
by lighting the parabola from the side (that is. 
putting the light source in the plane of the 
parabola). The photo above was made in that 



way. With front lighting, the shadow of the 
parabola can be projected onto graph paper 
for more precise measurement. 

Some heating of the doorbell coil results, 
so the striker should not be run continuously 
for long periods of time. 

Of course projectile trajectories can be 
photographed of any object thrown into the 
air using the electronic strobe and Polaroid 
Land camera. By fastening the camera (se- 
curely!) to a pair of carts, you can photograph 
the action from a moving frame of reference. 


Fire a projectile straight up from a cart or toy 
locomotive as shown in the photo below that is 
rolling across the floor with nearly uniform 
velocity. You can use a commercial device 
called a ballistic cart or make one yourself. A 
spring-loaded piston fires a steel ball when you 
pull a string attached to a trigger pin. Use the 
electronic strobe to photograph the path of the 


Here are three ways you can show how a mov- 
ing object would appear in a rotating reference 

Method I Attach a piece of paper to a phono- 
graph turntable. Draw a line across the paper 
as a turntable is turning (see Fig. 4-2 below), 
using as a guide a meter stick supported on 
books at either side of the turntable. The line 
should be drawn at a constant speed. 


Fig. 4-2 



Method II Place a Polaroid camera on the turn- 
table on the floor and let a tractor run along 
the edge of a table, with a flashlight bulb on a 
pencil taped to the tractor so that it sticks out 
over the edge of the table. 

I j'flljt source 

^ — Couv»"te»r- uitiqKt 

V^OtSIb pf=** ^tov -tractor 



Method III How would an elliptical path appear 
if you were to view it from a rotating reference 
system? You can find out by placing a Polaroid 
camera on a turntable on the floor, with the 
camera aimed upwards. (See Fig. 4-3 below.) 
For a pendulum, hang a flashhght bulb and 
an AA dry cell. Make the pendulum long enough 
so that the light is about 4 feet from the cam- 
era lens. 

/i^Vit Source 

on 5*"i 

different points in its swing by using a motor 
strobe in front of the camera, or by hanging 
a blinky. 


Bend a coat hanger into the shape shown in 
the sketch below in this right-hand column. 
Bend the end of the hook slightly with a pair of 
pliers so that it points to where the finger sup- 
ports the hanger. File the end of the hook flat. 
Balance a penny on the hook. Move your finger 
back and forth so that the hanger (and bal- 
anced penny) starts swinging Uke a pendulum. 
Some practice will enable you to swing the 
hanger in a vertical circle, or around your head 
and still keep the penny on the hook. The cen- 
tripetal force provided by the hanger keeps the 
penny from flying off" on a straight-line path. 
Some people have done this demonstration 
successfully with a pile of as many as five 
pennies at once. 

■Turns On -firiacr- here. 



Fig. 4-3 

With the hghts out, give the pendulum a 
swing so that it swings in an elliptical path. 
Hold the shutter open while the turntable 
makes one revolution. You can get an indi- 
cation of how fast the pendulum moves at 


Use a calibrated electronic stroboscope or a 
hand-stroboscope and stopwatch to measure 
the frequencies of various motions. Look for 
such examples as an electric fan, a doorbell 
clapper, and a banjo string. 

On page 108 of the Text you will find tables 
of frequencies of rotating objects. Notice the 
enormous range of frequencies listed, from the 
electron in the hydrogen atom to the rotation 
of our Milky Way galaxy. 





Two carts of equal mass collide in this film. 
Three sequences labeled Event A, Event B, and 
Event C are shown. Stop the projector after 
each event and describe these events in words, 
as they appear to you. View the loop now, be- 
fore reading further. 

Even though Events A, B, and C are visibly 
different to the observer, in each the carts 
interact similarly. The laws of motion apply 
for each case. Thus, these events could be the 
same event observed from different reference 
frames. They are closely similar events photo- 
graphed from different frames of reference, as 
you see after the initial sequence of the film. 

The three events are photographed by a 
camera on a cart which is on a second ramp 
parallel to the one on which the colliding carts 
move. The camera is your frame of reference, 
your coordinate system. This frame of refer- 
ence may or may not be in motion with respect 
to the ramp. As photographed, the three events 
appear to be quite different. Do such concepts 
as position and velocity have a meaning inde- 
pendently of a frame of reference, or do they 
take on a precise meaning only when a frame 
of reference is specified? Are these three events 
really similar events, viewed from different 
frames of reference? 

You might think that the question of which 
cart is in motion is resolved by sequences at 

B.C. by John Hart 

the end of the film in which an experimenter, 
Franklin Miller of Kenyon College, stands 
near the ramp to provide a reference object. 
Other visual clues may already have provided 
this information. The events may appear dif- 
ferent when this reference object is present. 
But is this fixed frame of reference any more 
fundamental than one of the moving frames 
of reference? fixed relative to what? Or is 
there a "completely" fixed frame of reference? 
If you have studied the concept of momen- 
tum, you can also consider each of these three 
events from the standpoint of momentum 
conservation. Does the total momentum de- 
pend on the frame of reference? Does it seem 
reasonable to assume that the carts would 
have the same mass in all the frames of refer- 
ence used in the film? 

By permission of John Hart and Field Enterprises, Inc. 


