special relativity— space and time 15 special relativity...

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THE BIG

IDEA ......

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SPECIAL RELATIVITY—SPACE AND TIME

Everyone knows that we move in time, at the rate of 24 hours per day. And everyone knows that we can move through space, at

rates ranging from a snail’s pace to those of super-sonic aircraft and space shuttles. But relatively few people know that motion through space is related to motion in time.

The first person to understand the relation-ship between space and time was Albert Einstein.15.0

Einstein went beyond common sense when he stated in 1905 that in moving through space we also change our rate of proceeding into the future—time itself is altered. This view was introduced to the world in his special theory of relativity. Ten years later Einstein announced a similar theory, called the general theory of relativity (discussed in the next chapter), that shows how gravity is related to space and time. These theories have enormously changed the way scientists view the workings of the universe.

Motion through space is related to motion in time.

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How are Speed and Length Contraction Related?1. Obtain six soda straws and determine the

length of a single soda straw in centimeters.

2. Multiply the length of a soda straw by the following factors: 1, 0.9999999999999978, 0.999999944, 0.995, 0.5, and 0.045.

3. Use scissors to cut soda straw segments to the lengths determined in Step 2. If you find it impossible to cut the straws to the required lengths, simply leave them uncut.

4. Compare the lengths of the soda straws by placing them one above the other.

Analyze and Conclude1. Observing The cut lengths represent the

effects of length contraction you would observe if a soda straw were moving past you at the following speeds (c represents the speed of light, or roughly 3 � 108 m/s): 0, 20 m/s, 100,000 m/s, 0.1c, 0.87c, and 0.999c.

2. Predicting What fraction of a soda straw’s length would you see if you were moving past a soda straw at a speed of 0.87c?

3. Making Generalizations When do length contraction effects become noticeable?

discover!

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SPECIAL RELATIVITY—SPACE AND TIMEObjectives• Introduce space-time and time

dilation. (15.1)

• Give examples of relative motion. (15.2)

• Discuss the constancy of the speed of light. (15.3)

• Describe time dilation. (15.4)

• Describe why space travel at relativistic speeds seems impossible. (15.5)

• Describe the conditions under which length contracts. (15.6)

discover!

MATERIALS soda straws, scissors

EXPECTED OUTCOME Students will create a model for length contraction.

ANALYZE AND CONCLUDE

Students will relate relativistic speed to length contraction for a straw.

You would see approximately half the soda straw’s length.

Relativistic effects become noticeable at speeds approaching the speed of light and hence are not observed under normal circumstances. Relativistic effects, such as length contraction, must be taken into account in the design of particle accelerators where particles routinely travel at extremely high speeds.

1.

2.

3.

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CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 283

15.1 Space-TimeNewton and other investigators before Einstein thought of space as an infinite expanse in which all things exist. It was never clear whether the universe exists in space, or space exists within the universe. Is there space outside the universe? Or is space only within the universe? The same question could be raised for time. Does the universe exist in time, or does time exist only within the universe? Einstein’s answer to these questions is that both space and time exist only within the uni-verse. There is no time or space “outside.” Einstein reasoned that space and time are two parts of one whole called space-time.

Einstein’s special theory of relativity describes how time is affected by motion in space at constant velocity, and how mass and energy are related. From the viewpoint of special relativity, you travel through a combination of space and time. You travel through space-time. The colorful cloud of gas and dust particles in Figure 15.1 moves through space-time. To begin to understand this, consider your present knowledge that you are moving through time at the rate of 24 hours per day. This is only half the story. To get the other half, con-vert your thinking from “moving through time” to “moving through space-time.” When you stand still, like the girl in Figure 15.2, then all your traveling is through time. When you move a bit, then some of your travel is through space and most of it is still through time. If you were somehow able to travel through space at the speed of light, all your traveling would be through space, with no travel through time!15.1

You would be as ageless as light, for light travels through space only and is timeless. From the frame of reference of a photon traveling from one part of the universe to another, the journey takes no time at all!

