motion in one dimension - central oregon...

Motion in one dimension

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a / Rotation.

b / Simultaneous rotation andmotion through space.

c / One person might say that thetipping chair was only rotating ina circle about its point of contactwith the floor, but another coulddescribe it as having both rotationand motion through space.

Chapter 2

Velocity and relativemotion

2.1 Types of motionIf you had to think consciously in order to move your body, youwould be severely disabled. Even walking, which we consider tobe no great feat, requires an intricate series of motions that yourcerebrum would be utterly incapable of coordinating. The task ofputting one foot in front of the other is controlled by the more prim-itive parts of your brain, the ones that have not changed much sincethe mammals and reptiles went their separate evolutionary ways.The thinking part of your brain limits itself to general directivessuch as “walk faster,” or “don’t step on her toes,” rather than mi-cromanaging every contraction and relaxation of the hundred or somuscles of your hips, legs, and feet.

Physics is all about the conscious understanding of motion, butwe’re obviously not immediately prepared to understand the mostcomplicated types of motion. Instead, we’ll use the divide-and-conquer technique. We’ll first classify the various types of motion,and then begin our campaign with an attack on the simplest cases.To make it clear what we are and are not ready to consider, we needto examine and define carefully what types of motion can exist.

Rigid-body motion distinguished from motion that changesan object’s shape

Nobody, with the possible exception of Fred Astaire, can simplyglide forward without bending their joints. Walking is thus an ex-ample in which there is both a general motion of the whole objectand a change in the shape of the object. Another example is themotion of a jiggling water balloon as it flies through the air. We arenot presently attempting a mathematical description of the way inwhich the shape of an object changes. Motion without a change inshape is called rigid-body motion. (The word “body” is often usedin physics as a synonym for “object.”)

Center-of-mass motion as opposed to rotation

A ballerina leaps into the air and spins around once before land-ing. We feel intuitively that her rigid-body motion while her feetare off the ground consists of two kinds of motion going on simul-

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e / No matter what point youhang the pear from, the stringlines up with the pear’s centerof mass. The center of masscan therefore be defined as theintersection of all the lines madeby hanging the pear in this way.Note that the X in the figureshould not be interpreted asimplying that the center of massis on the surface — it is actuallyinside the pear.

f / The circus performers hangwith the ropes passing throughtheir centers of mass.

taneously: a rotation and a motion of her body as a whole throughspace, along an arc. It is not immediately obvious, however, whatis the most useful way to define the distinction between rotationand motion through space. Imagine that you attempt to balance achair and it falls over. One person might say that the only motionwas a rotation about the chair’s point of contact with the floor, butanother might say that there was both rotation and motion downand to the side.

d / The leaping dancer’s motion is complicated, but the motion ofher center of mass is simple.

It turns out that there is one particularly natural and useful wayto make a clear definition, but it requires a brief digression. Everyobject has a balance point, referred to in physics as the center ofmass. For a two-dimensional object such as a cardboard cutout, thecenter of mass is the point at which you could hang the object froma string and make it balance. In the case of the ballerina (who islikely to be three-dimensional unless her diet is particularly severe),it might be a point either inside or outside her body, dependingon how she holds her arms. Even if it is not practical to attach astring to the balance point itself, the center of mass can be definedas shown in figure e.

Why is the center of mass concept relevant to the question ofclassifying rotational motion as opposed to motion through space?As illustrated in figures d and g, it turns out that the motion of anobject’s center of mass is nearly always far simpler than the motionof any other part of the object. The ballerina’s body is a large objectwith a complex shape. We might expect that her motion would bemuch more complicated than the motion of a small, simply-shaped

60 Chapter 2 Velocity and relative motion

h / An improperly balancedwheel has a center of mass thatis not at its geometric center.When you get a new tire, themechanic clamps little weights tothe rim to balance the wheel.

i / This toy was intentionallydesigned so that the mushroom-shaped piece of metal on topwould throw off the center ofmass. When you wind it up, themushroom spins, but the centerof mass doesn’t want to move,so the rest of the toy tends tocounter the mushroom’s motion,causing the whole thing to jumparound.

object, say a marble, thrown up at the same angle as the angle atwhich she leapt. But it turns out that the motion of the ballerina’scenter of mass is exactly the same as the motion of the marble. Thatis, the motion of the center of mass is the same as the motion theballerina would have if all her mass was concentrated at a point. Byrestricting our attention to the motion of the center of mass, we cantherefore simplify things greatly.

g / The same leaping dancer, viewed from above. Her center ofmass traces a straight line, but a point away from her center of mass,such as her elbow, traces the much more complicated path shown by thedots.

We can now replace the ambiguous idea of “motion as a wholethrough space” with the more useful and better defined conceptof “center-of-mass motion.” The motion of any rigid body can becleanly split into rotation and center-of-mass motion. By this defini-tion, the tipping chair does have both rotational and center-of-massmotion. Concentrating on the center of mass motion allows us tomake a simplified model of the motion, as if a complicated objectlike a human body was just a marble or a point-like particle. Sciencereally never deals with reality; it deals with models of reality.

Note that the word “center” in “center of mass” is not meantto imply that the center of mass must lie at the geometrical centerof an object. A car wheel that has not been balanced properly hasa center of mass that does not coincide with its geometrical center.An object such as the human body does not even have an obviousgeometrical center.

It can be helpful to think of the center of mass as the averagelocation of all the mass in the object. With this interpretation,we can see for example that raising your arms above your headraises your center of mass, since the higher position of the arms’mass raises the average. We won’t be concerned right now withcalculating centers of mass mathematically; the relevant equationsare in chapter 4 of Conservation Laws.

Ballerinas and professional basketball players can create an illu-sion of flying horizontally through the air because our brains intu-itively expect them to have rigid-body motion, but the body doesnot stay rigid while executing a grand jete or a slam dunk. The legsare low at the beginning and end of the jump, but come up higher at

Section 2.1 Types of motion 61

j / A fixed point on the dancer’s body follows a trajectory that is flat-ter than what we expect, creating an illusion of flight.

the middle. Regardless of what the limbs do, the center of mass willfollow the same arc, but the low position of the legs at the beginningand end means that the torso is higher compared to the center ofmass, while in the middle of the jump it is lower compared to thecenter of mass. Our eye follows the motion of the torso and triesto interpret it as the center-of-mass motion of a rigid body. Butsince the torso follows a path that is flatter than we expect, thisattempted interpretation fails, and we experience an illusion thatthe person is flying horizontally.

k / Example 1.

The center of mass as an average example 1. Explain how we know that the center of mass of each object isat the location shown in figure k.

. The center of mass is a sort of average, so the height of thecenters of mass in 1 and 2 has to be midway between the twosquares, because that height is the average of the heights of thetwo squares. Example 3 is a combination of examples 1 and2, so we can find its center of mass by averaging the horizontalpositions of their centers of mass. In example 4, each squarehas been skewed a little, but just as much mass has been movedup as down, so the average vertical position of the mass hasn’tchanged. Example 5 is clearly not all that different from example4, the main difference being a slight clockwise rotation, so just as


l / The high-jumper’s bodypasses over the bar, but hiscenter of mass passes under it.

m / Self-check B.

in example 4, the center of mass must be hanging in empty space,where there isn’t actually any mass. Horizontally, the center ofmass must be between the heels and toes, or else it wouldn’t bepossible to stand without tipping over.

Another interesting example from the sports world is the highjump, in which the jumper’s curved body passes over the bar, butthe center of mass passes under the bar! Here the jumper lowers hislegs and upper body at the peak of the jump in order to bring hiswaist higher compared to the center of mass.

Later in this course, we’ll find that there are more fundamentalreasons (based on Newton’s laws of motion) why the center of massbehaves in such a simple way compared to the other parts of anobject. We’re also postponing any discussion of numerical methodsfor finding an object’s center of mass. Until later in the course, wewill only deal with the motion of objects’ centers of mass.

Center-of-mass motion in one dimension

In addition to restricting our study of motion to center-of-massmotion, we will begin by considering only cases in which the centerof mass moves along a straight line. This will include cases suchas objects falling straight down, or a car that speeds up and slowsdown but does not turn.

Note that even though we are not explicitly studying the morecomplex aspects of motion, we can still analyze the center-of-massmotion while ignoring other types of motion that might be occurringsimultaneously . For instance, if a cat is falling out of a tree andis initially upside-down, it goes through a series of contortions thatbring its feet under it. This is definitely not an example of rigid-body motion, but we can still analyze the motion of the cat’s centerof mass just as we would for a dropping rock.

self-check AConsider a person running, a person pedaling on a bicycle, a personcoasting on a bicycle, and a person coasting on ice skates. In whichcases is the center-of-mass motion one-dimensional? Which cases areexamples of rigid-body motion? . Answer, p. 523

self-check BThe figure shows a gymnast holding onto the inside of a big wheel.From inside the wheel, how could he make it roll one way or the other?. Answer, p. 523

2.2 Describing distance and timeCenter-of-mass motion in one dimension is particularly easy to dealwith because all the information about it can be encapsulated in twovariables: x, the position of the center of mass relative to the origin,and t, which measures a point in time. For instance, if someone

Section 2.2 Describing distance and time 63

supplied you with a sufficiently detailed table of x and t values, youwould know pretty much all there was to know about the motion ofthe object’s center of mass.

A point in time as opposed to duration

In ordinary speech, we use the word “time” in two differentsenses, which are to be distinguished in physics. It can be used,as in “a short time” or “our time here on earth,” to mean a lengthor duration of time, or it can be used to indicate a clock reading, asin “I didn’t know what time it was,” or “now’s the time.” In sym-bols, t is ordinarily used to mean a point in time, while ∆t signifiesan interval or duration in time. The capital Greek letter delta, ∆,means “the change in...,” i.e. a duration in time is the change ordifference between one clock reading and another. The notation ∆tdoes not signify the product of two numbers, ∆ and t, but ratherone single number, ∆t. If a matinee begins at a point in time t = 1o’clock and ends at t = 3 o’clock, the duration of the movie was thechange in t,

∆t = 3 hours− 1 hour = 2 hours .

To avoid the use of negative numbers for ∆t, we write the clockreading “after” to the left of the minus sign, and the clock reading“before” to the right of the minus sign. A more specific definitionof the delta notation is therefore that delta stands for “after minusbefore.”

Even though our definition of the delta notation guarantees that∆t is positive, there is no reason why t can’t be negative. If tcould not be negative, what would have happened one second beforet = 0? That doesn’t mean that time “goes backward” in the sensethat adults can shrink into infants and retreat into the womb. Itjust means that we have to pick a reference point and call it t = 0,and then times before that are represented by negative values of t.An example is that a year like 2007 A.D. can be thought of as apositive t value, while one like 370 B.C. is negative. Similarly, whenyou hear a countdown for a rocket launch, the phrase “t minus tenseconds” is a way of saying t = −10 s, where t = 0 is the time ofblastoff, and t > 0 refers to times after launch.

Although a point in time can be thought of as a clock reading, itis usually a good idea to avoid doing computations with expressionssuch as “2:35” that are combinations of hours and minutes. Timescan instead be expressed entirely in terms of a single unit, such ashours. Fractions of an hour can be represented by decimals ratherthan minutes, and similarly if a problem is being worked in termsof minutes, decimals can be used instead of seconds.

self-check COf the following phrases, which refer to points in time, which refer totime intervals, and which refer to time in the abstract rather than as ameasurable number?


(1) “The time has come.”

(2) “Time waits for no man.”

(3) “The whole time, he had spit on his chin.” . Answer, p. 523

The Leibniz notation and infinitesimals

∆ is the Greek version of “D,” suggesting that there is a rela-tionship between ∆t and the notation dt from calculus. The “d”notation was invented by Leibniz around 1675 to suggest the word“difference.” The idea was that a dt would be like a ∆t that wasextremely small — smaller than any real number, and yet greaterthan zero. These infinitestimal numbers were the way the world’sgreatest mathematicians thought about calculus for the next twohundred years. For example, dy/dx meant the number you got whenyou divided dy by dx. The use of infinitesimal numbers was seenas a natural part of the process of generalization that had alreadyseen the invention of fractions and irrational numbers by the ancientGreeks, zero and negative numbers in India, and complex numbersin Renaissance Italy. By the end of the 19th century, mathemati-cians had begun making formal mathematical descriptions of num-ber systems, and they had succeeded in making nice tidy schemesout of all of these categories except for infinitesimals. Having runinto a brick wall, they decided to rebuild calculus using the notionof a limit. Depending on when and where you got your education incalculus, you may have been warned severely that dy and dx werenot numbers, and that dy/dx didn’t mean dividing one by another.

But in the 1960’s, the logician Abraham Robinson at Yale provedthat infinitesimals could be tamed and domesticated; they were nomore self-contradictory than negative numbers or fractions. Thereis a handy rule for making sure that you don’t come to incorrectconclusions by using infinitesimals. The rule is that you can applyany axiom of the real number system to infinitesimals, and the resultwill be correct, provided that the statement can be put in a formlike “for any number . . . ,” but not “for any set of numbers . . . ” Wecarry over the axiom, reinterpreting “number” to mean any memberof the enriched number system that includes both the real numbersand the infinitesimals.

Logic and infinitesimals example 2There is an axiom of the real number system that for any numbert , t + 0 = t . This applies to infinitesimals as well, so that dt + 0 =dt .

Position as opposed to change in position

As with time, a distinction should be made between a pointin space, symbolized as a coordinate x, and a change in position,symbolized as ∆x.

As with t,x can be negative. If a train is moving down the

Section 2.2 Describing distance and time 65

tracks, not only do you have the freedom to choose any point alongthe tracks and call it x = 0, but it’s also up to you to decide whichside of the x = 0 point is positive x and which side is negative x.