Film Loops 


This film is a partial actualization of an ex- 
periment described by Sagredo in Galileo's 
Two New Sciences: 

If it be true that the impetus with which 
the ship moves remains indelibly im- 
pressed in the stone after it is let fall from 
the mast; and if it be further true that 
this motion brings no impediment or 
retardment to the motion directly down- 
wards natural to the stone, then there 
ought to ensue an effect of a very won- 
drous nature. Suppose a ship stands still, 
and the time of the falling of a stone 
from the mast's round top to the deck is 
two beats of the pulse. Then afterwards 
have the ship under sail and let the same 
stone depart from the same place. Ac- 
cording to what has been premised, it 
shall take up the time of two pulses in its 
fall, in which time the ship will have 
gone, say, twenty yards. The true motion 
of the stone will then be a transverse 
line (i.e., a curved line in the vertical 
plane), considerably longer than the first 
straight and perpendicular line, the 
height of the mast, and yet nevertheless 
the stone will have passed it in the same 
time. Increase the ship's velocity as much 
as you will, the falling stone shall des- 
cribe its transverse hnes still longer and 
longer and yet shall pass them all in those 
selfsame two pulses. 
In the film a ball is dropped three times: 

Scene 1 : The ball is dropped from the 
mast. As in Galileo's discussion, the ball 
continues to move horizontally with the 
boat's velocity, and also it falls vertically 
relative to the mast. 

Scene 2: The ball is tipped off a stationary 
support as the boat goes by. It has no 
forward velocity, and it falls vertically 
relative to the water surface. 

Scene 3: The ball is picked up and held 
briefly before being released. 


The ship and earth are frames of reference 
in constant relative motion. Each of the three 
events can be described as viewed in either 
frame of reference. The laws of motion apply 
for all six descriptions. The fact that the laws 
of motion work for both frames of reference, 
one moving at constant velocity with respect 
to the other, is what is meant by "Galilean 
relativity." (The positions and velocities are 
relative to the frame of reference, but the laws 
of motion are not. A "relativity" principle also 
states what is not relative.) 

Scene 1 can be described from the boat 
frame as follows: "A ball, initially at rest, is 
released. It accelerates downward at 9.8 m/ 
sec- and strikes a point directly beneath the 
starting point." Scene 1 described differently 
from the earth frame is: "A ball is projected 
horizontally toward the left; its path is a par- 
abola and it strikes a point below and to the 
left of the starting point." 

To test your understanding of Galilean 
relativity, you should describe the following: 
Scene 2 from the boat frame; Scene 2 in earth 
frame; Scene 3 from the boat frame; Scene 3 
from the earth frame. 

Film Loops 189 


A Cessna 150 aircraft 23 feet long is moving 
about 100 ft/sec at an altitude of about 200 
feet. The action is filmed from the ground as 
a flare is dropped from the aircraft. Scene 1 
shows part of the flare's motion; Scene 2, shot 
from a greater distance, shows several flares 
dropping into a lake; Scene 3 shows the ver- 
tical motion viewed head-on. Certain frames 
of the film are "frozen" to allow measure- 
ments. The time interval between freeze 
frames is always the same. 

Seen from the earth's frame of reference, 
the motion is that of a projectile whose original 
velocity is the plane's velocity. If gravity is the 
only force acting on the flare, its motion should 
be a parabola. (Can you check this?) Relative 
to the airplane, the motion is that of a body 
falling freely from rest. In the frame of refer- 
ence of the plane, the motion is vertically 

The plane is flying approximately at uni- 
form speed in a straight line, but its path is not 
necessarily a horizontal line. The flare starts 
with the plane's velocity, in both magnitude 
and in direction. Since it also falls freely under 
the action of gravity, you expect the flare's 
downward displacement below the plane to be 
d = -g-at^. But the trouble is that you cannot 
be sure that the first freeze frame occurs at 

the very instant the flare is dropped. However, 
there is a way of getting around this difficulty. 
Suppose a time B has elapsed between the 
release of the flare and the first freeze frame. 
This time must be added to each of the freeze 
frame times (conveniently measured from the 
first freeze frame) and so you would have 

d = ^a(t + By 

To see if the flare follows an equation such as 
this, take the square root of each side: 

Vd = (constant) (t + B) 

Now if we plot Vd against t, we expect a 
straight line. Moreover, if B = 0, this straight 
line will also pass through the origin. 

Suggested Measurements 

(a) Vertical motion. Project Scene 1 on paper. 
At each freeze frame, when the motion on the 
screen is stopped briefly, mark the positions 
of the flare and of the aircraft cockpit. Measure 
the displacement d of the flare below the plane. 
Use any convenient units. The times can be 
taken as integers, t = 0, 1, 2, . . ., designating 
successive freeze frames. Plot Vd versus t. 
Is the graph a straight line? What would be 
the effect of air resistance, and how would this 
show up in your graph? Can you detect any 
signs of this? Does the graph pass through 
the origin? 

(b) Analyze Scene 2 In the same way. 

(c) Horizontal motion. Use another piece of 
graph paper with time (in intervals) plotted 
horizontally and displacements (in squares) 
plotted vertically. Using measurements from 
your record of the flare's path, make a graph 
of the two motions in Scene 2. What are the 
effects of air resistance in the horizontal 
motion? the vertical motion? Explain your 
findings between the effect of air friction on 
the horizontal and vertical motions. 

(d) Acceleration due to gravity. The "constant" 
in your equation, d = (constant) (t + B), is 
ja; this is the slope of the straight-line graph 
obtained in part (a). The square of the slope 
gives 2^ so the acceleration is twice the 


Film Loops 

square of the slope. In this way you can obtain 
the acceleration in squares/(interval)^ To 
convert your acceleration into ft/sec- or m/ 
sec^, you can estimate the size of a "square" 
from the fact that the length of the plane is 
23 ft (7 m). The time interval in seconds be- 
tween freeze frames can be found from the 
slow-motion factor. 