� FIGURE 15.1 The universe does not exist in a certain part of infinite space, nor does it exist during a certain era in time. It is the other way around: space and time exist within the universe.

FIGURE 15.2 �

When you stand still, you are traveling at the maximum rate in time: 24 hours per day. If you traveled at the maximum rate through space (the speed of light), time would stand still.

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If you must omit some of the chapters of the text, this and the following chapter can be skipped without consequence to the chapters that follow.

15.1 Space-Time

Key Termsspace-time, special theory of relativity, postulate

� Teaching Tip Explain that though time is one of those concepts with which we are all familiar, it is difficult to define. We can say it is what’s measured by a clock or that it is nature’s way of seeing that everything does not happen at once. Ask students for their thoughts on this.

� Teaching Tip After discussing Einstein and a broad overview of what special relativity is and is not, point out that the theory of relativity is grounded in experiment, and in its development it explained some very perplexing experimental facts (constancy of the speed of light, muon decay, solar energy, the nature of mass, etc.). It is not, as some people think, only the speculations of one man.

� Teaching Tip Challenge your students to think of space and time completely inside the universe as opposed to the universe being located in the vastness of space and time. No universe, no space; no universe, no time! Point out that the universe doesn’t occupy space; space (and time) are in the universe. Said another way, the expanding universe doesn’t spread out to fill a larger emptiness. The only space that exists is the space it creates as it goes.


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Motion in space affects motion in time. Whenever we move through space, we to some degree alter our rate of moving into the future. This is known as time dilation, or the stretching of time. If spacecraft of the future reach sufficient speed, people will be able to travel noticeably in time. They will be able to jump centuries ahead, just as today people can jump from Earth to the moon. The special theory of relativity that Einstein developed rests on two fundamental assumptions, or postulates.

CONCEPTCHECK ...

... How can you describe a person’s travel from the viewpoint of special relativity?

15.2 The First Postulate of Special Relativity

Einstein reasoned that there is no stationary hitching post in the universe relative to which motion should be measured. Instead, all motion is relative and all frames of reference are arbitrary. A space-ship, for example, cannot measure its speed relative to empty space, but only relative to other objects. Look at Figure 15.3. If spaceship A drifts past spaceship B in empty space, spaceman A and spacewoman B will each observe only the relative motion. From this observation each will be unable to determine who is moving and who is at rest, if either.

This is a familiar experience to a passenger in a car at rest waiting for the traffic light to change. If you look out the window and see the car in the next lane begin moving backward, you may be surprised to find that the car you’re observing is really at rest—your car is mov-ing forward. If you could not see out the windows, there would be no way to determine whether your car was moving with constant veloc-ity or was at rest.

Even empty space isn’t really empty. It’s filled with electromagnetic radiation and streams of subatomic particles.

FIGURE 15.3 �

Spaceman A considers himself at rest and sees spacewoman B pass by. But spacewoman B considers herself at rest and sees spaceman A pass by. Spaceman A and spacewoman B will both observe only the relative motion.


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Newspapers during the early part of the twentieth century used to report that there were only 12 people in the world who understood special relativity. This is inaccurate, for although in 1905 there was only one person to understand it, Einstein himself, after Einstein published his paper and explained it, large numbers of people in the community of physicists understood it.

From the viewpoint of special relativity,

you travel though a combination of space and time. You travel through space-time.

T e a c h i n g R e s o u r c e s

• Reading and Study Workbook

• PresentationEXPRESS

• Interactive Textbook

• Conceptual Physics Alive! DVDs Special Relativity I

15.2 The First Postulate of Special Relativity

Key Termfirst postulate of special relativity

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In the cabin of a high-speed jetliner, we flip a coin and catch it just as we would if the plane were at rest. If we swing a pendulum, it will move no differently when the plane is moving uniformly (constant velocity) than when not moving at all. There is no physi-cal experiment we can perform to determine our state of uniform motion. Of course, we can look outside and see Earth whizzing by, or send a radar signal out. However, no experiment confined within the cabin itself can determine whether or not there is uniform motion. The laws of physics within the uniformly moving cabin are the same as those in a stationary laboratory. The person playing pool in Figure 15.4 does not have to make adjustments to his game as long as the ship moves at a constant velocity.