Since we’ve defined the delta notation to mean “after minusbefore,” it is possible that ∆x will be negative, unlike ∆t which isguaranteed to be positive. Suppose we are describing the motionof a train on tracks linking Tucson and Chicago. As shown in thefigure, it is entirely up to you to decide which way is positive.

n / Two equally valid ways of de-scribing the motion of a train fromTucson to Chicago. In example 1,the train has a positive ∆x as itgoes from Enid to Joplin. In 2,the same train going forward inthe same direction has a negative∆x .

Note that in addition to x and ∆x, there is a third quantity wecould define, which would be like an odometer reading, or actualdistance traveled. If you drive 10 miles, make a U-turn, and driveback 10 miles, then your ∆x is zero, but your car’s odometer readinghas increased by 20 miles. However important the odometer readingis to car owners and used car dealers, it is not very important inphysics, and there is not even a standard name or notation for it.The change in position, ∆x, is more useful because it is so mucheasier to calculate: to compute ∆x, we only need to know the be-ginning and ending positions of the object, not all the informationabout how it got from one position to the other.

self-check DA ball falls vertically, hits the floor, bounces to a height of one meter,falls, and hits the floor again. Is the ∆x between the two impacts equalto zero, one, or two meters? . Answer, p. 523

Frames of reference

The example above shows that there are two arbitrary choicesyou have to make in order to define a position variable, x. You haveto decide where to put x = 0, and also which direction will be posi-tive. This is referred to as choosing a coordinate system or choosinga frame of reference. (The two terms are nearly synonymous, butthe first focuses more on the actual x variable, while the second ismore of a general way of referring to one’s point of view.) As long asyou are consistent, any frame is equally valid. You just don’t wantto change coordinate systems in the middle of a calculation.


o / Motion with constant ve-locity.

p / Motion that decreases xis represented with negativevalues of ∆x and v .

q / Motion with changing ve-locity. How can we find thevelocity at the time indicated bythe dot?

Have you ever been sitting in a train in a station when suddenlyyou notice that the station is moving backward? Most people woulddescribe the situation by saying that you just failed to notice thatthe train was moving — it only seemed like the station was moving.But this shows that there is yet a third arbitrary choice that goesinto choosing a coordinate system: valid frames of reference candiffer from each other by moving relative to one another. It mightseem strange that anyone would bother with a coordinate systemthat was moving relative to the earth, but for instance the frame ofreference moving along with a train might be far more convenientfor describing things happening inside the train.

2.3 Graphs of motion; velocity

Motion with constant velocity

In example o, an object is moving at constant speed in one di-rection. We can tell this because every two seconds, its positionchanges by five meters.

In algebra notation, we’d say that the graph of x vs. t showsthe same change in position, ∆x = 5.0 m, over each interval of∆t = 2.0 s. The object’s velocity or speed is obtained by calculatingv = ∆x/∆t = (5.0 m)/(2.0 s) = 2.5 m/s. In graphical terms, thevelocity can be interpreted as the slope of the line. Since the graphis a straight line, it wouldn’t have mattered if we’d taken a longertime interval and calculated v = ∆x/∆t = (10.0 m)/(4.0 s). Theanswer would still have been the same, 2.5 m/s.

Note that when we divide a number that has units of meters byanother number that has units of seconds, we get units of metersper second, which can be written m/s. This is another case wherewe treat units as if they were algebra symbols, even though they’renot.

In example p, the object is moving in the opposite direction: astime progresses, its x coordinate decreases. Recalling the definitionof the ∆ notation as “after minus before,” we find that ∆t is stillpositive, but ∆x must be negative. The slope of the line is thereforenegative, and we say that the object has a negative velocity, v =∆x/∆t = (−5.0 m)/(2.0 s) = −2.5 m/s. We’ve already seen thatthe plus and minus signs of ∆x values have the interpretation oftelling us which direction the object moved. Since ∆t is alwayspositive, dividing by ∆t doesn’t change the plus or minus sign, andthe plus and minus signs of velocities are to be interpreted in thesame way. In graphical terms, a positive slope characterizes a linethat goes up as we go to the right, and a negative slope tells us thatthe line went down as we went to the right.

Section 2.3 Graphs of motion; velocity 67

Motion with changing velocity

Now what about a graph like figure q? This might be a graphof a car’s motion as the driver cruises down the freeway, then slowsdown to look at a car crash by the side of the road, and then speedsup again, disappointed that there is nothing dramatic going on suchas flames or babies trapped in their car seats. (Note that we arestill talking about one-dimensional motion. Just because the graphis curvy doesn’t mean that the car’s path is curvy. The graph is notlike a map, and the horizontal direction of the graph represents thepassing of time, not distance.)

If we apply the equation v = ∆x/∆t to this example, we willget the wrong answer, because the ∆x/∆t gives a single number,but the velocity is clearly changing. This is an example of a goodgeneral rule that tells you when you need to use your differentialcalculus. Any time a rate of change is measured by an expression ofthe form ∆ . . . /∆ . . ., the result will only be right when the rate ofchange is constant. When the rate of change is varying, we need togeneralize the expression by making it into a derivative. Just as aninfinitesimally small1 ∆t is notated dt, an infinitesimally small ∆xis a dx. The velocity is then the derivative dx/dt.

Units of velocity example 3. Verify that the units of v = dx/dt make sense.

. We expect the velocity to have units of meters per second, andit does come out to have those units, since dx has units of me-ters and dt seconds. This ability to check the units of derivativesis one of the main reason that Leibniz designed his notation forderivatives the way he did.

An insect pest example 4. An insect pest from the United States is inadvertently releasedin a village in rural China. The pests spread outward at a rateof s kilometers per year, forming a widening circle of contagion.Find the number of square kilometers per year that become newlyinfested. Check that the units of the result make sense. Interpretthe result.

. Let t be the time, in years, since the pest was introduced. Theradius of the circle is r = st , and its area is a = πr2 = π(st)2. Thederivative is

dadt

= (2πs2)t

The units of s are km/year, so squaring it gives km2/year2. The 2and the π are unitless, and multiplying by t gives units of km2/year,which is what we expect for da/dt , since it represents the numberof square kilometers per year that become infested.

1see p. 65


Interpreting the result, we notice a couple of things. First, the rateof infestation isn’t constant; it’s proportional to t , so people mightnot pay so much attention at first, but later on the effort requiredto combat the problem will grow more and more quickly. Second,we notice that the result is proportional to s2. This suggests thatanything that could be done to reduce s would be very helpful.For instance, a measure that cut s in half would reduce da/dt bya factor of four.

2.4 The principle of inertiaPhysical effects relate only to a change in velocity

Consider two statements of a kind that was at one time madewith the utmost seriousness:

People like Galileo and Copernicus who say the earth is ro-tating must be crazy. We know the earth can’t be moving.Why, if the earth was really turning once every day, then ourwhole city would have to be moving hundreds of leagues inan hour. That’s impossible! Buildings would shake on theirfoundations. Gale-force winds would knock us over. Treeswould fall down. The Mediterranean would come sweepingacross the east coasts of Spain and Italy. And furthermore,what force would be making the world turn?

All this talk of passenger trains moving at forty miles an houris sheer hogwash! At that speed, the air in a passenger com-partment would all be forced against the back wall. People inthe front of the car would suffocate, and people at the backwould die because in such concentrated air, they wouldn’t beable to expel a breath.

Some of the effects predicted in the first quote are clearly justbased on a lack of experience with rapid motion that is smooth andfree of vibration. But there is a deeper principle involved. In eachcase, the speaker is assuming that the mere fact of motion musthave dramatic physical effects. More subtly, they also believe that aforce is needed to keep an object in motion: the first person thinksa force would be needed to maintain the earth’s rotation, and thesecond apparently thinks of the rear wall as pushing on the air tokeep it moving.

Common modern knowledge and experience tell us that thesepeople’s predictions must have somehow been based on incorrectreasoning, but it is not immediately obvious where the fundamentalflaw lies. It’s one of those things a four-year-old could infuriateyou by demanding a clear explanation of. One way of getting atthe fundamental principle involved is to consider how the modernconcept of the universe differs from the popular conception at thetime of the Italian Renaissance. To us, the word “earth” implies

Section 2.4 The principle of inertia 69

s / Why does Aristotle lookso sad? Has he realized thathis entire system of physics iswrong?

t / The earth spins. Peoplein Shanghai say they’re at restand people in Los Angeles aremoving. Angelenos say the sameabout the Shanghainese.

u / The jets are at rest. TheEmpire State Building is moving.

a planet, one of the nine planets of our solar system, a small ballof rock and dirt that is of no significance to anyone in the universeexcept for members of our species, who happen to live on it. ToGalileo’s contemporaries, however, the earth was the biggest, mostsolid, most important thing in all of creation, not to be comparedwith the wandering lights in the sky known as planets. To us, theearth is just another object, and when we talk loosely about “howfast” an object such as a car “is going,” we really mean the car-object’s velocity relative to the earth-object.

r / This Air Force doctor volunteered to ride a rocket sled as a medicalexperiment. The obvious effects on his head and face are not because ofthe sled’s speed but because of its rapid changes in speed: increasingin 2 and 3, and decreasing in 5 and 6. In 4 his speed is greatest, butbecause his speed is not increasing or decreasing very much at thismoment, there is little effect on him.

Motion is relative

According to our modern world-view, it really isn’t that reason-able to expect that a special force should be required to make theair in the train have a certain velocity relative to our planet. Afterall, the “moving” air in the “moving” train might just happen tohave zero velocity relative to some other planet we don’t even knowabout. Aristotle claimed that things “naturally” wanted to be atrest, lying on the surface of the earth. But experiment after exper-iment has shown that there is really nothing so special about beingat rest relative to the earth. For instance, if a mattress falls out ofthe back of a truck on the freeway, the reason it rapidly comes to


Discussion question A.

Discussion question B.

rest with respect to the planet is simply because of friction forcesexerted by the asphalt, which happens to be attached to the planet.

Galileo’s insights are summarized as follows:

The principle of inertiaNo force is required to maintain motion with constant velocity in

a straight line, and absolute motion does not cause any observablephysical effects.

There are many examples of situations that seem to disprove theprinciple of inertia, but these all result from forgetting that frictionis a force. For instance, it seems that a force is needed to keep asailboat in motion. If the wind stops, the sailboat stops too. Butthe wind’s force is not the only force on the boat; there is alsoa frictional force from the water. If the sailboat is cruising andthe wind suddenly disappears, the backward frictional force stillexists, and since it is no longer being counteracted by the wind’sforward force, the boat stops. To disprove the principle of inertia,we would have to find an example where a moving object sloweddown even though no forces whatsoever were acting on it. Over theyears since Newton’s lifetime, physicists have done more and moreprecise experiments to search for such a counterexample, but theresults have always been negative. Section 4.7 describes one suchobservation that achieved a precision of one part in 107.

self-check EWhat is incorrect about the following supposed counterexamples to theprinciple of inertia?

(1) When astronauts blast off in a rocket, their huge velocity does causea physical effect on their bodies — they get pressed back into theirseats, the flesh on their faces gets distorted, and they have a hard timelifting their arms.

(2) When you’re driving in a convertible with the top down, the wind inyour face is an observable physical effect of your absolute motion. .

Answer, p. 523

. Solved problem: a bug on a wheel page 93, problem 13

Discussion questions

A A passenger on a cruise ship finds, while the ship is docked, thathe can leap off of the upper deck and just barely make it into the poolon the lower deck. If the ship leaves dock and is cruising rapidly, will thisadrenaline junkie still be able to make it?

B You are a passenger in the open basket hanging under a heliumballoon. The balloon is being carried along by the wind at a constantvelocity. If you are holding a flag in your hand, will the flag wave? If so,which way? [Based on a question from PSSC Physics.]

C Aristotle stated that all objects naturally wanted to come to rest, withthe unspoken implication that “rest” would be interpreted relative to the

Section 2.4 The principle of inertia 71

Discussion question D.

surface of the earth. Suppose we go back in time and transport Aristotleto the moon. Aristotle knew, as we do, that the moon circles the earth; hesaid it didn’t fall down because, like everything else in the heavens, it wasmade out of some special substance whose “natural” behavior was to goin circles around the earth. We land, put him in a space suit, and kickhim out the door. What would he expect his fate to be in this situation? Ifintelligent creatures inhabited the moon, and one of them independentlycame up with the equivalent of Aristotelian physics, what would they thinkabout objects coming to rest?

D The glass is sitting on a level table in a train’s dining car, but thesurface of the water is tilted. What can you infer about the motion of thetrain?

2.5 Addition of velocitiesAddition of velocities to describe relative motion

Since absolute motion cannot be unambiguously measured, theonly way to describe motion unambiguously is to describe the motionof one object relative to another. Symbolically, we can write vPQfor the velocity of object P relative to object Q.

Velocities measured with respect to different reference points canbe compared by addition. In the figure below, the ball’s velocityrelative to the couch equals the ball’s velocity relative to the truckplus the truck’s velocity relative to the couch:

vBC = vBT + vTC

= 5 cm/s + 10 cm/s

= 15 cm/s

The same equation can be used for any combination of threeobjects, just by substituting the relevant subscripts for B, T, andC. Just remember to write the equation so that the velocities beingadded have the same subscript twice in a row. In this example, ifyou read off the subscripts going from left to right, you get BC . . . =. . .BTTC. The fact that the two “inside” subscripts on the right arethe same means that the equation has been set up correctly. Noticehow subscripts on the left look just like the subscripts on the right,but with the two T’s eliminated.

Negative velocities in relative motion

My discussion of how to interpret positive and negative signs ofvelocity may have left you wondering why we should bother. Whynot just make velocity positive by definition? The original reasonwhy negative numbers were invented was that bookkeepers decidedit would be convenient to use the negative number concept for pay-ments to distinguish them from receipts. It was just plain easier thanwriting receipts in black and payments in red ink. After adding up


v / These two highly competent physicists disagree on absolute ve-locities, but they would agree on relative velocities. Purple Dinoconsiders the couch to be at rest, while Green Dino thinks of the truck asbeing at rest. They agree, however, that the truck’s velocity relative to thecouch is vTC = 10 cm/s, the ball’s velocity relative to the truck is vBT = 5cm/s, and the ball’s velocity relative to the couch is vBC = vBT + vTC = 15cm/s.

your month’s positive receipts and negative payments, you either gota positive number, indicating profit, or a negative number, showinga loss. You could then show that total with a high-tech “+” or “−”sign, instead of looking around for the appropriate bottle of ink.