A rocket tube is mounted on bearings that 
leave the tube free to turn in any direction. 
When the tube is hauled along the snow- 
covered surface of a frozen lake by a "ski-doo," 
the bearings allow the tube to remain pointing 
vertically upward in spite of some roughness 
of path. Equally spaced lamps along the path 
allow you to judge whether the ski-doo has 
constant velocity or whether it is accelerating. 
A preliminary run shows the entire scene; the 
setting is in the Laurentian Mountains in the 
Province of Quebec at dusk. 

Four scenes are photographed. In each 
case a rocket flare is fired vertically upward. 
With care you can trace a record of the tra- 

Scene 1: The ski-doo is stationary relative to 
the earth. How does the flare move? 

Scene 2: The ski-doo moves at uniform velocity 
relative to the earth. Describe the motion of 

the flare relative to the earth; describe the 
motion of the flare relative to the ski-doo. 

Scenes 3 and 4: The ski-doo's speed changes 
after the shot is fired. In each case describe 
the motion of the ski-doo and describe the 
flare's motion relative to the earth and relative 
to the ski-doo. In which cases are the motions 
a parabola? 

How do the events shown in this film illus- 
trate the principle of Galilean relativity? In 
which frames of reference does the rocket 
flare behave the way you would expect it to 
behave in all four scenes knowing that the 
force is constant, and assuming Newton's laws 
of motion? In which systems do Newton's laws 
fail to predict the correct motion in some of 
the scenes? 


The initial scenes in this film show a regula- 
tion hurdle race, with 1-meter-high hurdles 
spaced 9 meters apart. (Judging from the 
number of hurdles knocked over, the com- 
petitors were of something less than Olympic 
caliber!) Next, a runner, Frank White, a 75- 
kg student at McGill University, is shown in 
medium slow-motion (slow-motion factor 3) 
during a 50-meter run. His time was 8.1 sec- 
onds. Finally, the beginning of the run is 
shown in extreme slow motion (slow-motion 
factor of 80). "Analysis of a Hurdle Race 11" 
has two more extreme slow-motion sequences. 

To study the runner's motion, measure 
the average speed for each of the 1-meter 
intervals in the slow-motion scene. A "drag- 
strip" chart recorder is particularly convenient 
for recording the data on a single viewing of 
the loop. Whatever method you use for measur- 
ing time, the small but significant variations 
in speed will be lost in experimental uncer- 
tainty unless you work very carefully. Repeat 
each measurement several times. 

The extreme slow-motion sequence shows 
the runner from m to 6 m. The seat of the 
runner's white shorts might serve as a refer- 
ence mark. (What are other reference points 
on the runner that could be used? Are all ref- 

Film Loops 




erence points equally useful?) Measure the 
time to cover each of the distances, 0-1, 1-2, 
2-3, 3-4, 4-5, and 5-6 m. Repeat the measure- 
ments several times, viewing the film over 
again, and average your results for each in- 
terval. Your accuracy might be improved by 
forming a grand average that combines your 
average with others in the class. (Should you 
use all the measurements in the class?) Cal- 
culate the average speed for each interval, 
and plot a graph of speed versus displacement. 
Draw a smooth graph through the points. 
Discuss any interesting features of the graph. 

You might assume that the runner's legs 
push between the time when a foot is directly 
beneath his hip and the time when that foot 
is off the ground. Is there any relationship 
between your graph of speed and the way the 
runner's feet push on the track? 

The initial acceleration of the runner can 
be estimated from the time to move from the 
starting point to the 1 -meter mark. You can 
use a watch with a sweep second hand. Calcu- 
late the average acceleration, in m/sec^ during 
this initial interval. How does this forward 
acceleration compare with the magnitude of 
the acceleration of a falling body? How much 
force was required to give the runner this 
acceleration? What was the origin of this 


This film loop, which is a continuation of "An- 

alysis of a Hurdle Race I," shows two scenes 
of a hurdle race which was photographed at a 
slow-motion factor of 80. 

In Scene 1 the hurdler moves from 20 m to 
26 m, clearing a hurdle at 23 m. (See photo- 
graph.) In Scene 2 the runner moves from 
40 m to 50 m, clearing a hurdle at 41 m and 
sprinting to the finish line at 50 m. Plot graphs 
of these motions, and discuss any interesting 
features. The seat of the runner's pants fur- 
nishes a convenient reference point for mea- 
surements. (See the film-notes about the 
"Analysis of a Hurdle Race I" for further de- 

No measurement is entirely precise; mea- 
surement error is always present, and it cannot 
be ignored. Thus it may be difficult to teU if 
the small changes in the runner's speed are 
significant, or are only the result of measure- 
ment uncertainties. You are in the best tradi- 
tion of experimental science when you pay 
close attention to errors. 

It is often useful to display the experimen- 
tal uncertainty graphically, along with the 
measured or computed values. 

For example, say that the dragstrip timer 
was used to make three different measure- 
ments of the time required for the first meter 
of the run: 13.7 units, 12.9 units, and 13.5 
units, which give an average time of 13.28 

"2 I- 

^ 7 





— -^ 

J L 

I I I 

\ 2 3 ^ S 

clfsp\acc.rrefrr ^^eje^) 

192 Film Loops 

units. (If you wish to convert the dragstrip units 
to seconds, it will be easier to wait until the 
graph has been plotted using just units, and 
then add a seconds scale to the graph.) The 
lowest and highest values are about 0.4 units 
on either side of the average, so we could report 
the time as 13.3 + 0.4 units. The uncertainty 
0.4 is about 3% of 13.3, therefore the percent- 
age uncertainty in the time is 3%. If we assume 
that the distance was exactly one meter, so 
that all the uncertainty is in the time, then the 
percentage uncertainty in the speed will be 
the same as for the time— 3%. The slow-motion 
speed is 100 cm/ 13.3 time units, which equals 
7.53 cm/unit. Since 3% of 7.53 is 0.23, the 
speed can be reported as 7.53 + 0.23 cm/unit. 
In graphing this speed value, you plot a point 
at 7.53 and draw an "error bar" extending 
0.23 above and below the point. Now estimate 
the limit of error for a typical point on your 
graph and add error bars showing the range 
to each plotted point. 