Einstein’s first postulate of special relativity assumes our inability to detect a state of uniform motion. The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference. Many experiments can detect accelerated motion, but none can, according to Einstein, detect the state of uniform motion. No experiment can be performed that will determine whether a closed cabin is at rest or moving at constant velocity.

CONCEPTCHECK ...

... What does the first postulate of special relativity state?

15.3 The Second Postulate of Special Relativity

One of the questions that Einstein as a youth asked himself was, “What would a light beam look like if you traveled along beside it?” According to classical physics, the beam would be at rest to such an observer. The more Einstein thought about this, the more convinced he became of its impossibility. He came to the conclusion that if anobserver could travel close to the speed of light, he would measure the light as moving away from him at 300,000 km/s.15.3

The postulates them-selves don’t have to make “common” sense. As with all postulates in science, the test of their validity is that they lead to predictions that we can test.

� FIGURE 15.4A person playing pool on a smooth and fast-moving ocean liner does not have to make adjustments to compensate for the speed of the ship. The laws of physics are the same for the ship whether it is moving uniformly or is at rest.

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� Teaching Tip Explain that the laws of physics are the same in all uniformly moving reference frames: A bee inside a fast-moving jet plane executes the same flying maneuvers regardless of the speed of the plane, a coin dropped to the floor of the moving plane will fall as if the plane is at rest, and a flight attendant need make no adjustments in pouring tea because of the plane’s high speed. The fact that physical experiments produce the same results in all uniformly moving frames leads to one of the fundamentals of special relativity: The speed of light is seen to be the same by all observers.

� Teaching Tip Emphasize that uniform motion is motion with no change in either speed or direction.

The first postulate of special relativity

states that all the laws of nature are the same in all uniformly moving frames of reference.





15.3 The Second Postulate of Special Relativity

Key Termsecond postulate of special relativity

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Einstein’s second postulate of special relativity assumes that the speed of light is constant. The second postulate of special relativ-ity states that the speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer. As Figure 15.5 shows, the speed of light is constant regardless of the speed of the flashlight or the source.

The speed of light in all reference frames is always the same. Consider, for example, a spaceship departing from the space station shown in Figure 15.6. A flash of light is emitted from the station at 300,000 km/s—a speed we’ll simply call c. No matter what the speed of the spaceship relative to the space station is, an observer on the spaceship will measure the speed of the flash of light passing her as c. If she sends a flash of her own to the space station, observers on the station will measure the speed of these flashes as c. The speed of the flashes will be no different if the spaceship stops or turns around and approaches. All observers who measure the speed of light will find it has the same value, c.

As Figure 15.7 shows, the constancy of the speed of light is what unifies space and time. And for any observation of motion through space, there is a corresponding passage of time. The ratio of space to time for light is the same for all who measure it. The speed of light is a constant.

CONCEPTCHECK ...

... What does the second postulate of special relativity state?

‘

FIGURE 15.5 �

The speed of light is con-stant regardless of the speed of the flashlight or observer.

FIGURE 15.6 �

The speed of a light flash emitted by either the spaceship or the space station is mea-sured as c by observers on the ship or the space station. Everyone who measures the speed of light will get the same value c.

FIGURE 15.7 �

All space and time mea-surements of light are unified by c.

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As you stand still, toss a piece of chalk in the air and catch it. Ask the class to suppose that all measurements show the chalk to have a constant average speed. Call this constant speed c. Proceed to walk at a brisk pace across the room and toss the chalk again. State that from your frame of reference the measured speed is again the same. Ask if the speed looked different to them. They should respond that the chalk was moving faster this time.