Nowadays we use positive and negative numbers for all kindsof things, but in every case the point is that it makes sense toadd and subtract those things according to the rules you learnedin grade school, such as “minus a minus makes a plus, why this istrue we need not discuss.” Adding velocities has the significanceof comparing relative motion, and with this interpretation negativeand positive velocities can be used within a consistent framework.For example, the truck’s velocity relative to the couch equals thetruck’s velocity relative to the ball plus the ball’s velocity relativeto the couch:

vTC = vTB + vBC

= −5 cm/s + 15 cm/s

= 10 cm/s

If we didn’t have the technology of negative numbers, we would havehad to remember a complicated set of rules for adding velocities: (1)if the two objects are both moving forward, you add, (2) if one ismoving forward and one is moving backward, you subtract, but (3)

Section 2.5 Addition of velocities 73

if they’re both moving backward, you add. What a pain that wouldhave been.

. Solved problem: two dimensions page 93, problem 10

Airspeed example 5On June 1, 2009, Air France flight 447 disappeared without warn-ing over the Atlantic Ocean. All 232 people aboard were killed.Investigators believe the disaster was triggered because the pilotslost the ability to accurately determine their speed relative to theair. This is done using sensors called Pitot tubes, mounted out-side the plane on the wing. Automated radio signals showed thatthese sensors gave conflicting readings before the crash, possi-bly because they iced up. For fuel efficiency, modern passengerjets fly at a very high altitude, but in the thin air they can only flywithin a very narrow range of speeds. If the speed is too low, theplane stalls, and if it’s too high, it breaks up. If the pilots can’t tellwhat their airspeed is, they can’t keep it in the safe range.

Many people’s reaction to this story is to wonder why planes don’tjust use GPS to measure their speed. One reason is that GPStells you your speed relative to the ground, not relative to the air.Letting P be the plane, A the air, and G the ground, we have

vPG = vPA + vAG ,

where vPG (the “true ground speed”) is what GPS would measure,vPA (“airspeed”) is what’s critical for stable flight, and vAG is thevelocity of the wind relative to the ground 9000 meters below.Knowing vPG isn’t enough to determine vPA unless vAG is alsoknown.

w / 1. The same aircraft before the disaster. 2. A Pitot tube. 3. The flight path of flight 447. 4. Wreck-age being recovered.


A Interpret the general rule vAB = −vBA in words.

B Wa-Chuen slips away from her father at the mall and walks up thedown escalator, so that she stays in one place. Write this in terms ofsymbols.


x / This Global PositioningSystem (GPS) system, runningon a smartphone attached to abike’s handlebar, depends onEinstein’s theory of relativity.Time flows at a different ratesaboard a GPS satellite than itdoes on the bike, and the GPSsoftware has to take this intoaccount.

y / The clock took up two seats,and two tickets were bought for itunder the name of “Mr. Clock.”

z / Newton’s laws do not dis-tinguish past from future. Thefootball could travel in eitherdirection while obeying Newton’slaws.

2.6 ? RelativityTime is not absolute

So far we’ve been discussing relativity according to Galileo andNewton, but there is also relativity according to Einstein. WhenEinstein first began to develop the theory of relativity, around 1905,the only real-world observations he could draw on were ambiguousand indirect. Today, the evidence is part of everyday life. For ex-ample, every time you use a GPS receiver, x, you’re using Einstein’stheory of relativity. Somewhere between 1905 and today, technol-ogy became good enough to allow conceptually simple experimentsthat students in the early 20th century could only discuss in termslike “Imagine that we could. . . ” A good jumping-on point is 1971.In that year, J.C. Hafele and R.E. Keating brought atomic clocksaboard commercial airliners, y, and went around the world, oncefrom east to west and once from west to east. Hafele and Keatingobserved that there was a discrepancy between the times measuredby the traveling clocks and the times measured by similar clocksthat stayed home at the U.S. Naval Observatory in Washington.The east-going clock lost time, ending up off by −59 ± 10 nanosec-onds, while the west-going one gained 273± 7 ns.

The correspondence principle

This establishes that time doesn’t work the way Newton believedit did when he wrote that “Absolute, true, and mathematical time,of itself, and from its own nature flows equably without regard toanything external. . . ” We are used to thinking of time as absoluteand universal, so it is disturbing to find that it can flow at a differentrate for observers in different frames of reference. Nevertheless, theeffects that Hafele and Keating observed were small. This makessense: Newton’s laws have already been thoroughly tested by ex-periments under a wide variety of conditions, so a new theory likerelativity must agree with Newton’s to a good approximation, withinthe Newtonian theory’s realm of applicability. This requirement ofbackward-compatibility is known as the correspondence principle.

Causality

It’s also reassuring that the effects on time were small comparedto the three-day lengths of the plane trips. There was therefore noopportunity for paradoxical scenarios such as one in which the east-going experimenter arrived back in Washington before he left andthen convinced himself not to take the trip. A theory that maintainsthis kind of orderly relationship between cause and effect is said tosatisfy causality.

Causality is like a water-hungry front-yard lawn in Los Angeles:we know we want it, but it’s not easy to explain why. Even in plainold Newtonian physics, there is no clear distinction between past

Section 2.6 ? Relativity 75

aa / All three clocks are movingto the east. Even though thewest-going plane is moving to thewest relative to the air, the airis moving to the east due to theearth’s rotation.

and future. In figure z, number 18 throws the football to number25, and the ball obeys Newton’s laws of motion. If we took a videoof the pass and played it backward, we would see the ball flying from25 to 18, and Newton’s laws would still be satisfied. Nevertheless,we have a strong psychological impression that there is a forwardarrow of time. I can remember what the stock market did last year,but I can’t remember what it will do next year. Joan of Arc’s mil-itary victories against England caused the English to burn her atthe stake; it’s hard to accept that Newton’s laws provide an equallygood description of a process in which her execution in 1431 causedher to win a battle in 1429. There is no consensus at this pointamong physicists on the origin and significance of time’s arrow, andfor our present purposes we don’t need to solve this mystery. In-stead, we merely note the empirical fact that, regardless of whatcausality really means and where it really comes from, its behavioris consistent. Specifically, experiments show that if an observer in acertain frame of reference observes that event A causes event B, thenobservers in other frames agree that A causes B, not the other wayaround. This is merely a generalization about a large body of ex-perimental results, not a logically necessary assumption. If Keatinghad gone around the world and arrived back in Washington beforehe left, it would have disproved this statement about causality.

Time distortion arising from motion and gravity

Hafele and Keating were testing specific quantitative predictionsof relativity, and they verified them to within their experiment’serror bars. Let’s work backward instead, and inspect the empiricalresults for clues as to how time works.

The two traveling clocks experienced effects in opposite direc-tions, and this suggests that the rate at which time flows dependson the motion of the observer. The east-going clock was moving inthe same direction as the earth’s rotation, so its velocity relative tothe earth’s center was greater than that of the clock that remainedin Washington, while the west-going clock’s velocity was correspond-ingly reduced. The fact that the east-going clock fell behind, andthe west-going one got ahead, shows that the effect of motion is tomake time go more slowly. This effect of motion on time was pre-dicted by Einstein in his original 1905 paper on relativity, writtenwhen he was 26.

If this had been the only effect in the Hafele-Keating experiment,then we would have expected to see effects on the two flying clocksthat were equal in size. Making up some simple numbers to keep thearithmetic transparent, suppose that the earth rotates from west toeast at 1000 km/hr, and that the planes fly at 300 km/hr. Then thespeed of the clock on the ground is 1000 km/hr, the speed of theclock on the east-going plane is 1300 km/hr, and that of the west-going clock 700 km/hr. Since the speeds of 700, 1000, and 1300


ab / The correspondence princi-ple requires that the relativisticdistortion of time become smallfor small velocities.

km/hr have equal spacing on either side of 1000, we would expectthe discrepancies of the moving clocks relative to the one in the labto be equal in size but opposite in sign.

In fact, the two effects are unequal in size: −59 ns and 273 ns.This implies that there is a second effect involved, simply due to theplanes’ being up in the air. Relativity predicts that time’s rate offlow also changes with height in a gravitational field. Einstein didn’tfigure out how to incorporate gravity into relativity until 1915, aftermuch frustration and many false starts. The simpler version of thetheory without gravity is known as special relativity, the full versionas general relativity. We’ll restrict ourselves to special relativity inthis book, and that means that what we want to focus on right nowis the distortion of time due to motion, not gravity.

We can now see in more detail how to apply the correspondenceprinciple. The behavior of the three clocks in the Hafele-Keatingexperiment shows that the amount of time distortion increases asthe speed of the clock’s motion increases. Newton lived in an erawhen the fastest mode of transportation was a galloping horse, andthe best pendulum clocks would accumulate errors of perhaps aminute over the course of several days. A horse is much slowerthan a jet plane, so the distortion of time would have had a relativesize of only ∼ 10−15 — much smaller than the clocks were capableof detecting. At the speed of a passenger jet, the effect is about10−12, and state-of-the-art atomic clocks in 1971 were capable ofmeasuring that. A GPS satellite travels much faster than a jet air-plane, and the effect on the satellite turns out to be ∼ 10−10. Thegeneral idea here is that all physical laws are approximations, andapproximations aren’t simply right or wrong in different situations.Approximations are better or worse in different situations, and thequestion is whether a particular approximation is good enough in agiven situation to serve a particular purpose. The faster the motion,the worse the Newtonian approximation of absolute time. Whetherthe approximation is good enough depends on what you’re tryingto accomplish. The correspondence principle says that the approxi-mation must have been good enough to explain all the experimentsdone in the centuries before Einstein came up with relativity.

By the way, don’t get an inflated idea of the importance of theHafele-Keating experiment. Special relativity had already been con-firmed by a vast and varied body of experiments decades before 1971.The only reason I’m giving such a prominent role to this experiment,which was actually more important as a test of general relativity, isthat it is conceptually very direct.


ac / Two events are given aspoints on a graph of positionversus time. Joan of Arc helps torestore Charles VII to the throne.At a later time and a differentposition, Joan of Arc is sentencedto death.

ad / A change of units dis-torts an x-t graph. This graphdepicts exactly the same eventsas figure ac. The only change isthat the x and t coordinates aremeasured using different units,so the grid is compressed in tand expanded in x .

ae / A convention we’ll useto represent a distortion of timeand space.

Distortion of space and time

The Lorentz transformation

Relativity says that when two observers are in different frames ofreference, each observer considers the other one’s perception of timeto be distorted. We’ll also see that something similar happens totheir observations of distances, so both space and time are distorted.What exactly is this distortion? How do we even conceptualize it?

The idea isn’t really as radical as it might seem at first. Wecan visualize the structure of space and time using a graph withposition and time on its axes. These graphs are familiar by now,but we’re going to look at them in a slightly different way. Before, weused them to describe the motion of objects. The grid underlyingthe graph was merely the stage on which the actors played theirparts. Now the background comes to the foreground: it’s time andspace themselves that we’re studying. We don’t necessarily needto have a line or a curve drawn on top of the grid to represent aparticular object. We may, for example, just want to talk aboutevents, depicted as points on the graph as in figure ac. A distortionof the Cartesian grid underlying the graph can arise for perfectlyordinary reasons that Isaac Newton would have readily accepted.For example, we can simply change the units used to measure timeand position, as in figure ad.

We’re going to have quite a few examples of this type, so I’lladopt the convention shown in figure ae for depicting them. Figureae summarizes the relationship between figures ac and ad in a morecompact form. The gray rectangle represents the original coordinategrid of figure ac, while the grid of black lines represents the newversion from figure ad. Omitting the grid from the gray rectanglemakes the diagram easier to decode visually.

Our goal of unraveling the mysteries of special relativity amountsto nothing more than finding out how to draw a diagram like aein the case where the two different sets of coordinates representmeasurements of time and space made by two different observers,each in motion relative to the other. Galileo and Newton thoughtthey knew the answer to this question, but their answer turnedout to be only approximately right. To avoid repeating the samemistakes, we need to clearly spell out what we think are the basicproperties of time and space that will be a reliable foundation forour reasoning. I want to emphasize that there is no purely logicalway of deciding on this list of properties. The ones I’ll list are simplya summary of the patterns observed in the results from a large bodyof experiments. Furthermore, some of them are only approximate.For example, property 1 below is only a good approximation whenthe gravitational field is weak, so it is a property that applies tospecial relativity, not to general relativity.


af / A Galilean version of therelationship between two framesof reference. As in all suchgraphs in this chapter, the originalcoordinates, represented by thegray rectangle, have a time axisthat goes to the right, and aposition axis that goes straightup.

Experiments show that:

1. No point in time or space has properties that make it differentfrom any other point.

2. Likewise, all directions in space have the same properties.

3. Motion is relative, i.e., all inertial frames of reference areequally valid.

4. Causality holds, in the sense described on page 75.

5. Time depends on the state of motion of the observer.

Most of these are not very subversive. Properties 1 and 2 dateback to the time when Galileo and Newton started applying thesame universal laws of motion to the solar system and to the earth;this contradicted Aristotle, who believed that, for example, a rockwould naturally want to move in a certain special direction (down)in order to reach a certain special location (the earth’s surface).Property 3 is the reason that Einstein called his theory “relativity,”but Galileo and Newton believed exactly the same thing to be true,as dramatized by Galileo’s run-in with the Church over the questionof whether the earth could really be in motion around the sun.Property 4 would probably surprise most people only because itasserts in such a weak and specialized way something that they feeldeeply must be true. The only really strange item on the list is 5,but the Hafele-Keating experiment forces it upon us.