Your graph for this experiment may well 
look like some commonly obtained in scientific 
research. For example, in the figure at the right 
a research team has plotted its experimental 
data; they published their results in spite of 

3.:5- 3.\i, S.17 3.)g 3.19 B.2C 5.2.1 3.2Z 

the considerable scattering of plotted points 
and even though some of the plotted points 
have errors as large as 5%. 

How would you represent the uncertainty 
in measuring distance, if there were signifi- 
cant errors here also? 

Acceleration, 28-30, 68, 70 

as a vector, 75 

average, 29 

centripetal, 109-113, 114-115 

constant (uniform), 29-30, 47-49, 
50. 51-52, 54, 58 

defined, 75 

direction and, 28, 29, 75 

due to gravity, 80 

force and, 79-80 

instantaneous, 29 

linear, 28 

mass and, 80-81 
Aether (see quintessence) 
Agena rocket, 80-89 
Alouette I (satellite), 114,. 116 
Angle of inclination, 54, 57 
Aristotelian cosmology, 47, 58-59 
Aristotle, 38-42, 46, 58, 59, 69-70 

air resistance, 40 

his theory of motion, 38, 40, 69-70 

his theory of motion attacked, 41, 
46, 47 

his theory of motion refuted, 58 

time line chart, 39 
Atom, diameter of, 6 
Atoms in the Family: My Life with 
Enrico Fermi (Laura Fermi), 

Brahe, Tycho, 119 

Centripetal acceleration 

force, 109-110 
Circular motion, 107-115 
Copernicus, Nicolaus, 119 

Aristotelian, 38 

medieval, 38 
Curie, Irene, 1 
Curie, Marie, 1 
Curie, Pierre, 1 

De Medici, Prince Giovanni, 53 

De Montbeillard, 24-25 

Delta (A), 17 

Dialogue on the Two Great World 

Systems (GalUeo), 43 

acceleration and, 28, 29, 75 
constant, 76 
of vectors, 74 
velocity and, 75 
Discourses and Mathematical 
Demonstrations Concerning 
Two New Sciences Pertaining 
to Mechanics and Local Motion 
(Galileo), 43-49, 53, 56-57, 60, 
Discoveries and Opinions of Galileo 
(tr. Drake), 104 


Distance, measuring, 13-15 
Dry ice, 11-15, 75-76 
Dynamics, 67 

"Dynamism of a Cyclist" (Boccioni), 


diameter of, 6 

motion of, 9 

precession of the axis of, 7 
Earth satellites, 113-116 
Einstein, Albert, 107 
Elements, Aristotle's four, 37-38 
Equilibrium, 73, 77 

forces in, 70-73 
Ether (see quintessence) 
Extrapolation. 22-23 

Fermi, Enrico, 1-5 
Fermi, Laura, 1-4 
Force(s), 67, 69-70, 81-83, 86-88 

acceleration and, 79-80 

centripetal, 109-113 

directional nature of, 71 

equilibrium in, 70-73 

friction al, 91 

nature's basic, 90-92 

net, 71, 72, 73 

resultant (see net force) 

total (see net force) 

vector sum of, 72 
Free fall, all Chapter 2, 83-86 

defined, 45 
Frame(s) of reference, 77-78, 105- 

Frequency of circular motion. 107, 

Friction, 91 

Galaxies, distance to, 6 
Galilean relativity principle, 106-107 
Galileo, 30, 36, all Chapter 2, 76-77, 
85, 100, 104. 105-107. 118. 119 

consequences of works of, 58-60 

Dialogue on the Two Great World 
Systems, 43 

idea of a straight line, 77 

time line chart, 42 

Two New Sciences, 43 
Gemini spacecraft, 88-89 
Graphs, 18-23, 24-25, 29 

extrapolations of, 22-23 

interpolations of, 22 
Gravity, 30, 83-86, 91-92 

Huygens, Christian, 57, 

Hypothesis, 59 
direct test of, 59-50 
explanations, 67-68 
indirect test of, 53-54, 
of Galileo, 49-50 



proven, 58 
Inclined plane, 54, 57 
Inertia, 77, 78, 80, 85 

and Newton's second law, 79-80 

law of, 77 

measured, 81-82 

principle of, 77 
International Bureau of Weights 

and Measures. 82 
Interpolation, 22 
Instantaneous acceleration, 29 
Instantaneous speed, 23 
Instantaneous velocity, 25 

Kepler, Johannes, 1 19 
Kilogram (unit of mass), 82-83 

concepts of, 67 

defined, 67 

of uniform circular motion, 114 

Laws of motion, 
Newton's 1st (inertia), 75-78 
2nd (force). 79-83 
3rd (reaction), 86-88 