Suppose instead that their measurement of speed was the same. Write on the board, with uniformly sized letters,

c 5 SPACE TIME

This represents speed as seen by you in your frame of reference. State that from the frame of reference of the class, the space covered by the tossed chalk was seen to be greater, so write the word SPACE in correspondingly larger letters. Underline it. State that if they measure the same ratio of space to time, then the greater space can be accounted for if the measured time is also greater. Write the enlarged word TIME beneath the underline, equating it to c. All observers measure the same ratio of space to time for light waves in free space.

The speed of light in empty space will

always have the same value.

DemonstrationDemonstration

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15.4 Time DilationEinstein proposed that time can be stretched depending on the motion between the observer and the events being observed. The stretching of time is time dilation. Time dilation occurs ever so slightly for everyday speeds, but significantly for speeds approach-ing the speed of light.

We measure time with a clock. A clock can be any device that measures periodic intervals, such as the swings of a pendulum, the oscillations of a balance wheel, or the vibrations of a quartz crystal. We are going to consider a “light clock,” a rather impractical device, but one that will help to describe time dilation.

A Moving Light Clock Imagine an empty tube with a mirror at each end as shown in Figure 15.8. A flash of light bounces back and forth between the parallel mirrors. The mirrors are perfect reflectors, so the flash bounces indefinitely. If the tube is 300,000 km in length, each bounce will take 1 s in the frame of reference of the light clock. If the tube is 3 km long, each bounce will take 0.00001 s.

Suppose we view the light clock as it whizzes past us in a high-speed spaceship as shown in Figure 15.9. We see the light flash bouncing up and down along a longer diagonal path.

think!But remember the second postulate of relativity: The speed will

be measured by any observer as c. Since the speed of light will not increase, we must measure more time between bounces! For us, look-ing in from the outside, one tick of the light clock takes longer than it takes for occupants of the spaceship. The spaceship’s clock, according to our observations, has slowed down—although, for occupants of the spaceship, it has not slowed at all!

FIGURE 15.8 �A stationary light clock is shown here. Light bounces between paral-lel mirrors and “ticks off” equal intervals of time.

b

a

FIGURE 15.9 �The moving ship contains a light clock. a. An observer moving with the spaceship observes the light flash moving vertically. b. An observer who is passed by the moving ship observes the flash moving along a diagonal path.

Does time dilation mean that time really passes more slowly in moving systems or that it only seems to pass more slowly? Explain.Answer: 15.4.1

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15.4 Time Dilation

Key Termtime dilation

� Teaching Tip Relate the analogy of your chalk-tossing sequence to the light clock discussed in the text and in Figures 15.8, 15.9, 15.10, and 15.11, and also to the material on page 289.

� Teaching Tip Discuss the prospects of “century hopping,” a scenario in which future space travelers may take relatively short trips of a few years or so, and return in decades, centuries, or even thousands of years. This is, of course, pending the solution to two major problems: durable rocket engines and sufficient fuel supplies for prolonged voyages, and a means of shielding astronauts from the radiation that would be produced by impact with interstellar matter.

� Teaching Tip While discussing Figure 15.9, explain that the time for light to go from one mirror to another is shorter for the person on the spaceship than for the person watching light move along the longer diagonal path from the rest frame.


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Einstein showed that the relation between the time t0 (proper

time) in the observer’s own frame of reference and the relative time tmeasured in another frame of reference is

FIGURE 15.10 �The longer distance taken by the light flash in following the diagonal path must be divided by a correspondingly longer time interval to yield an unvarying value for the speed of light.

FIGURE 15.11 �A light clock moves to the right at a constant speed, v.

If you were moving in a spaceship at a high speed relative to Earth, would you notice a difference in your pulse rate? In the pulse rate of the people back on Earth? Explain.Answer: 15.4.2

think!

vc

t1 ( )2

t0

where v represents the relative velocity between the observer and the observed and c is the speed of light. As the equation for time and Figure 15.10 show, the speed of the light clock has no effect on the speed of light.

The slowing of time is not peculiar to the light clock. It is time itself in the moving frame of reference, as viewed from our frame of reference, that slows. The heartbeats of the spaceship occupants will have a slower rhythm. All events on the moving ship will be observed by us as slower. We say that time is stretched—it is dilated.