If it were not for property 5, we could imagine that figure afwould give the correct transformation between frames of referencein motion relative to one another. Let’s say that observer 1, whosegrid coincides with the gray rectangle, is a hitch-hiker standing bythe side of a road. Event A is a raindrop hitting his head, andevent B is another raindrop hitting his head. He says that A and Boccur at the same location in space. Observer 2 is a motorist whodrives by without stopping; to him, the passenger compartment ofhis car is at rest, while the asphalt slides by underneath. He saysthat A and B occur at different points in space, because during thetime between the first raindrop and the second, the hitch-hiker hasmoved backward. On the other hand, observer 2 says that events Aand C occur in the same place, while the hitch-hiker disagrees. Theslope of the grid-lines is simply the velocity of the relative motionof each observer relative to the other.

Figure af has familiar, comforting, and eminently sensible behav-ior, but it also happens to be wrong, because it violates property5. The distortion of the coordinate grid has only moved the verticallines up and down, so both observers agree that events like B andC are simultaneous. If this was really the way things worked, thenall observers could synchronize all their clocks with one another for


ag / A transformation that leadsto disagreements about whethertwo events occur at the sametime and place. This is not justa matter of opinion. Either thearrow hit the bull’s-eye or it didn’t.

ah / A nonlinear transforma-tion.

once and for all, and the clocks would never get out of sync. Thiscontradicts the results of the Hafele-Keating experiment, in whichall three clocks were initially synchronized in Washington, but laterwent out of sync because of their different states of motion.

It might seem as though we still had a huge amount of wiggleroom available for the correct form of the distortion. It turns out,however, that properties 1-5 are sufficient to prove that there is onlyone answer, which is the one found by Einstein in 1905. To see whythis is, let’s work by a process of elimination.

Figure ag shows a transformation that might seem at first glanceto be as good a candidate as any other, but it violates property 3,that motion is relative, for the following reason. In observer 2’sframe of reference, some of the grid lines cross one another. Thismeans that observers 1 and 2 disagree on whether or not certainevents are the same. For instance, suppose that event A marks thearrival of an arrow at the bull’s-eye of a target, and event B is thelocation and time when the bull’s-eye is punctured. Events A andB occur at the same location and at the same time. If one observersays that A and B coincide, but another says that they don’t, wehave a direct contradiction. Since the two frames of reference infigure ag give contradictory results, one of them is right and oneis wrong. This violates property 3, because all inertial frames ofreference are supposed to be equally valid. To avoid problems likethis, we clearly need to make sure that none of the grid lines evercross one another.

The next type of transformation we want to kill off is shown infigure ah, in which the grid lines curve, but never cross one another.The trouble with this one is that it violates property 1, the unifor-mity of time and space. The transformation is unusually “twisty”at A, whereas at B it’s much more smooth. This can’t be correct,because the transformation is only supposed to depend on the rela-tive state of motion of the two frames of reference, and that giveninformation doesn’t single out a special role for any particular pointin spacetime. If, for example, we had one frame of reference rotatingrelative to the other, then there would be something special aboutthe axis of rotation. But we’re only talking about inertial frames ofreference here, as specified in property 3, so we can’t have rotation;each frame of reference has to be moving in a straight line at con-stant speed. For frames related in this way, there is nothing thatcould single out an event like A for special treatment compared toB, so transformation ah violates property 1.

The examples in figures ag and ah show that the transformationwe’re looking for must be linear, meaning that it must transformlines into lines, and furthermore that it has to take parallel lines toparallel lines. Einstein wrote in his 1905 paper that “. . . on accountof the property of homogeneity [property 1] which we ascribe to time


aj / In the units that are mostconvenient for relativity, the trans-formation has symmetry about a45-degree diagonal line.

and space, the [transformation] must be linear.”2 Applying this toour diagrams, the original gray rectangle, which is a special typeof parallelogram containing right angles, must be transformed intoanother parallelogram. There are three types of transformations,figure ai, that have this property. Case I is the Galilean transfor-mation of figure af on page 79, which we’ve already ruled out.

ai / Three types of transformations that preserve parallelism. Theirdistinguishing feature is what they do to simultaneity, as shown by whathappens to the left edge of the original rectangle. In I, the left edgeremains vertical, so simultaneous events remain simultaneous. In II, theleft edge turns counterclockwise. In III, it turns clockwise.

Case II can also be discarded. Here every point on the grid ro-tates counterclockwise. What physical parameter would determinethe amount of rotation? The only thing that could be relevant wouldbe v, the relative velocity of the motion of the two frames of referencewith respect to one another. But if the angle of rotation was pro-portional to v, then for large enough velocities the grid would haveleft and right reversed, and this would violate property 4, causality:one observer would say that event A caused a later event B, butanother observer would say that B came first and caused A.

The only remaining possibility is case III, which I’ve redrawn infigure aj with a couple of changes. This is the one that Einsteinpredicted in 1905. The transformation is known as the Lorentztransformation, after Hendrik Lorentz (1853-1928), who partiallyanticipated Einstein’s work, without arriving at the correct inter-pretation. The distortion is a kind of smooshing and stretching, assuggested by the hands. Also, we’ve already seen in figures ac-ae onpage 78 that we’re free to stretch or compress everything as much aswe like in the horizontal and vertical directions, because this simplycorresponds to choosing different units of measurement for time anddistance. In figure aj I’ve chosen units that give the whole drawinga convenient symmetry about a 45-degree diagonal line. Ordinarilyit wouldn’t make sense to talk about a 45-degree angle on a graphwhose axes had different units. But in relativity, the symmetric ap-pearance of the transformation tells us that space and time ought

2A. Einstein, “On the Electrodynamics of Moving Bodies,” Annalen derPhysik 17 (1905), p. 891, tr. Saha and Bose, 1920


ak / Interpretation of the Lorentztransformation. The slope in-dicated in the figure gives therelative velocity of the two framesof reference. Events A and B thatwere simultaneous in frame 1are not simultaneous in frame 2,where event A occurs to the rightof the t = 0 line represented bythe left edge of the grid, but eventB occurs to its left.

to be treated on the same footing, and measured in the same units.

As in our discussion of the Galilean transformation, slopes areinterpreted as velocities, and the slope of the near-horizontal lines infigure ak is interpreted as the relative velocity of the two observers.The difference between the Galilean version and the relativistic oneis that now there is smooshing happening from the other side aswell. Lines that were vertical in the original grid, representing si-multaneous events, now slant over to the right. This tells us that, asrequired by property 5, different observers do not agree on whetherevents that occur in different places are simultaneous. The Hafele-Keating experiment tells us that this non-simultaneity effect is fairlysmall, even when the velocity is as big as that of a passenger jet,and this is what we would have anticipated by the correspondenceprinciple. The way that this is expressed in the graph is that if wepick the time unit to be the second, then the distance unit turns outto be hundreds of thousands of miles. In these units, the velocityof a passenger jet is an extremely small number, so the slope v infigure ak is extremely small, and the amount of distortion is tiny —it would be much too small to see on this scale.

The only thing left to determine about the Lorentz transforma-tion is the size of the transformed parallelogram relative to the sizeof the original one. Although the drawing of the hands in figure ajmay suggest that the grid deforms like a framework made of rigidcoat-hanger wire, that is not the case. If you look carefully at thefigure, you’ll see that the edges of the smooshed parallelogram areactually a little longer than the edges of the original rectangle. Infact what stays the same is not lengths but areas. The proof of thisfact is straightforward, but a little lengthy, so I’ve relegated it tosection 2.6.3. Oversimplifying a little, the basic idea of the proof isthat it wouldn’t make sense if the area was increased by the Lorentztransformation, because then area would have to be decreased bya Lorentz transformation corresponding to motion in the oppositedirection, and this would violate property 2 on page 79, which statesthat all directions in space have the same properties.

The G factor

With a little algebra and geometry (homework problem 18, page94), one can use the equal-area property to show that the factor G(Greek letter gamma) defined in figure al is given by the equation

G =1√

1− v2.

If you’ve had good training in physics, the first thing you probablythink when you look at this equation is that it must be nonsense,


al / The G factor.

because its units don’t make sense. How can we take somethingwith units of velocity squared, and subtract it from a unitless 1?But remember that this is expressed in our special relativistic units,in which the same units are used for distance and time. In thissystem, velocities are always unitless. This sort of thing happensfrequently in physics. For instance, before James Joule discoveredconservation of energy, nobody knew that heat and mechanical en-ergy were different forms of the same thing, so instead of measuringthem both in units of joules as we would do now, they measuredheat in one unit (such as calories) and mechanical energy in another(such as foot-pounds). In ordinary metric units, we just need anextra convension factor c, and the equation becomes

G =1√

1−(vc

)2 .

Here’s why we care about G. Figure al defines it as the ratio oftwo times: the time between two events as expressed in one coordi-nate system, and the time between the same two events as measuredin the other one. The interpretation is:

Time dilationA clock runs fastest in the frame of reference of an observerwho is at rest relative to the clock. An observer in motionrelative to the clock at speed v perceives the clock as runningmore slowly by a factor of G.

Since the Lorentz transformation treats time and distance entirelysymmetrically, we could just as well have defined G using the uprightx axis in figure al, and we therefore have a similar interpretation interms of space:

Length contractionA meter-stick appears longest to an observer who is at restrelative to it. An observer moving relative to the meter-stickat v observes the stick to be shortened by a factor of G.

self-check FWhat is G when v = 0? What does this mean? . Answer, p. 523


an / Muons accelerated tonearly c undergo radioactive de-cay much more slowly than theywould according to an observerat rest with respect to the muons.The first two data-points (unfilledcircles) were subject to largesystematic errors.

am / Apparatus used for the testof relativistic time dilation de-scribed in example 6. The promi-nent black and white blocks arelarge magnets surrounding a cir-cular pipe with a vacuum inside.(c) 1974 by CERN.

Large time dilation example 6The time dilation effect in the Hafele-Keating experiment was verysmall. If we want to see a large time dilation effect, we can’t doit with something the size of the atomic clocks they used; the ki-netic energy would be greater than the total megatonnage of allthe world’s nuclear arsenals. We can, however, accelerate sub-atomic particles to speeds at which G is large. For experimentalparticle physicists, relativity is something you do all day beforeheading home and stopping off at the store for milk. An early, low-precision experiment of this kind was performed by Rossi and Hallin 1941, using naturally occurring cosmic rays. Figure am showsa 1974 experiment3 of a similar type which verified the time di-lation predicted by relativity to a precision of about one part perthousand.

Particles called muons (named after the Greek letter µ, “myoo”)were produced by an accelerator at CERN, near Geneva. A muonis essentially a heavier version of the electron. Muons undergoradioactive decay, lasting an average of only 2.197 µs before theyevaporate into an electron and two neutrinos. The 1974 experi-ment was actually built in order to measure the magnetic proper-ties of muons, but it produced a high-precision test of time dilationas a byproduct. Because muons have the same electric chargeas electrons, they can be trapped using magnetic fields. Muonswere injected into the ring shown in figure am, circling aroundit until they underwent radioactive decay. At the speed at whichthese muons were traveling, they had G = 29.33, so on the av-erage they lasted 29.33 times longer than the normal lifetime. Inother words, they were like tiny alarm clocks that self-destructedat a randomly selected time. Figure an shows the number of ra-dioactive decays counted, as a function of the time elapsed af-ter a given stream of muons was injected into the storage ring.3Bailey at al., Nucl. Phys. B150(1979) 1


ap / In the garage’s frame ofreference, 1, the bus is moving,and can fit in the garage. In thebus’s frame of reference, thegarage is moving, and can’t holdthe bus.

The two dashed lines show the rates of decay predicted with andwithout relativity. The relativistic line is the one that agrees withexperiment.

An example of length contraction example 7Figure ao shows an artist’s rendering of the length contractionfor the collision of two gold nuclei at relativistic speeds in theRHIC accelerator in Long Island, New York, which went on linein 2000. The gold nuclei would appear nearly spherical (or justslightly lengthened like an American football) in frames movingalong with them, but in the laboratory’s frame, they both appeardrastically foreshortened as they approach the point of collision.The later pictures show the nuclei merging to form a hot soup, inwhich experimenters hope to observe a new form of matter.

ao / Colliding nuclei show rela-tivistic length contraction.

The garage paradox example 8One of the most famous of all the so-called relativity paradoxeshas to do with our incorrect feeling that simultaneity is well de-fined. The idea is that one could take a schoolbus and drive it atrelativistic speeds into a garage of ordinary size, in which it nor-mally would not fit. Because of the length contraction, the buswould supposedly fit in the garage. The paradox arises when weshut the door and then quickly slam on the brakes of the bus.An observer in the garage’s frame of reference will claim that thebus fit in the garage because of its contracted length. The driver,however, will perceive the garage as being contracted and thuseven less able to contain the bus. The paradox is resolved whenwe recognize that the concept of fitting the bus in the garage “allat once” contains a hidden assumption, the assumption that itmakes sense to ask whether the front and back of the bus cansimultaneously be in the garage. Observers in different framesof reference moving at high relative speeds do not necessarilyagree on whether things happen simultaneously. The person inthe garage’s frame can shut the door at an instant he perceives tobe simultaneous with the front bumper’s arrival at the back wall ofthe garage, but the driver would not agree about the simultaneityof these two events, and would perceive the door as having shutlong after she plowed through the back wall.


aq / A proof that causalityimposes a universal speed limit.In the original frame of reference,represented by the square, eventA happens a little before event B.In the new frame, shown by theparallelogram, A happens aftert = 0, but B happens before t = 0;that is, B happens before A. Thetime ordering of the two eventshas been reversed. This can onlyhappen because events A and Bare very close together in timeand fairly far apart in space. Theline segment connecting A andB has a slope greater than 1,meaning that if we wanted to bepresent at both events, we wouldhave to travel at a speed greaterthan c (which equals 1 in theunits used on this graph). You willfind that if you pick any two pointsfor which the slope of the linesegment connecting them is lessthan 1, you can never get them tostraddle the new t = 0 line in thisfunny, time-reversed way. Sincedifferent observers disagree onthe time order of events like Aand B, causality requires thatinformation never travel fromA to B or from B to A; if it did,then we would have time-travelparadoxes. The conclusion is thatc is the maximum speed of causeand effect in relativity.