Mach, E., 83, 85 
Magnitude of vectors, 74 
Mass, 67, 81-86 

acceleration and, 80-81 

compared with weight. 80. 84-85 

defined, 82 

force and, 81 

standard of, 82 
Mathematical Principles of Natu- 
ral Philosophy (Newton), 68, 69, 
Measurement, 5, 6, 7 

accuracy of, 14 

distance, 13-15 

mass, 82 

speed, 12-15 

time, 13-15. 56-57 

weight, 84 
Medieval world system, 37-38 
Meter, 82-83 
Midpoint, speed at, 24 
Moon, a trip to, 99-100 
Motion pictures, 26. 27 

Natural motion, 69 
Newton, (unit of force), 83 
Newton, Isaac, 59, 67-92, 98, 100, 
117, 118, 119 
first law of motion (inertia), 75-78, 

88-92, 90-91 
idea of a straight line, 77 
The Principia, 68 
second law of motion, (force). 79- 

83. 85, 88-92, 1 10 
third law of motion, (reaction), 86- 

Nucleus, diameter of, 6 


of Earth satellites, 113-116 
Oresme, Nicolas, 47 
Oscillation, 116-117 

Parabola, 104 

Parallelogram method of adding 

vectors, 74 
Parsimony, rule of, 48 
alpha, 1-2 

tracks in a bubble chamber, 7 
People and Particles (documentary 

film), 5 
Period of circular motion, 107, 108 
Philoponus, John, 41 
Philosophiae Naturalis Principia 
Mathematica (Newton), 68, 69, 
Photography, 11-12, 26-27 
Physics, definitions of, 5 
Plato, 38 

Projectiles, 101-107 
defined, 101 

trajectory of, 101, 103-105 
Pythagorean Theorem, 1 1 1 

Quintessence, 38 

Radioactivity, 1-4 

Raphael, 38 

Reference frame(s), 77-78, 105-106 

Relativity principle, Gahlean, 106- 

Relativity theory, 81, 85 
Rest, 68, 70, 73, 76, 77, 78 
Revolution, 107-109 

defined, 107 

frequency of, 107 

period of, 107 

Rockets, 88-89, 99-100 
Rotation, defined, 107 
Rule of Parsimony, 48 

Sagredo, 48 

Salviati, 48-49 

Simplicio, 47-48 

Satelhtes of the earth, 113-116 
Scalar quantities, 75 
Simple harmonic motion, 117 
Slope, of graph, 24, 25 

finding, 18-22 
absolute, 106 
acceleration and, 28-30 
average, 12, 15-17, 23, 24, 29, 108 

defined, 17 
constant (uniform), 15, 29, 47, 48, 

76, 77 
defined, 12 

distinguished from velocity, 25 
expressions of, 12 
instantaneous, 12, 23-27, 29, 108 
measuring, 12-15 
nonuniform, 15 
point, at a, 23 
relative, 106 
Speedometer(s), 12, 25 
Stars, distance to, 6 
Stroboscopic, 13-14 

photography, 27 
distance to, 6 
radius of, 21 
System of the World (Newton), 98 

Tangent of graph line, 29 
Time, 7 

measuring, 13-15, 56-57 
Time Line chart 

Aristotle, 39 

Galileo, 42 

Thought Experiment (Galileo's), 

Trajectory of a projectile, 101, 103- 

Two New Sciences (Galileo), 43-49, 

53, 56-57, 60, 104 

Unbalanced force, 79 
Uniform motion, 68. 70 
U.S. Bureau of Standards. 82 

Vacuum. 45-46 

Van Goph, Vincent, 42 

Vector, 73-75 

defined, 75 

direction of, 74 

displacement, 73 

magnitude of, 74 

resultant, 74 

sum of forces, 72 
Velocity, 98, 106 

average, 108 

circular motion, 107-108 

constant. 76 

distinguished from speed, 25 

frames of reference, 105-106 

instantaneous, 25 

two ways of changing. 75-78 

unchanging. 76, 77 

uniform, 70 
Verne, Jules, 99 
Violent motion, 69 

Water clock, 56-57 
Weight, 83-86 

compared with mass, 80, 84-85 

defined, 84 

measuring. 84 
Weightlessness. 84 
World of Enrico Fermi, The (docu- 
mentary film), 5 



Accelerated motion, 152 
centripetal, 181 

due to gravity— I (film loop), 164 
due to gravity— II (film loop), 165 
Acceleration (of gravity) 
from falling water drops, 161 
from a pendulum, 159-60 
measurement by direct fall, 

measurement by slow-motion 

photography, 160-61 
with falling ball and turntable, 

with strobe photography, 162 
Accelerometers (activity), 170-73 
automobile, 172-73 
calibration of , 172 
damped-pendulum, 173 
liquid-surface, 170-71 
ballistic cart projectile, 185-86 
beaker and hammer, 170 
checker snapping, 170 
experiencing Newton's second 

law, 170 
extrapolation, 163 
falling weights, 163 
make one of these accelerometers, 

making a frictionless puck, 151 
measuring unknown frequencies, 

measuring your reaction time, 

motion in a rotating reference 

frame, 185-86 
penny and coat hanger, 186 
photographing a waterdrop 

parabola, 184-85 
projectile motion demonstration, 

pulls and jerks, 170 
speed of a stream of water, 184 
using the electronic stroboscope, 

when is air resistance important, 
Air resistance 

importance of (activity), 163 

of object, 134 
Archytas, 129 
Astrolabe, 137 
naked eye (experiment), 134-41 
references in 6-7, 134-35 
Azimuth, 134, 137 

Ballistic cart projectiles (activity), 

Beaker and hammer (activity), 170 
Black Cloud, The, 133 
Big Dipper, 135 

Camera, Polaroid, 132 

Celestial Calendar and Handbook, 

Centripetal acceleration, 182 
Centripetal force (experiment), 181 
on a turntable (experiment), 