How do the occupants on the spaceship view their own time? Time for them is the same as when they do not appear to us to be moving at all. Recall Einstein’s first postulate: All laws of nature are the same in all uniformly moving frames of reference. There is no way the spaceship occupants can tell uniform motion from rest. They have no clues that events on board are seen to be dilated when viewed from other frames of reference.


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� Teaching Tip Sketch a simple version of the art below on the board.

Explain that for a ship at rest relative to the two observers on the distant planets, light flashes emitted at 6-min intervals would be seen by both to also be at 6-min intervals. But with motion, the situation is different. Give examples of the Doppler effect: the changing pitch of a car horn when it approaches and when it recedes; the pitter patter of a slanting rain when you run into the rain vs. when you run away from it.


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The mathematical derivation of this equa-tion for time dilation is included here mainly to show that it involves only a bit of geometry and elementary algebra. It is not expected that you master it!

How do occupants on the spaceship view our time?From their frame of reference, it appears that we are the ones who are moving. They see our time running slow, just as we see their time running slow. There is no con-tradiction here. It is physically impossible for observers in different frames of reference to refer to one and the same realm of space-time. The measurements in one frame of reference need not agree with the measurements made in another reference frame. There is only one measurement they will always agree on: the speed of light.

How Can You Derive the Time Dilation Equation?15.4

Figure 15.11 shows three successive positions of the light clock as it moves to the right at constant speed v. The diagonal lines represent the path of the light flash as it starts from the lower mirror at posi-tion 1, moves to the upper mirror at position 2, and then back to the lower mirror at position 3.

The symbol to represents the time it takes for the flash to move between the mirrors as measured from a frame of reference fixed to the light clock. Since the speed of light is always c, the light flash is observed to move a vertical distance cto in the frame of reference of the light clock. This is the distance between mirrors. This vertical dis-tance is the same in both reference frames.

The symbol t represents the time it takes the flash to move from one mirror to the other as measured from a frame of reference in which the light clock moves to the right with speed v. Since the speed of the flash is c and the time to go from position 1 to posi-tion 2 is t, the diagonal distance traveled is ct. During this time t, the clock moves a horizontal distance vt from position 1 to position 2.

These three distances make up a right triangle in the figure, in which ct is the hypotenuse, and cto and vt are legs. A well-known theorem of geometry (the Pythagorean theorem) states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. If we apply this to the figure, we obtain

(ct)2

(ct)2 (vt)2

t2[1 (v2/c2)]

t2

t

(ct )2 (vt)2

(ct )2

t 2

t 2

1 (v2/c2)

t

1 (v2/c2)

do the math!FIGURE 15.12 �Physicist Ken Ford empha-sizes the meaning of the time dilation equation with his ninth-grade high school students.

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� Teaching Tip Ask your class to suppose the ship in your sketch moves so fast toward the right observer that the flashes reach the observer at twice the frequency—with flashes closer together so they appear at 3-min intervals. The time between flashes is cut in half. Ask how the time between flashes would be seen by the observer on the left—who sees the source receding. Is it reasonable to say the opposite occurs? That instead of being squeezed together, the flashes are spread apart twice as much so that the time between flashes is doubled? That if 6-min flash intervals are cut to 3-min for approach, they’ll be stretched to 12-min intervals for recession? Once this is acceptable to your class, go on to discuss the Twin Trip.


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The Twin Trip A dramatic illustration of time dilation is afforded by identical twins, one an astronaut who takes a high-speed round-trip journey while the other stays home on Earth. As Figure 15.13 shows, when the traveling twin returns, he is younger than the stay-at-home twin. How much younger depends on the relative speeds involved. If the traveling twin maintains a speed of 50% the speed of light for one year (according to clocks aboard the spaceship), 1.15 years will have elapsed on Earth. If the traveling twin maintains a speed of 87% the speed of light for a year, then 2 years will have elapsed on Earth. At 99.5% the speed of light, 10 Earth years would pass in one spaceship year. At this speed the traveling twin would age a single year while the stay-at-home twin ages 10 years.