The universal speed c

Let’s think a little more about the role of the 45-degree diagonalin the Lorentz transformation. Slopes on these graphs are inter-preted as velocities. This line has a slope of 1 in relativistic units,but that slope corresponds to c in ordinary metric units. We al-ready know that the relativistic distance unit must be extremelylarge compared to the relativistic time unit, so c must be extremelylarge. Now note what happens when we perform a Lorentz transfor-mation: this particular line gets stretched, but the new version ofthe line lies right on top of the old one, and its slope stays the same.In other words, if one observer says that something has a velocityequal to c, every other observer will agree on that velocity as well.(The same thing happens with −c.)

. Velocities don’t simply add and subtract.

This is surprising, since we expect, as in section 2.5.1, that avelocity c in one frame should become c + v in a frame movingat velocity v relative to the first one. But velocities are measuredby dividing a distance by a time, and both distance and time aredistorted by relativistic effects, so we actually shouldn’t expect theordinary arithmetic addition of velocities to hold in relativity; it’san approximation that’s valid at velocities that are small comparedto c. Problem 22 on p. 96 shows that relativistically, combiningvelocities u and v gives not u + v but (u + v)/(1 + uv) (in unitswhere c = 1).

. A universal speed limit

For example, suppose Janet takes a trip in a spaceship, andaccelerates until she is moving at 0.6c relative to the earth. Shethen launches a space probe in the forward direction at a speedrelative to her ship of 0.6c. We might think that the probe was thenmoving at a velocity of 1.2c, but in fact the answer is still less thanc (problem 21, page 95). This is an example of a more general factabout relativity, which is that c represents a universal speed limit.This is required by causality, as shown in figure aq.

. Light travels at c.

Now consider a beam of light. We’re used to talking casuallyabout the “speed of light,” but what does that really mean? Motionis relative, so normally if we want to talk about a velocity, we haveto specify what it’s measured relative to. A sound wave has a certainspeed relative to the air, and a water wave has its own speed relativeto the water. If we want to measure the speed of an ocean wave, forexample, we should make sure to measure it in a frame of referenceat rest relative to the water. But light isn’t a vibration of a physicalmedium; it can propagate through the near-perfect vacuum of outerspace, as when rays of sunlight travel to earth. This seems like aparadox: light is supposed to have a specific speed, but there is no


way to decide what frame of reference to measure it in. The wayout of the paradox is that light must travel at a velocity equal to c.Since all observers agree on a velocity of c, regardless of their frameof reference, everything is consistent.

. The Michelson-Morley experiment

The constancy of the speed of light had in fact already beenobserved when Einstein was an 8-year-old boy, but because nobodycould figure out how to interpret it, the result was largely ignored.In 1887 Michelson and Morley set up a clever apparatus to measureany difference in the speed of light beams traveling east-west andnorth-south. The motion of the earth around the sun at 110,000km/hour (about 0.01% of the speed of light) is to our west during theday. Michelson and Morley believed that light was a vibration of amysterious medium called the ether, so they expected that the speedof light would be a fixed value relative to the ether. As the earthmoved through the ether, they thought they would observe an effecton the velocity of light along an east-west line. For instance, if theyreleased a beam of light in a westward direction during the day, theyexpected that it would move away from them at less than the normalspeed because the earth was chasing it through the ether. They weresurprised when they found that the expected 0.01% change in thespeed of light did not occur.

ar / The Michelson-Morley experi-ment, shown in photographs, anddrawings from the original 1887paper. 1. A simplified draw-ing of the apparatus. A beam oflight from the source, s, is par-tially reflected and partially trans-mitted by the half-silvered mirrorh1. The two half-intensity parts ofthe beam are reflected by the mir-rors at a and b, reunited, and ob-served in the telescope, t. If theearth’s surface was supposed tobe moving through the ether, thenthe times taken by the two lightwaves to pass through the mov-ing ether would be unequal, andthe resulting time lag would bedetectable by observing the inter-ference between the waves whenthey were reunited. 2. In the realapparatus, the light beams werereflected multiple times. The ef-fective length of each arm wasincreased to 11 meters, whichgreatly improved its sensitivity tothe small expected difference inthe speed of light. 3. In anearlier version of the experiment,they had run into problems withits “extreme sensitiveness to vi-bration,” which was “so great thatit was impossible to see the in-terference fringes except at briefintervals . . . even at two o’clockin the morning.” They thereforemounted the whole thing on amassive stone floating in a pool ofmercury, which also made it pos-sible to rotate it easily. 4. A photoof the apparatus.


Discussion question B


A A person in a spaceship moving at 99.99999999% of the speedof light relative to Earth shines a flashlight forward through dusty air, sothe beam is visible. What does she see? What would it look like to anobserver on Earth?

B A question that students often struggle with is whether time andspace can really be distorted, or whether it just seems that way. Comparewith optical illusions or magic tricks. How could you verify, for instance,that the lines in the figure are actually parallel? Are relativistic effects thesame, or not?

C On a spaceship moving at relativistic speeds, would a lecture seemeven longer and more boring than normal?

D Mechanical clocks can be affected by motion. For example, it wasa significant technological achievement to build a clock that could sailaboard a ship and still keep accurate time, allowing longitude to be deter-mined. How is this similar to or different from relativistic time dilation?

E Figure ao from page 85, depicting the collision of two nuclei at theRHIC accelerator, is reproduced below. What would the shapes of the twonuclei look like to a microscopic observer riding on the left-hand nucleus?To an observer riding on the right-hand one? Can they agree on what ishappening? If not, why not — after all, shouldn’t they see the same thingif they both compare the two nuclei side-by-side at the same instant intime?

as / Discussion question E: colliding nuclei show relativistic lengthcontraction.

F If you stick a piece of foam rubber out the window of your car whiledriving down the freeway, the wind may compress it a little. Does it makesense to interpret the relativistic length contraction as a type of strainthat pushes an object’s atoms together like this? How does this relate todiscussion question E?


Proof that the Lorentz transformation preserves area

On page 82, I claimed that area must be preserved by Lorentztransformations. Here is the proof of that fact.

First, consider a process, P, in which we start with a square ofunit area, and perform two successive Lorentz transformations, onewith v and one with −v. Let the first transformation scale the areaof the square by a factor R, which we want to prove equals 1. Sincethe second transformation undoes the effect of the first one, its effectis to scale the area of the intermediate parallelogram back down bya factor of 1/R.

Now consider a slightly different process Q, involving the threesteps shown in the figure below: (1) perform a Lorentz transforma-tion with velocity v; (2) chop the resulting parallelogram into littlesquares; and (3) transform each square with a −v transformation.We already know how a +v transformation acts on the area of asquare: it makes it into a parallelogram with an area that is greaterby a factor of R. In step 3, we are also acting on a square, but witha −v transformation. Property 2 of spacetime on page 79 statesthat all directions in space have the same properties, and thereforea −v transformation acting on a square must have the same effecton area as a +v transformation. (By property 1, homogeneity, itdoesn’t matter that these squares are located in different places.)The result is that after completing process Q, we have scaled thearea by a factor of R2.

Comparing processes P and Q, we see that they must both re-store the original square to itself. (In step 2 of process Q, there willplaces near the edges of the parallelogram where we would only havea fraction of a square, but the total area of this ragged edge regioncan be made as small as desired by making the squares sufficientlysmall.) Since P scales the area by 1, and Q scales the area by R2, weconclude that R2 = 1. This means that either R = +1 or R = −1,and R must be independent of v. But for v = 0 we have R = +1,so R = +1 for all values of v.


SummarySelected vocabularycenter of mass . . the balance point of an objectvelocity . . . . . . the rate of change of position; the slope of the

tangent line on an x− t graph.

Notationx . . . . . . . . . . a point in spacet . . . . . . . . . . a point in time, a clock reading∆ . . . . . . . . . “change in;” the value of a variable afterwards

minus its value before∆x . . . . . . . . a distance, or more precisely a change in x,

which may be less than the distance traveled;its plus or minus sign indicates direction

∆t . . . . . . . . . a duration of timev . . . . . . . . . . velocityvAB . . . . . . . . the velocity of object A relative to object B

Other terminology and notationdisplacement . . a name for the symbol ∆xspeed . . . . . . . the absolute value of the velocity, i.e., the ve-

locity stripped of any information about itsdirection

Summary

An object’s center of mass is the point at which it can be bal-anced. For the time being, we are studying the mathematical de-scription only of the motion of an object’s center of mass in casesrestricted to one dimension. The motion of an object’s center ofmass is usually far simpler than the motion of any of its other parts.

It is important to distinguish location, x, from distance, ∆x,and clock reading, t, from time interval ∆t. When an object’s x− tgraph is linear, we define its velocity as the slope of the line, ∆x/∆t.When the graph is curved, we generalize the definition so that thevelocity is the derivative dx/dt.

Galileo’s principle of inertia states that no force is required tomaintain motion with constant velocity in a straight line, and abso-lute motion does not cause any observable physical effects. Thingstypically tend to reduce their velocity relative to the surface of ourplanet only because they are physically rubbing against the planet(or something attached to the planet), not because there is anythingspecial about being at rest with respect to the earth’s surface. Whenit seems, for instance, that a force is required to keep a book slidingacross a table, in fact the force is only serving to cancel the contraryforce of friction.

Absolute motion is not a well-defined concept, and if two ob-servers are not at rest relative to one another they will disagreeabout the absolute velocities of objects. They will, however, agree


about relative velocities. If object A is in motion relative to objectB, and B is in motion relative to C, then A’s velocity relative to Cis given by vAC = vAB + vBC . Positive and negative signs are usedto indicate the direction of an object’s motion.

Modern experiments show that space and time only approxi-mately have the properties claimed by Galileo and Newton. Timeand space as seen by one observer are distorted compared to anotherobserver’s perceptions if they are moving relative to each other. Thisdistortion is quantified by the factor

γ =1√

1− v2

c2

,

where v is the relative velocity of the two observers, and c is auniversal velocity that is the same in all frames of reference. Lighttravels at c. A clock appears to run fastest to an observer who isnot in motion relative to it, and appears to run too slowly by afactor of γ to an observer who has a velocity v relative to the clock.Similarly, a meter-stick appears longest to an observer who sees itat rest, and appears shorter to other observers. Time and space arerelative, not absolute. In particular, there is no well-defined conceptof simultaneity. Velocities don’t add according to u + v but rather(u+ v)/(1 + uv) (in units where c = 1).

All of these strange effects, however, are very small when the rel-ative velocities are small compared to c. This makes sense, becauseNewton’s laws have already been thoroughly tested by experimentsat such speeds, so a new theory like relativity must agree with theold one in their realm of common applicability. This requirement ofbackwards-compatibility is known as the correspondence principle.

Summary 91

Problem 1.

ProblemsKey√

A computerized answer check is available online.∫A problem that requires calculus.

? A difficult problem.

1 The graph represents the motion of a ball that rolls up a hilland then back down. When does the ball return to the location ithad at t = 0? . Solution, p. 510

2 The graph represents the velocity of a bee along a straightline. At t = 0, the bee is at the hive. (a) When is the bee farthestfrom the hive? (b) How far is the bee at its farthest point from thehive? (c) At t = 13 s, how far is the bee from the hive?

√

3 (a) Let θ be the latitude of a point on the Earth’s surface.Derive an algebra equation for the distance, L, traveled by thatpoint during one rotation of the Earth about its axis, i.e., over oneday, expressed in terms of θ and R, the radius of the earth. Check:Your equation should give L = 0 for the North Pole.(b) At what speed is Fullerton, at latitude θ = 34◦, moving withthe rotation of the Earth about its axis? Give your answer in unitsof mi/h. [See the table in the back of the book for the relevantdata.]

√

4 A honeybee’s position as a function of time is given by x =10t− t3, where t is in seconds and x in meters. What is its velocityat t = 3.0 s?

√

5 Freddi Fish(TM) has a position as a function of time given byx = a/(b + t2). (a) Infer the units of the constants a and b. (b)Find her maximum speed. (c) Check that your answer has the rightunits.

√

6 A metal square expands and contracts with temperature, thelengths of its sides varying according to the equation ` = (1+αT )`o.Infer the units of α. Find the rate of change of its surface area withrespect to temperature. That is, find dA/dT . Check that youranswer has the right units, as in example 4 on page 68.

√


7 Let t be the time that has elapsed since the Big Bang. Inthat time, one would imagine that light, traveling at speed c, hasbeen able to travel a maximum distance ct. (In fact the distance isseveral times more than this, because according to Einstein’s theoryof general relativity, space itself has been expanding while the ray oflight was in transit.) The portion of the universe that we can observewould then be a sphere of radius ct, with volume v = (4/3)πr3 =(4/3)π(ct)3. Compute the rate dv/dt at which the volume of theobservable universe is increasing, and check that your answer hasthe right units, as in example 4 on page 68.

√

8 (a) Express the chain rule in Leibniz notation, and show thatit always results in an answer whose units make sense.(b) An object has a position as a function of time given by x =A cos(bt), where A and b are constants. Infer the units of A and b,and interpret their physical meanings.(c) Find the velocity of this object, and check that the chain rulehas indeed given an answer with the right units.

. Solution, p. 511

9 (a) Translate the following information into symbols, usingthe notation with two subscripts introduced in section 2.5. Eowynis riding on her horse at a velocity of 11 m/s. She twists around inher saddle and fires an arrow backward. Her bow fires arrows at 25m/s. (b) Find the velocity of the arrow relative to the ground.