Checker snapping (activity), 170 
Compass, magnetic, 134 
Constant speed, 167 
Constellations, 135, 136 


recording of, 156 

variations in (experiment), 144 
Direct fall 

acceleration by, 158-59 

Earth satellite, 181 
Einstein, Albert, 169 
Experimental errors, 167 
a seventeenth century experiment 

centripetal force, 181 
centripetal force on a turntable, 

curves of trajectories, 176-78 
mass and weight, 169 
measuring the acceleration of 

gravity, 158-62 
measuring uniform motion, 145- 

naked eye astronomy, 134-41 
Newton's second law, 166-68 
prediction of trajectories, 179-80 
regularity and time, 142-43 
twentieth-century version of 
Galileo's experiment, 157 
variations in data, 144 
Extrapolation (activity), 163 

Falling weights (activity), 163 
Film loops 

a matter of relative motion, 187 

acceleration due to gravity I and 
II, 164-65 
analysis of a hurdle race I and II, 

Galilean relativity, 188-90 

vector addition, 174-75 
Free Fall 

approximation of, 152-56 

measuring unknown(activity), 186 

of test event, 142 

on a rotating disc, 182-83 

Galilean relativity (film loop), 189-91 

Galileo, 153, 156, 157 
his relativity (film loops), 189-91 
his Two New Sciences, 153, 189 

drawing, 150 


acceleration of , 158-62 
measuring acceleration of 
(experiment), 158-62 

How to Lie with Statistics, 163 
Hurdle race 
analysis of (film loops), 190-192 

Inertia, 169 
Instantaneous speed, 164 

Laboratory exercises 

keeping records of, 129, 130-31 
Little Dipper, 135 

Mach, Ernst, 169 
Magentic declination 

angle of, 134 

and weight (comparison), 169 

and weight (experiment), 169 

measuring, 169 
Measurement, precise, 150 

observation of, 139 
Meteor showers 

observation of (table), 141 

eclipse of, 139 

observation of, 138-39 

accelerated, 152 

in rotating reference frame 
(activity), 185-86 

relative (film loop), 187 

uniform measurement of 
(experiment), 145-50 

Newton, Isaac 
experiencing his second law 

(activity), 170 
his second law of motion 
(experiment), 166-68 
North-south line, 134-35, 137 
North Star (Polaris), 134-35 

Parabola, waterdrop 

photograph of (activity), 184-85 

acceleration from a, 159-60 
Penny and coat hanger (activity), 


of waterdrop parabola, 184-85 

slow-motion, 160-61 

stroboscopic, 132, 146, 162 
Physics Teacher, The, 173 

and eclipse observations (table), 

observation of, 139 
Polaris (North Star), 134-35 
Polaroid camera 

use of, 132 
Project Physics Reader, 133 

ballistic cart (activity), 185-86 
motion demonstration (activity), 
Making a frictionless (activity), 
Pulls and jerks (activity), 170 

Reaction time 

measurement of (activity), 163 

in astronomy, 134-35 

North-south line, 134-135, 137 

and time (experiment), 142-43 

of an event, 142 

Satellite, earth, 181 
Seventeenth-century experiment, 

Sky and Telescope, 139 


and measurement of motion, 

constant, 166 

instantaneous, 164 
Standard event, 142 

chart of, 136 

observation of, 139 
Stroboscope, electronic (activity), 

Stroboscopic photography, 132, 

146, 162 

observation of, 138 

f avorability of observing meteor 

showers, 141 
guide for planet and eclipse 

observations, 140 


and regularity (experiment), 

curves of (experiment), 176-78 

prediction of (experiment), 
Twentieth-century version 

of Galileo's experiment, 157 
Two New Sciences, 153, 188 

drawing by, 180 

addition of (film loop), 174-75 
diagrams, 174, 175 

Water clock, 153-56 
and mass (experiment), 169 


Answers to End of Section Question 

Chapter 1 

Q1 We have no way of knowing the lengths of time 

involved in going the observed distances. 

Q2 No; the time between stroboscope flashes is 

constant and the distance intervals shown are not 


Q3 An object has a uniform speed if it travels 

equal distances in equal time intervals; or, if the 

distance traveled = constant, regardless of the 

particular distances and times chosen. 

Q4 Average speed is equal to the distance travelled 

divided by the elapsed time while going that 



(entries in brackets are those 
already given in the text) 

Q6 Hint: to determine location of left edge of puck 
relative to readings on the meter stick, line up a 
straight edge with the edge of puck and both marks 
on meter stick corresponding to a given reading. 


Ad /At 































5 vds 
— = 0.6 yd/sec from the table 

Q7 The one on the left has the larger slope 

mathematically; it corresponds to 100 miles/hr 

whereas the one on the right corresponds to 50 


Q8 Most rapidly at the beginning when the slope 

is steepest; most slowly toward the end where the 

slope is most shallow. 

Ad 2.5 yds. 

aF " 4 sec " yd/sec from the graph 


At 8.6 sec 
Q10 Interpolation means estimating values 
between data points; extrapolation means estimating 
values beyond data points. 

Q11 An estimate for an additional lap (extrapola- 

Q12 Instantaneous speed means the limit 
approached by the average speed as the time interval 
involved gets smaller and smaller. 


V = limit ^r- as At approaches zero. 


Q13 Instantaneous speed is just a special case of 
average speed in which the ratio Ad/Af does not 
change as Af is made smaller and smaller. However, 
Ad/ At always gives average speed no matter how 
large or how small Af is. 

_ final speed - initial speed _ 60 — mph 
time elapsed 5 sec 

= 12 mph/sec 



2 mph — 4 mph _ 

-8 mph/hr, or -0.13 

1/4 hr 
No, not since average is specified. 

Chapter 2 

Q1 Composition: terrestrial objects are composed 

of combinations of earth, water, air and fire; celestial 

objects of nothing but a unique fifth element. 

Motion: terrestrial objects seek their natural positions 

of rest depending on their relative contents of 

earth (heaviest), water, air and fire (lightest); 

celestial objects moved endlessly in circles. 

Q2 (a), (b), and (c) 

Q3 Aristotle: the nail is heavier than the toothpick 

so it falls faster. 

Galileo: air resistance slows down the toothpick 

more than the nail. 

04 See Q3 of Chapter 1 p. 15 


Q5 An object is uniformly accelerated if its speed 

increases by equal amounts during equal time 

intervals. Av/At = constant 

Q6 The definition should (1) be mathematically 

simple and (2) correspond to actual free fall motion. 

07 (b) 

Q8 Distances are relatively easy to measure as 

compared with speeds; measuring short time 

intervals remained a problem, however. 

Q9 The expression d = v t can only be used if v is 

constant. The second equation refers to accelerated 

motion in which v is not constant. Therefore the two 

equations cannot be applied to the same event. 

Q10 (c) and (e) 

Oil (d) 

Q12 (a), (c) and (d) 

Chapter 3 

Q1 kinematic — (a), (b), (d) 

dynamic — (c), (e) 

Q2 A continuously applied force 

Q3 The air pushed aside by the puck moves around 

to fill the space left behind the puck as it moves 

along and so provides the propelling force needed. 

Q4 The force of gravity downward and an upward 

force of equal size exerted by the table. 

The sum of the forces must be zero because the 

vase is not accelerating. 

Q5 The first three. 

Q6 No, in many cases equilibrium involves 

frictional forces which depend on the fact that the 

object is in motion. 

Q7 Vector quantities (1) have magnitude and 


(2) can be represented graphically by arrows 

(3) can be combined to form a single resultant vector 
by using either the head to tail or the parallelogram 
method. (Note: only vectors of the same kind are 
combined in this way; that is, we add force vectors 
to force vectors, not force vectors to velocity 
vectors, for example.) 

Q8 Direction is now taken into account, (we must 

now consider a change of direction to be as valid a 

case of acceleration as speeding up or slowing 


Q9 W downward, 0,0,0 

Q10 Galileo's "straight line forever " motion may 

have meant at a constant height above the earth 

whereas Newton's meant moving in a straight line 

through empty space. 

Q11 Meter, Kilogram and Second 


10 N 

= L = 
a 4m/sec^ 

2.5 kg 

Q13 False; (frictional forces must be taken into 
account in determining the actual net force 

Q14 Acceleration = 

— 10 m/sec 
5 sec 

= —2 m/sec'^ 

Force = ma = 2 kg x (-2 m/sec^) = -4 Newtons 


(the minus sign arises because the force and the 

acceleration are opposite in direction to the original 

motion. Since the question asks only for the 

magnitude of the force it may be disregarded.) 

Q15 10 m/sec^ 

150 m/sec- 

60 m/sec^ 

0.67 m/seC 

10 m 

0.4 m 

Q16 (c) and (f) 

Q17 (e) and (f) 

Q18 (1) appear in pairs 

(2) are equal in magnitude 

(3) opposite in direction 

(4) act on two different objects 

Q19 The horse pushes against the earth, the earth 
pushes against the horse causing the horse to 
accelerate forward. (The earth accelerates also but 
can you measure it?) The swimmer pushes backward 
against the water; the water, according to the third 
law, pushes forward against the swimmer; however, 
there is also a backward frictional force of drag 
exerted by the water on the swimmer. The two 
forces acting on the swimmer add up to zero, since 
he is not accelerating. 

Q20 No, the force "pulling the string apart " is still 
only 300 N; the 500 N would have to be exerted at 
both ends to break the line. 
Q21 See text p. 68 

Chapter 4 

Q1 The same acceleration a^, its initial horizontal 

speed has no effect on its vertical accelerated 


Q2 (a), (c) and (e) 

Q3 They must be moving with a uniform speed 

relative to each other. 

Q4 (a) T = 1/f = 1/45 = 2.2 X 10-= minutes 

(b) 2.2 X ^0~^ minutes x 60 seconds/minute 
= 1.32 sec. 

(c) f = 45 rpm x 1/60 minutes/sec = 0.75 rps 
Q5 T = 1 hour = 60 minutes 

277-R _ 2 X 3.14 X3 
T 60 

= .31 cm/minute 

Q6 f = 80 vibrations/minute = 1.3 vib/sec 

T = 1/f = 1/1.3 = .75 sec 
Q7 (a) and (b) 

Q8 Along a tangent to the wheel at the point where 
the piece broke loose. 



Q10 A-rrmR 

Q11 The value of the gravitational acceleration and 
the radius of the moon (to which 70 miles is added 
to determine R). 