10 Our full discussion of two- and three-dimensional motion ispostponed until the second half of the book, but here is a chance touse a little mathematical creativity in anticipation of that general-ization. Suppose a ship is sailing east at a certain speed v, and apassenger is walking across the deck at the same speed v, so thathis track across the deck is perpendicular to the ship’s center-line.What is his speed relative to the water, and in what direction is hemoving relative to the water? . Solution, p. 511

11 You’re standing in a freight train, and have no way to see out.If you have to lean to stay on your feet, what, if anything, does thattell you about the train’s velocity? Explain. . Solution, p. 511

12 Driving along in your car, you take your foot off the gas,and your speedometer shows a reduction in speed. Describe a frameof reference in which your car was speeding up during that sameperiod of time. (The frame of reference should be defined by anobserver who, although perhaps in motion relative to the earth, isnot changing her own speed or direction of motion.)

13 The figure shows the motion of a point on the rim of a rollingwheel. (The shape is called a cycloid.) Suppose bug A is riding onthe rim of the wheel on a bicycle that is rolling, while bug B is onthe spinning wheel of a bike that is sitting upside down on the floor.Bug A is moving along a cycloid, while bug B is moving in a circle.Both wheels are doing the same number of revolutions per minute.

Problems 93

Problem 13.

Problem 18.

Which bug has a harder time holding on, or do they find it equallydifficult? . Solution, p. 511

14 Astronauts in three different spaceships are communicatingwith each other. Those aboard ships A and B agree on the rate atwhich time is passing, but they disagree with the ones on ship C.(a) Alice is aboard ship A. How does she describe the motion of herown ship, in its frame of reference?(b) Describe the motion of the other two ships according to Alice.(c) Give the description according to Betty, whose frame of referenceis ship B.(d) Do the same for Cathy, aboard ship C.

15 What happens in the equation for G when you put in anegative number for v? Explain what this means physically, andwhy it makes sense.

16 The Voyager 1 space probe, launched in 1977, is movingfaster relative to the earth than any other human-made object, at17,000 meters per second.(a) Calculate the probe’s G.(b) Over the course of one year on earth, slightly less than one yearpasses on the probe. How much less? (There are 31 million secondsin a year.)

√

17 The earth is orbiting the sun, and therefore is contractedrelativistically in the direction of its motion. Compute the amountby which its diameter shrinks in this direction.

√

18 In this homework problem, you’ll fill in the steps of the alge-bra required in order to find the equation for G on page 82. To keepthe algebra simple, let the time t in figure al equal 1, as suggestedin the figure accompanying this homework problem. The originalsquare then has an area of 1, and the transformed parallelogrammust also have an area of 1. (a) Prove that point P is at x = vG, sothat its (t,x) coordinates are (G, vG). (b) Find the (t,x) coordinatesof point Q. (c) Find the length of the short diagonal connecting Pand Q. (d) Average the coordinates of P and Q to find the coordi-nates of the midpoint C of the parallelogram, and then find distanceOC. (e) Find the area of the parallelogram by computing twice thearea of triangle PQO. [Hint: You can take PQ to be the base ofthe triangle.] (f) Set this area equal to 1 and solve for G to proveG = 1/

√1− v2.

√

19 (a) Show that for v = (3/5)c, γ comes out to be a simplefraction.(b) Find another value of v for which γ is a simple fraction.

20 In the equation for the relativistic addition of velocities uand v, consider the limit in which u approaches 1, but v simulta-neously approaches −1. Give both a physical and a mathematicalinterpretation.


21 The figure illustrates a Lorentz transformation using theconventions employed in section 2.6.2. For simplicity, the transfor-mation chosen is one that lengthens one diagonal by a factor of 2.Since Lorentz transformations preserve area, the other diagonal isshortened by a factor of 2. Let the original frame of reference, de-picted with the square, be A, and the new one B. (a) By measuringwith a ruler on the figure, show that the velocity of frame B rela-tive to frame A is 0.6c. (b) Print out a copy of the page. With aruler, draw a third parallelogram that represents a second succes-sive Lorentz transformation, one that lengthens the long diagonalby another factor of 2. Call this third frame C. Use measurementswith a ruler to determine frame C’s velocity relative to frame A.Does it equal double the velocity found in part a? Explain why itshould be expected to turn out the way it does. A general equationfor this type of calculation is derived in problem 22.

√

Problems 95

22 The purpose of this problem is to prove the general re-sult w = (u + v)/(1 + uv) (in units where c = 1) for the kind ofcombination of velocities found graphically in problem 21. Supposethat we perform two Lorentz transformations, with velocities u andv, one after the other. Representing these transformations as dis-tortions of parallelograms, we stretch the stretching diagonals byfactors S(u) and S(v) (and compress the compressing ones by theinverses of these factors), so that the combined result is a stretchingby S(u)S(v). We want to prove that S(w) = S(u)S(v) gives the ex-pression claimed above for w. One can easily show by fiddling withthe result of part c of problem 18 that S(x) =

√(1 + x)/(1− x).

(a) Use these facts to write down an equation relating u, v, and w.(b) Solve for w in terms of u and v. (c) Show that your answerto part b satisfies the correspondence principle. (d) Show that it isconsistent with the constancy of c.

23 Sometimes doors are built with mechanisms that automat-ically close them after they have been opened. The designer canset both the strength of the spring and the amount of friction. Ifthere is too much friction in relation to the strength of the spring,the door takes too long to close, but if there is too little, the doorwill oscillate. For an optimal design, we get motion like x = cte−bt,where x is the position of some point on the door, and c and b arepositive constants. (Similar systems are used for other mechanicaldevices, such as stereo speakers and the recoil mechanisms of guns.)(a) Infer the units of the constants b and c.(b) Find the door’s maximum speed (i.e., the greatest absolute valueof its velocity) for t > 0.

√

(c) Show that your answer has units that make sense.


Galileo’s contradiction of Aristotle had serious consequences. He wasinterrogated by the Church authorities and convicted of teaching that theearth went around the sun as a matter of fact and not, as he had promisedpreviously, as a mere mathematical hypothesis. He was placed under per-manent house arrest, and forbidden to write about or teach his theories.Immediately after being forced to recant his claim that the earth revolvedaround the sun, the old man is said to have muttered defiantly “and yetit does move.” The story is dramatic, but there are some omissions inthe commonly taught heroic version. There was a rumor that the Sim-plicio character represented the Pope. Also, some of the ideas Galileoadvocated had controversial religious overtones. He believed in the exis-tence of atoms, and atomism was thought by some people to contradictthe Church’s doctrine of transubstantiation, which said that in the Catholicmass, the blessing of the bread and wine literally transformed them intothe flesh and blood of Christ. His support for a cosmology in which theearth circled the sun was also disreputable because one of its support-ers, Giordano Bruno, had also proposed a bizarre synthesis of Christianitywith the ancient Egyptian religion.

Chapter 3

Acceleration and free fall

3.1 The motion of falling objectsThe motion of falling objects is the simplest and most commonexample of motion with changing velocity. The early pioneers of

97

a / Galileo dropped a cannonballand a musketball simultaneouslyfrom a tower, and observed thatthey hit the ground at nearly thesame time. This contradictedAristotle’s long-accepted ideathat heavier objects fell faster.

physics had a correct intuition that the way things drop was a mes-sage directly from Nature herself about how the universe worked.Other examples seem less likely to have deep significance. A walkingperson who speeds up is making a conscious choice. If one stretch ofa river flows more rapidly than another, it may be only because thechannel is narrower there, which is just an accident of the local ge-ography. But there is something impressively consistent, universal,and inexorable about the way things fall.

Stand up now and simultaneously drop a coin and a bit of paperside by side. The paper takes much longer to hit the ground. That’swhy Aristotle wrote that heavy objects fell more rapidly. Europeansbelieved him for two thousand years.

Now repeat the experiment, but make it into a race between thecoin and your shoe. My own shoe is about 50 times heavier thanthe nickel I had handy, but it looks to me like they hit the ground atexactly the same moment. So much for Aristotle! Galileo, who hada flair for the theatrical, did the experiment by dropping a bulletand a heavy cannonball from a tall tower. Aristotle’s observationshad been incomplete, his interpretation a vast oversimplification.

It is inconceivable that Galileo was the first person to observe adiscrepancy with Aristotle’s predictions. Galileo was the one whochanged the course of history because he was able to assemble theobservations into a coherent pattern, and also because he carriedout systematic quantitative (numerical) measurements rather thanjust describing things qualitatively.

Why is it that some objects, like the coin and the shoe, have sim-ilar motion, but others, like a feather or a bit of paper, are different?Galileo speculated that in addition to the force that always pulls ob-jects down, there was an upward force exerted by the air. Anyonecan speculate, but Galileo went beyond speculation and came upwith two clever experiments to probe the issue. First, he experi-mented with objects falling in water, which probed the same issuesbut made the motion slow enough that he could take time measure-ments with a primitive pendulum clock. With this technique, heestablished the following facts:

• All heavy, streamlined objects (for example a steel rod droppedpoint-down) reach the bottom of the tank in about the sameamount of time, only slightly longer than the time they wouldtake to fall the same distance in air.

• Objects that are lighter or less streamlined take a longer timeto reach the bottom.

This supported his hypothesis about two contrary forces. Heimagined an idealized situation in which the falling object did not

98 Chapter 3 Acceleration and free fall

c / The v − t graph of a fallingobject is a line.

have to push its way through any substance at all. Falling in airwould be more like this ideal case than falling in water, but evena thin, sparse medium like air would be sufficient to cause obviouseffects on feathers and other light objects that were not streamlined.Today, we have vacuum pumps that allow us to suck nearly all theair out of a chamber, and if we drop a feather and a rock side byside in a vacuum, the feather does not lag behind the rock at all.

How the speed of a falling object increases with time

Galileo’s second stroke of genius was to find a way to make quan-titative measurements of how the speed of a falling object increasedas it went along. Again it was problematic to make sufficiently accu-rate time measurements with primitive clocks, and again he found atricky way to slow things down while preserving the essential physi-cal phenomena: he let a ball roll down a slope instead of dropping itvertically. The steeper the incline, the more rapidly the ball wouldgain speed. Without a modern video camera, Galileo had inventeda way to make a slow-motion version of falling.

b / Velocity increases more gradually on the gentle slope, but themotion is otherwise the same as the motion of a falling object.

Although Galileo’s clocks were only good enough to do accurateexperiments at the smaller angles, he was confident after makinga systematic study at a variety of small angles that his basic con-clusions were generally valid. Stated in modern language, what hefound was that the velocity-versus-time graph was a line. In the lan-guage of algebra, we know that a line has an equation of the formy = ax+ b, but our variables are v and t, so it would be v = at+ b.(The constant b can be interpreted simply as the initial velocity ofthe object, i.e., its velocity at the time when we started our clock,which we conventionally write as vo.)

self-check AAn object is rolling down an incline. After it has been rolling for a shorttime, it is found to travel 13 cm during a certain one-second interval.During the second after that, if goes 16 cm. How many cm will it travelin the second after that? . Answer, p. 524

Section 3.1 The motion of falling objects 99

d / Galileo’s experiments showthat all falling objects have thesame motion if air resistance isnegligible.

e / 1. Aristotle said that heavierobjects fell faster than lighterones. 2. If two rocks are tiedtogether, that makes an extra-heavy rock, which should fallfaster. 3. But Aristotle’s theorywould also predict that the lightrock would hold back the heavyrock, resulting in a slower fall.

A contradiction in Aristotle’s reasoning

Galileo’s inclined-plane experiment disproved the long-acceptedclaim by Aristotle that a falling object had a definite “natural fallingspeed” proportional to its weight. Galileo had found that the speedjust kept on increasing, and weight was irrelevant as long as airfriction was negligible. Not only did Galileo prove experimentallythat Aristotle had been wrong, but he also pointed out a logicalcontradiction in Aristotle’s own reasoning. Simplicio, the stupidcharacter, mouths the accepted Aristotelian wisdom:

SIMPLICIO: There can be no doubt but that a particular body. . . has a fixed velocity which is determined by nature. . .

SALVIATI: If then we take two bodies whose natural speedsare different, it is clear that, [according to Aristotle], on unit-ing the two, the more rapid one will be partly held back bythe slower, and the slower will be somewhat hastened by theswifter. Do you not agree with me in this opinion?

SIMPLICIO: You are unquestionably right.

SALVIATI: But if this is true, and if a large stone moves with aspeed of, say, eight [unspecified units] while a smaller moveswith a speed of four, then when they are united, the systemwill move with a speed less than eight; but the two stoneswhen tied together make a stone larger than that which beforemoved with a speed of eight. Hence the heavier body moveswith less speed than the lighter; an effect which is contrary toyour supposition. Thus you see how, from your assumptionthat the heavier body moves more rapidly than the lighter one,I infer that the heavier body moves more slowly.

What is gravity?

The physicist Richard Feynman liked to tell a story about howwhen he was a little kid, he asked his father, “Why do things fall?”As an adult, he praised his father for answering, “Nobody knows whythings fall. It’s a deep mystery, and the smartest people in the worlddon’t know the basic reason for it.” Contrast that with the averageperson’s off-the-cuff answer, “Oh, it’s because of gravity.” Feynmanliked his father’s answer, because his father realized that simplygiving a name to something didn’t mean that you understood it.The radical thing about Galileo’s and Newton’s approach to sciencewas that they concentrated first on describing mathematically whatreally did happen, rather than spending a lot of time on untestablespeculation such as Aristotle’s statement that “Things fall becausethey are trying to reach their natural place in contact with theearth.” That doesn’t mean that science can never answer the “why”questions. Over the next month or two as you delve deeper intophysics, you will learn that there are more fundamental reasons whyall falling objects have v − t graphs with the same slope, regardless


f / Example 1.

g / Example 2.

of their mass. Nevertheless, the methods of science always imposelimits on how deep our explanation can go.

3.2 AccelerationDefinition of acceleration for linear v − t graphs

Galileo’s experiment with dropping heavy and light objects froma tower showed that all falling objects have the same motion, and hisinclined-plane experiments showed that the motion was described byv = at+vo. The initial velocity vo depends on whether you drop theobject from rest or throw it down, but even if you throw it down,you cannot change the slope, a, of the v − t graph.