Brief Answers to Study Guide 

Chapter 1 

1.1 Information 

1.2 (a) discussion (b) 58.3 mph (c) 
discussion (d) discussion (e) 

1.3 (a) 6 cm/sec (b) 15 mi. (c) 0.25 
min. (d) 3 cm/sec 24 cm (e) 30 mi/hr 
(f) 30 mi/hr? 120 mi? (g) 5.5 sec (h) 
8.8 m 

1.4 22xl03mi 

1.5 (a) 9.5 X 10'-^ m (b) 2.7 x 10" sec 
or 8.5 years 

1.6 1.988 mph or 2 mph 

1.7 (a) 1.7 m/sec (b) 3.0 m/sec 

1.8 discussion 

1.9 discussion 
.10 discussion 

1.11 (a) 0.5, 1.0, 1.5, and 2.0 (b) 

1.12 Answer 

1.13 25.6 meters; 4:00 for men, 4:30 
for women 

1.14 discussion 

1.15 graph 

1.16 graphs 

d vs f: d = 0,9,22,39,5,60.5,86cm 
(approx) at intervals of 0.2 sec 

vwst:v = 45,65,87.5,105,127 cm/sec 
(approx) at intervals of 0.2 sec 

1.17 (a) Between 1 and 4.5 sec; 1.3 
m/sec (b) 0.13 m/sec (c) 0.75 m/sec 
(d) 1.0 m/sec (e) 0.4 m (approx) 

1.18 (a) 14.1 m/sec (b) 6.3 m/sec^ 

1.19 315,000 in/sec 

1.20 discussion 

1.21 discussion 

Chapter 2 

2.1 Information 

2.2 discussion 

2.3 discussion 

2.4 discussion 

2.5 discussion 

2.6 discussion 

2.7 proof 

2.8 (a), (b), (c) 

2.9 discussion 

2.10 discussion 

2.11 proof 

2.12 17 years $7000 

2.13 discussion 

2.14 (a) 57 m/sec- (b) 710 m (c) 
-190 m/sec^ 

2.15 proof 

2.16 discussion 

2.17 (a) true (b) true (based on 
measurements of 6 lower positions) 
(c) true (d) true (e) true 

2.18 proof 

2.19 (a) Position 















(b) proof (c) discussion 

2.20 discussion 

2.21 (a) 5.0 m (b) 10 m/sec (c) 15 m 

2.22 (a) 10 m/sec (b) 15 m (c) 2 sec 
(d) 20 m (e) -20 m/sec 

2.23 (a) 20 m/sec (b) -20 m/sec (c) 
4 sec (d) 80 m (e) mi/sec (f) -40 

2.24 (a) -2 m/sec- (b) 2 m/sec (c) 
2 m/sec (d) 4 m (e) —2 m/sec 

(f) 4 sec 

2.25 discussion 

2.26 (a) 4.3 welfs/surg2 (b) 9.8 

2.27 proof 

2.28 proof 

2.29 proof 

2.30 discussion 

2.31 discussion 

2.32 discussion 

2.33 discussion 

Chapter 3 

3.1 information 

3.2 discussion 

3.3 (a) construction (b) 2.4 units, 

3.4 proof 

3.5 discussion 

3.6 discussion 

3.7 discussion 

3.8 discussion 

3.9 discussion 

3.10 discussion 

3.11 discussion 
3.12- 2.8 X 10-^ hr/sec 

3.13 6/1 

3.14 discussion 

3.15 discussion 

3.16 discussion 

3.17 proof 

3.18 discussion 

3.19 (c) 24N (d) 14.8N (e) 0.86N (f) 
9.0 Kg (g) 0.30 Kg (h) 0.20 Kg (i) 3 
m/sec^ (j) 2.5 m/sec- (k) 2.50 m/sec- 

3.20 (a) 2.0 X 10^ m/sec^ 7.8 x 10'^ 
m/sec (b) discussion (c) 2.4 x 10- 

3.21 discussion 

3.22 discussion 

3.23 2.0 Kg 

3.24 discussion 

3.25 discussion 

3.26 (a) 1 Kg. 9.81 N in Paris, 9.80N 
in Washington (b) individual 

3.27 individual calculation 

3.28 discussion 

3.29 (a) -5 X lO-" m/sec^ (b) 10 
m/sec (c) 1 X 10"" m/sec 

3.30 discussion 

3.31 discussion 

3.32 (a) diagram (b) 1.7 X lO-^-" 

m/sec^ (c) ^ ^^^'' (d) diagram 

3.33 (a) 862N, 750N, 638N (b) The 
same as in (a) for scale calibrated in 
Newtons (c) discussion 

3.34 hints for solving motion 

Chapter 4 

4.1 Information 

4.2 13.6 m/sec-; 2.71 sec; mass 

4.3 discussion 

4.4 derivation 

4.5 proof 

4.6 1 .3 m; at an angle of 67° below 
the horizontal; 5.1 m/sec, 78° below 
the horizontal 

4.7 discussion 

4.8 discussion 

4.9 discussion 

4.10 discussion 

4.11 6.0 X 10-2 min, 3.0 x lO"'^ min, 
1.3 X 10"- min 

4.12 (a) 1.9 sec (b) 32 rpm (c) 50 
cm/sec (d) 35 cm/sec (e) (f) 
190Vsec, yes (g) 120 cm/sec^ (h) 160 
cm/sec'^ (i) discussion 

4.13 discussion 

4.14 discussion 

4.15 table completion 

4.16 (a) 2.2 X 10"'" m/secMb) 4 X 
10-" N (c) approximately 1/100 

4.17 approximately 10'* N 

4.18 discussion 

4.19 (a) Syncom 2 (b) Lunik 3 (c) 
Luna 4 (d) dosent change 

4.20 5.1 X 10' sec or 85 min 
7.9 X 10^ m/sec 

4.21 discussion 

4.22 7.1 X 10^ sec or 120 min 

4.23 (a) 3.6 x 10^ sec (b) 36 Km (c) 

4.24 t = (m/F)(Vo - V) 

4.25 discussion 

4.26 essay