Since these experiments show that all falling objects have lin-ear v − t graphs with the same slope, the slope of such a graph isapparently an important and useful quantity. We use the word accel-eration, and the symbol a, for the slope of such a graph. In symbols,a = ∆v/∆t. The acceleration can be interpreted as the amount ofspeed gained in every second, and it has units of velocity divided bytime, i.e., “meters per second per second,” or m/s/s. Continuing totreat units as if they were algebra symbols, we simplify “m/s/s” toread “m/s2.” Acceleration can be a useful quantity for describingother types of motion besides falling, and the word and the symbol“a” can be used in a more general context. We reserve the morespecialized symbol “g” for the acceleration of falling objects, whichon the surface of our planet equals 9.8 m/s2. Often when doingapproximate calculations or merely illustrative numerical examplesit is good enough to use g = 10 m/s2, which is off by only 2%.

Finding final speed, given time example 1. A despondent physics student jumps off a bridge, and falls forthree seconds before hitting the water. How fast is he going whenhe hits the water?

. Approximating g as 10 m/s2, he will gain 10 m/s of speed eachsecond. After one second, his velocity is 10 m/s, after two sec-onds it is 20 m/s, and on impact, after falling for three seconds,he is moving at 30 m/s.

Extracting acceleration from a graph example 2. The x − t and v − t graphs show the motion of a car startingfrom a stop sign. What is the car’s acceleration?

. Acceleration is defined as the slope of the v-t graph. The graphrises by 3 m/s during a time interval of 3 s, so the acceleration is(3 m/s)/(3 s) = 1 m/s2.

Incorrect solution #1: The final velocity is 3 m/s, and accelerationis velocity divided by time, so the acceleration is (3 m/s)/(10 s) =0.3 m/s2.

Section 3.2 Acceleration 101

The solution is incorrect because you can’t find the slope of agraph from one point. This person was just using the point at theright end of the v-t graph to try to find the slope of the curve.

Incorrect solution #2: Velocity is distance divided by time so v =(4.5 m)/(3 s) = 1.5 m/s. Acceleration is velocity divided by time,so a = (1.5 m/s)/(3 s) = 0.5 m/s2.

The solution is incorrect because velocity is the slope of thetangent line. In a case like this where the velocity is changing,you can’t just pick two points on the x-t graph and use them tofind the velocity.

Converting g to different units example 3. What is g in units of cm/s2?

. The answer is going to be how many cm/s of speed a fallingobject gains in one second. If it gains 9.8 m/s in one second, thenit gains 980 cm/s in one second, so g = 980 cm/s2. Alternatively,we can use the method of fractions that equal one:

9.8��ms2 × 100 cm

1��m=

980 cms2

. What is g in units of miles/hour2?

.

9.8 ms2 × 1 mile

1600 m×(

3600 s1 hour

)2

= 7.9× 104 mile/hour2

This large number can be interpreted as the speed, in miles perhour, that you would gain by falling for one hour. Note that we hadto square the conversion factor of 3600 s/hour in order to cancelout the units of seconds squared in the denominator.

. What is g in units of miles/hour/s?

.

9.8 ms2 × 1 mile

1600 m× 3600 s

1 hour= 22 mile/hour/s

This is a figure that Americans will have an intuitive feel for. Ifyour car has a forward acceleration equal to the acceleration of afalling object, then you will gain 22 miles per hour of speed everysecond. However, using mixed time units of hours and secondslike this is usually inconvenient for problem-solving. It would belike using units of foot-inches for area instead of ft2 or in2.

The acceleration of gravity is different in different locations.

Everyone knows that gravity is weaker on the moon, but actu-ally it is not even the same everywhere on Earth, as shown by thesampling of numerical data in the following table.


location latitude elevation (m) g (m/s2)north pole 90◦N 0 9.8322Reykjavik, Iceland 64◦N 0 9.8225Guayaquil, Ecuador 2◦S 0 9.7806Mt. Cotopaxi, Ecuador 1◦S 5896 9.7624Mt. Everest 28◦N 8848 9.7643

The main variables that relate to the value of g on Earth are latitudeand elevation. Although you have not yet learned how g wouldbe calculated based on any deeper theory of gravity, it is not toohard to guess why g depends on elevation. Gravity is an attractionbetween things that have mass, and the attraction gets weaker withincreasing distance. As you ascend from the seaport of Guayaquilto the nearby top of Mt. Cotopaxi, you are distancing yourself fromthe mass of the planet. The dependence on latitude occurs becausewe are measuring the acceleration of gravity relative to the earth’ssurface, but the earth’s rotation causes the earth’s surface to fallout from under you. (We will discuss both gravity and rotation inmore detail later in the course.)

h / This false-color map showsvariations in the strength of theearth’s gravity. Purple areas havethe strongest gravity, yellow theweakest. The overall trend towardweaker gravity at the equator andstronger gravity at the poles hasbeen artificially removed to al-low the weaker local variations toshow up. The map covers onlythe oceans because of the tech-nique used to make it: satelliteslook for bulges and depressionsin the surface of the ocean. Avery slight bulge will occur over anundersea mountain, for instance,because the mountain’s gravita-tional attraction pulls water to-ward it. The US government orig-inally began collecting data likethese for military use, to correctfor the deviations in the paths ofmissiles. The data have recentlybeen released for scientific andcommercial use (e.g., searchingfor sites for off-shore oil wells).

Much more spectacular differences in the strength of gravity canbe observed away from the Earth’s surface:

Section 3.2 Acceleration 103

location g (m/s2)asteroid Vesta (surface) 0.3Earth’s moon (surface) 1.6Mars (surface) 3.7Earth (surface) 9.8Jupiter (cloud-tops) 26Sun (visible surface) 270typical neutron star (surface) 1012

black hole (center) infinite according to some theo-ries, on the order of 1052 accord-ing to others

A typical neutron star is not so different in size from a large asteroid,but is orders of magnitude more massive, so the mass of a bodydefinitely correlates with the g it creates. On the other hand, aneutron star has about the same mass as our Sun, so why is its gbillions of times greater? If you had the misfortune of being on thesurface of a neutron star, you’d be within a few thousand miles of allits mass, whereas on the surface of the Sun, you’d still be millionsof miles from most of its mass.


A What is wrong with the following definitions of g?

(1) “g is gravity.”

(2) “g is the speed of a falling object.”

(3) “g is how hard gravity pulls on things.”

B When advertisers specify how much acceleration a car is capableof, they do not give an acceleration as defined in physics. Instead, theyusually specify how many seconds are required for the car to go from restto 60 miles/hour. Suppose we use the notation “a” for the acceleration asdefined in physics, and “acar ad” for the quantity used in advertisements forcars. In the US’s non-metric system of units, what would be the units ofa and acar ad? How would the use and interpretation of large and small,positive and negative values be different for a as opposed to acar ad?

C Two people stand on the edge of a cliff. As they lean over the edge,one person throws a rock down, while the other throws one straight upwith an exactly opposite initial velocity. Compare the speeds of the rockson impact at the bottom of the cliff.

3.3 Positive and negative accelerationGravity always pulls down, but that does not mean it always speedsthings up. If you throw a ball straight up, gravity will first slowit down to v = 0 and then begin increasing its speed. When Itook physics in high school, I got the impression that positive signsof acceleration indicated speeding up, while negative accelerationsrepresented slowing down, i.e., deceleration. Such a definition wouldbe inconvenient, however, because we would then have to say thatthe same downward tug of gravity could produce either a positive


i / The ball’s acceleration staysthe same — on the way up, at thetop, and on the way back down.It’s always negative.

or a negative acceleration. As we will see in the following example,such a definition also would not be the same as the slope of the v− tgraph

Let’s study the example of the rising and falling ball. In the ex-ample of the person falling from a bridge, I assumed positive velocityvalues without calling attention to it, which meant I was assuminga coordinate system whose x axis pointed down. In this example,where the ball is reversing direction, it is not possible to avoid neg-ative velocities by a tricky choice of axis, so let’s make the morenatural choice of an axis pointing up. The ball’s velocity will ini-tially be a positive number, because it is heading up, in the samedirection as the x axis, but on the way back down, it will be a neg-ative number. As shown in the figure, the v − t graph does not doanything special at the top of the ball’s flight, where v equals 0. Itsslope is always negative. In the left half of the graph, there is anegative slope because the positive velocity is getting closer to zero.On the right side, the negative slope is due to a negative velocitythat is getting farther from zero, so we say that the ball is speedingup, but its velocity is decreasing!

To summarize, what makes the most sense is to stick with theoriginal definition of acceleration as the slope of the v − t graph,∆v/∆t. By this definition, it just isn’t necessarily true that thingsspeeding up have positive acceleration while things slowing downhave negative acceleration. The word “deceleration” is not usedmuch by physicists, and the word “acceleration” is used unblush-ingly to refer to slowing down as well as speeding up: “There was ared light, and we accelerated to a stop.”

Numerical calculation of a negative acceleration example 4. In figure i, what happens if you calculate the acceleration be-tween t = 1.0 and 1.5 s?

. Reading from the graph, it looks like the velocity is about−1 m/sat t = 1.0 s, and around −6 m/s at t = 1.5 s. The acceleration,figured between these two points, is

a =∆v∆t

=(−6 m/s)− (−1 m/s)

(1.5 s)− (1.0 s)= −10 m/s2 .

Even though the ball is speeding up, it has a negative accelera-tion.

Another way of convincing you that this way of handling the plusand minus signs makes sense is to think of a device that measuresacceleration. After all, physics is supposed to use operational defini-tions, ones that relate to the results you get with actual measuringdevices. Consider an air freshener hanging from the rear-view mirrorof your car. When you speed up, the air freshener swings backward.Suppose we define this as a positive reading. When you slow down,the air freshener swings forward, so we’ll call this a negative reading

Section 3.3 Positive and negative acceleration 105

on our accelerometer. But what if you put the car in reverse andstart speeding up backwards? Even though you’re speeding up, theaccelerometer responds in the same way as it did when you weregoing forward and slowing down. There are four possible cases:

motion of car accelerometerswings

slope ofv-t graph

directionof forceacting oncar

forward, speeding up backward + forwardforward, slowing down forward − backwardbackward, speeding up forward − backwardbackward, slowing down backward + forward

Note the consistency of the three right-hand columns — nature istrying to tell us that this is the right system of classification, notthe left-hand column.

Because the positive and negative signs of acceleration dependon the choice of a coordinate system, the acceleration of an objectunder the influence of gravity can be either positive or negative.Rather than having to write things like “g = 9.8 m/s2 or −9.8 m/s2”every time we want to discuss g’s numerical value, we simply defineg as the absolute value of the acceleration of objects moving underthe influence of gravity. We consistently let g = 9.8 m/s2, but wemay have either a = g or a = −g, depending on our choice of acoordinate system.

Acceleration with a change in direction of motion example 5. A person kicks a ball, which rolls up a sloping street, comes toa halt, and rolls back down again. The ball has constant accel-eration. The ball is initially moving at a velocity of 4.0 m/s, andafter 10.0 s it has returned to where it started. At the end, it hassped back up to the same speed it had initially, but in the oppositedirection. What was its acceleration?

. By giving a positive number for the initial velocity, the statementof the question implies a coordinate axis that points up the slopeof the hill. The “same” speed in the opposite direction shouldtherefore be represented by a negative number, -4.0 m/s. Theacceleration is

a = ∆v/∆t= (vf − vo)/10.0 s= [(−4.0 m/s)− (4.0 m/s)]/10.0s

= −0.80 m/s2 .

The acceleration was no different during the upward part of theroll than on the downward part of the roll.

Incorrect solution: Acceleration is ∆v/∆t, and at the end it’s notmoving any faster or slower than when it started, so ∆v=0 and


Discussion question C.

a = 0.

The velocity does change, from a positive number to a negativenumber.

Discussion question B.


A A child repeatedly jumps up and down on a trampoline. Discuss thesign and magnitude of his acceleration, including both the time when he isin the air and the time when his feet are in contact with the trampoline.

B The figure shows a refugee from a Picasso painting blowing on arolling water bottle. In some cases the person’s blowing is speeding thebottle up, but in others it is slowing it down. The arrow inside the bottleshows which direction it is going, and a coordinate system is shown at thebottom of each figure. In each case, figure out the plus or minus signs ofthe velocity and acceleration. It may be helpful to draw a v − t graph ineach case.

C Sally is on an amusement park ride which begins with her chair beinghoisted straight up a tower at a constant speed of 60 miles/hour. Despitestern warnings from her father that he’ll take her home the next time shemisbehaves, she decides that as a scientific experiment she really needsto release her corndog over the side as she’s on the way up. She doesnot throw it. She simply sticks it out of the car, lets it go, and watches itagainst the background of the sky, with no trees or buildings as referencepoints. What does the corndog’s motion look like as observed by Sally?Does its speed ever appear to her to be zero? What acceleration doesshe observe it to have: is it ever positive? negative? zero? What wouldher enraged father answer if asked for a similar description of its motionas it appears to him, standing on the ground?

D Can an object maintain a constant acceleration, but meanwhilereverse the direction of its velocity?

E Can an object have a velocity that is positive and increasing at thesame time that its acceleration is decreasing?

Section 3.3 Positive and negative acceleration 107

3.4 Varying accelerationSo far we have only been discussing examples of motion for whichthe acceleration is constant. As always, an expression of the form∆ . . . /∆ . . . for a rate of change must be generalized to a derivativewhen the rate of change isn’t constant. We therefore define the ac-celeration as a = dv/dt, which is the same as the second derivative,which Leibniz notated as

a =d2x

dt2.

The seemingly inconsistent placement of the twos on the top andbottom confuses all beginning calculus students. The motivationfor this funny notation is that acceleration has units of m/s2, andthe notation correctly suggests that: the top looks like it has units ofmeters, the bottom seconds2. The notation is not meant, however,to suggest that t is really squared.

3.5 Algebraic results for constant accelerationWhen an object is accelerating, the variables x, v, and t are allchanging continuously. It is often of interest to eliminate one ofthese and relate the other two to each other.

Constant acceleration example 6. How high does a diving board have to be above the water if thediver is to have as much as 1.0 s in the air?

. The diver starts at rest, and has an acceleration of 9.8 m/s2.We need to find a connection between the distance she travelsand time it takes. In other words, we’re looking for informationabout the function x(t), given information about the acceleration.To go from acceleration to position, we need to integrate twice:

x =∫ ∫

a dt dt

=∫

(at + vo) dt [vo is a constant of integration.]

=∫

at dt [vo is zero because she’s dropping from rest.]

=12

at2 + xo [xo is a constant of integration.]

=12

at2 [xo can be zero if we define it that way.]

Note some of the good problem-solving habits demonstrated here.We solve the problem symbolically, and only plug in numbers atthe very end, once all the algebra and calculus are done. Oneshould also make a habit, after finding a symbolic result, of check-ing whether the dependence on the variables make sense. Agreater value of t in this expression would lead to a greater value


for x ; that makes sense, because if you want more time in theair, you’re going to have to jump from higher up. A greater ac-celeration also leads to a greater height; this also makes sense,because the stronger gravity is, the more height you’ll need in or-der to stay in the air for a given amount of time. Now we plug innumbers.

x =12

(9.8 m/s2

)(1.0 s)2

= 4.9 m

Note that when we put in the numbers, we check that the unitswork out correctly,

(m/s2) (s)2 = m. We should also check that

the result makes sense: 4.9 meters is pretty high, but not unrea-sonable.

Under conditions of constant acceleration, we can relate velocityand time,

a =∆v

∆t,

or, as in the example 6, position and time,

x =1

2at2 + vot+ xo .

It can also be handy to have a relation involving velocity and posi-tion, eliminating time. Straightforward algebra gives

v2f = v2

o + 2a∆x ,

where vf is the final velocity, vo the initial velocity, and ∆x thedistance traveled.

. Solved problem: Dropping a rock on Mars page 112, problem 13

. Solved problem: The Dodge Viper page 112, problem 11

Section 3.5 Algebraic results for constant acceleration 109

SummarySelected vocabularygravity . . . . . . A general term for the phenomenon of attrac-

tion between things having mass. The attrac-tion between our planet and a human-sized ob-ject causes the object to fall.

acceleration . . . The rate of change of velocity; the slope of thetangent line on a v − t graph.

Notationvo . . . . . . . . . initial velocityvf . . . . . . . . . final velocitya . . . . . . . . . . accelerationg . . . . . . . . . . the acceleration of objects in free fall; the

strength of the local gravitational field

Summary

Galileo showed that when air resistance is negligible all fallingbodies have the same motion regardless of mass. Moreover, theirv− t graphs are straight lines. We therefore define a quantity calledacceleration as the derivative dv/dt. This definition has the advan-tage that a force with a given sign, representing its direction, alwaysproduces an acceleration with the same sign. The acceleration of ob-jects in free fall varies slightly across the surface of the earth, andgreatly on other planets.

For motion with constant acceleration, the following three equa-tions hold:

∆x = vo∆t+1

2a∆t2

v2f = v2

o + 2a∆x

a =∆v

∆t

They are not valid if the acceleration is changing.


Problem 6.

Problem 7.

Problem 9.

ProblemsKey√

A computerized answer check is available online.∫A problem that requires calculus.

? A difficult problem.

1 On New Year’s Eve, a stupid person fires a pistol straight up.The bullet leaves the gun at a speed of 100 m/s. How long does ittake before the bullet hits the ground? . Solution, p. 511

2 What is the acceleration of a car that moves at a steadyvelocity of 100 km/h for 100 seconds? Explain your answer. [Basedon a problem by Hewitt.]

3 You are looking into a deep well. It is dark, and you cannotsee the bottom. You want to find out how deep it is, so you dropa rock in, and you hear a splash 3.0 seconds later. How deep is thewell?

√

4 A honeybee’s position as a function of time is given by x =10t− t3, where t is in seconds and x in meters. What is its acceler-ation at t = 3.0 s? . Solution, p. 511

5 Alice drops a rock off a cliff. Bubba shoots a gun straightdown from the edge of the same cliff. Compare the accelerations ofthe rock and the bullet while they are in the air on the way down.[Based on a problem by Serway and Faughn.]

6 The top part of the figure shows the position-versus-time graphfor an object moving in one dimension. On the bottom part of thefigure, sketch the corresponding v-versus-t graph.

. Solution, p. 512

7 (a) The ball is released at the top of the ramp shown in thefigure. Friction is negligible. Use physical reasoning to draw v − tand a− t graphs. Assume that the ball doesn’t bounce at the pointwhere the ramp changes slope. (b) Do the same for the case wherethe ball is rolled up the slope from the right side, but doesn’t quitehave enough speed to make it over the top. . Solution, p. 512

8 You throw a rubber ball up, and it falls and bounces sev-eral times. Draw graphs of position, velocity, and acceleration asfunctions of time. . Solution, p. 512

9 A ball rolls down the ramp shown in the figure, consisting of acurved knee, a straight slope, and a curved bottom. For each part ofthe ramp, tell whether the ball’s velocity is increasing, decreasing,or constant, and also whether the ball’s acceleration is increasing,decreasing, or constant. Explain your answers. Assume there is noair friction or rolling resistance.

Problems 111

Problem 12.

10 Consider the following passage from Alice in Wonderland, inwhich Alice has been falling for a long time down a rabbit hole:

Down, down, down. Would the fall never come to an end? “Iwonder how many miles I’ve fallen by this time?” she said aloud.“I must be getting somewhere near the center of the earth. Let mesee: that would be four thousand miles down, I think” (for, you see,Alice had learned several things of this sort in her lessons in theschoolroom, and though this was not a very good opportunity forshowing off her knowledge, as there was no one to listen to her, stillit was good practice to say it over)...

Alice doesn’t know much physics, but let’s try to calculate theamount of time it would take to fall four thousand miles, startingfrom rest with an acceleration of 10 m/s2. This is really only a lowerlimit; if there really was a hole that deep, the fall would actuallytake a longer time than the one you calculate, both because thereis air friction and because gravity gets weaker as you get deeper (atthe center of the earth, g is zero, because the earth is pulling youequally in every direction at once).

√

11 In July 1999, Popular Mechanics carried out tests to findwhich car sold by a major auto maker could cover a quarter mile(402 meters) in the shortest time, starting from rest. Because thedistance is so short, this type of test is designed mainly to favor thecar with the greatest acceleration, not the greatest maximum speed(which is irrelevant to the average person). The winner was theDodge Viper, with a time of 12.08 s. The car’s top (and presumablyfinal) speed was 118.51 miles per hour (52.98 m/s). (a) If a car,starting from rest and moving with constant acceleration, coversa quarter mile in this time interval, what is its acceleration? (b)What would be the final speed of a car that covered a quarter milewith the constant acceleration you found in part a? (c) Based onthe discrepancy between your answer in part b and the actual finalspeed of the Viper, what do you conclude about how its accelerationchanged over time? . Solution, p. 513

12 The photo shows Apollo 16 astronaut John Young jumpingon the moon and saluting at the top of his jump. The video footageof the jump shows him staying aloft for 1.45 seconds. Gravity onthe moon is 1/6 as strong as on the earth. Compute the height ofthe jump.

√

13 If the acceleration of gravity on Mars is 1/3 that on Earth,how many times longer does it take for a rock to drop the samedistance on Mars? Ignore air resistance. . Solution, p. 513

14 You climb half-way up a tree, and drop a rock. Then youclimb to the top, and drop another rock. How many times greateris the velocity of the second rock on impact? Explain. (The answeris not two times greater.)


Problem 21.

15 Starting from rest, a ball rolls down a ramp, traveling adistance L and picking up a final speed v. How much of the distancedid the ball have to cover before achieving a speed of v/2? [Basedon a problem by Arnold Arons.] . Solution, p. 513

16 A toy car is released on one side of a piece of track that isbent into an upright U shape. The car goes back and forth. Whenthe car reaches the limit of its motion on one side, its velocity iszero. Is its acceleration also zero? Explain using a v − t graph.[Based on a problem by Serway and Faughn.]

17 A physics homework question asks, “If you start from restand accelerate at 1.54 m/s2 for 3.29 s, how far do you travel by theend of that time?” A student answers as follows:

1.54× 3.29 = 5.07 m

His Aunt Wanda is good with numbers, but has never taken physics.She doesn’t know the formula for the distance traveled under con-stant acceleration over a given amount of time, but she tells hernephew his answer cannot be right. How does she know?

18 Find the error in the following calculation. A student wantsto find the distance traveled by a car that accelerates from rest for5.0 s with an acceleration of 2.0 m/s2. First he solves a = ∆v/∆t for∆v = 10 m/s. Then he multiplies to find (10 m/s)(5.0 s) = 50 m.Do not just recalculate the result by a different method; if that wasall you did, you’d have no way of knowing which calculation wascorrect, yours or his.

19 Acceleration could be defined either as ∆v/∆t or as the slopeof the tangent line on the v − t graph. Is either one superior as adefinition, or are they equivalent? If you say one is better, give anexample of a situation where it makes a difference which one youuse.

20 If an object starts accelerating from rest, we have v2 =2a∆x for its speed after it has traveled a distance ∆x. Explain inwords why it makes sense that the equation has velocity squared, butdistance only to the first power. Don’t recapitulate the derivationin the book, or give a justification based on units. The point isto explain what this feature of the equation tells us about the wayspeed increases as more distance is covered.

21 The graph shows the acceleration of a chipmunk in a TVcartoon. It consists of two circular arcs and two line segments.At t = 0.00 s, the chipmunk’s velocity is −3.10 m/s. What is itsvelocity at t = 10.00 s?

√

Problems 113

22 You take a trip in your spaceship to another star. Setting off,you increase your speed at a constant acceleration. Once you gethalf-way there, you start decelerating, at the same rate, so that bythe time you get there, you have slowed down to zero speed. You seethe tourist attractions, and then head home by the same method.(a) Find a formula for the time, T , required for the round trip, interms of d, the distance from our sun to the star, and a, the magni-tude of the acceleration. Note that the acceleration is not constantover the whole trip, but the trip can be broken up into constant-acceleration parts.(b) The nearest star to the Earth (other than our own sun) is Prox-ima Centauri, at a distance of d = 4× 1016 m. Suppose you use anacceleration of a = 10 m/s2, just enough to compensate for the lackof true gravity and make you feel comfortable. How long does theround trip take, in years?(c) Using the same numbers for d and a, find your maximum speed.Compare this to the speed of light, which is 3.0× 108 m/s. (Laterin this course, you will learn that there are some new things goingon in physics when one gets close to the speed of light, and that itis impossible to exceed the speed of light. For now, though, just usethe simpler ideas you’ve learned so far.)

√?

Problem 23. This spectacular series of photos from a 2011 paper by Bur-rows and Sutton (“Biomechanics of jumping in the flea,” J. Exp. Biology214:836) shows the flea jumping at about a 45-degree angle, but for thesake of this estimate just consider the case of a flea jumping vertically.

23 Some fleas can jump as high as 30 cm. The flea only has ashort time to build up speed — the time during which its center ofmass is accelerating upward but its feet are still in contact with theground. Make an order-of-magnitude estimate of the accelerationthe flea needs to have while straightening its legs, and state youranswer in units of g, i.e., how many “g’s it pulls.” (For comparison,fighter pilots black out or die if they exceed about 5 or 10 g’s.)


24 The speed required for a low-earth orbit is 7.9×103 m/s (seech. 10). When a rocket is launched into orbit, it goes up a little atfirst to get above almost all of the atmosphere, but then tips overhorizontally to build up to orbital speed. Suppose the horizontalacceleration is limited to 3g to keep from damaging the cargo (orhurting the crew, for a crewed flight). (a) What is the minimumdistance the rocket must travel downrange before it reaches orbitalspeed? How much does it matter whether you take into account theinitial eastward velocity due to the rotation of the earth? (b) Ratherthan a rocket ship, it might be advantageous to use a railgun design,in which the craft would be accelerated to orbital speeds along arailroad track. This has the advantage that it isn’t necessary to lifta large mass of fuel, since the energy source is external. Based onyour answer to part a, comment on the feasibility of this design forcrewed launches from the earth’s surface.

25 A person is parachute jumping. During the time betweenwhen she leaps out of the plane and when she opens her chute, heraltitude is given by an equation of the form

y = b− c(t+ ke−t/k

),

where e is the base of natural logarithms, and b, c, and k are con-stants. Because of air resistance, her velocity does not increase at asteady rate as it would for an object falling in vacuum.(a) What units would b, c, and k have to have for the equation tomake sense?(b) Find the person’s velocity, v, as a function of time. [You willneed to use the chain rule, and the fact that d(ex)/dx = ex.]

√

(c) Use your answer from part (b) to get an interpretation of theconstant c. [Hint: e−x approaches zero for large values of x.](d) Find the person’s acceleration, a, as a function of time.

√

(e) Use your answer from part (b) to show that if she waits longenough to open her chute, her acceleration will become very small.

Problems 115

Problem 26.

26 The figure shows a practical, simple experiment for determin-ing g to high precision. Two steel balls are suspended from electro-magnets, and are released simultaneously when the electric currentis shut off. They fall through unequal heights ∆x1 and ∆x2. Acomputer records the sounds through a microphone as first one balland then the other strikes the floor. From this recording, we canaccurately determine the quantity T defined as T = ∆t2 −∆t1, i.e.,the time lag between the first and second impacts. Note that sincethe balls do not make any sound when they are released, we haveno way of measuring the individual times ∆t2 and ∆t1.(a) Find an equation for g in terms of the measured quantities T ,∆x1 and ∆x2.

√

(b) Check the units of your equation.(c) Check that your equation gives the correct result in the casewhere ∆x1 is very close to zero. However, is this case realistic?(d) What happens when ∆x1 = ∆x2? Discuss this both mathemat-ically and physically.


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