The question arises, since motion is relative, why isn’t it just as well the other way around—why wouldn’t the traveling twin return to find his stay-at-home twin younger than himself? Aha, there’s a fundamental difference here. The space-traveling twin experiences two frames of reference in his round trip—one receding from Earth, and the other approaching Earth. He has been in two realms of space-time, separated by the event of turning around. The stay-at-home twin, on the other hand, experiences a single frame of refer-ence—one realm of space-time.

Please do the practice pages on The Twin Trip in the Concept Development Practice Book. You’ll see that the twins can meet again at the same place in space only at the expense of time.

FIGURE 15.13 �The traveling twin does not age as fast as the stay-at-home twin.

Cosmonaut Sergei Avdeyev spent more than two years aboard the orbiting Mir space station, and due to time dilation is today two-hundredths of a second younger than he would be if he’d never been in space!

For:Visit:Web Code: –

Links on relativity of time www.SciLinks.org csn 1504

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Be patient with your students as they ponder the Twin Trip. It may take more than a class period to develop these ideas.



Clockwatching on a Trolley-Car Ride Pretend you are Einstein in a trolley car that provided the high-speed travel back then. Suppose the trolley car, like the one shown in Figure 15.14, is moving in a direction away from a huge clock displayed in a village square. The clock reads 12 noon. To say it reads 12 noon is to say that light carrying the information “12 noon” is reflected by the clock and trav-els toward you along your line of sight. If you suddenly move your head to the side, instead of meeting your eye, the light carrying the information continues past, presumably out into space. Out there an observer who later receives the light says, “Oh, it’s 12 noon on Earth now” (or more correctly, “light left the clock at 12 noon on Earth”). You and the distant observer will see 12 noon at different times. You wonder more about this idea. If the trolley car traveled as fast as the light, then it would keep up with the information that says “12 noon.” Traveling at the speed of light, then, tells the time is always 12 noon at the village square. Time at the village square is frozen!

If the trolley car is not moving, you see the village-square clock move into the future at the rate of 60 seconds per minute; if you move at the speed of light, you see seconds on the clock taking infi-nite time. These are the two extremes. What’s in between? What hap-pens for speeds less than the speed of light?

A little thought will show that you will receive the message “1 o’clock” anywhere from 60 minutes to an infinity of time after you receive the message “12 noon,” depending on what your speed is between the extremes of zero and the speed of light. From your high- speed (but less than c) moving frame of reference, you see all events taking place in the reference frame of the clock on Earth as happen-ing in slow motion. As Figure 15.15 shows, 1 second on a stationary clock is stretched out, as measured on a moving clock. If you reverse direction and travel at high-speed back toward the clock, you’ll see all events occurring in the clock’s frame of reference as being speeded up. When you return and are once again sitting in the square, will the effects of going and coming compensate each other? Amazingly, no! Time will be stretched. The wristwatch you were wearing the whole time and the village clock will disagree. This is time dilation.

CONCEPTCHECK ...

... How does time dilation at everyday speeds compare with time dilation at light speed?

FIGURE 15.14 �Light that carries the information “12 noon” is reflected by the clock and travels toward the trolley.

Will observers A and B agree on measurements of time if A moves at half the speed of light relative to B? If both A and B move together at 0.5c relative to Earth? Explain. Answer: 15.4.3

think! FIGURE 15.15 �

The graph shows how 1 second on a stationary clock is stretched out, as measured on a moving clock.

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Time dilation occurs ever so slightly for

everyday speeds, but significantly for speeds approaching the speed of light.



• Concept-Development Practice Book 15-1, 15-2

• Problem-Solving Exercises in Physics 9-1



• Next-Time Questions 15-1

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15.5 Space and Time TravelBefore the theory of special relativity was introduced, it was argued that humans would never be able to venture to the stars. It was thought that our life span is too short to cover such great distances—at least for the distant stars. Alpha Centauri is the nearest star to Earth, after the sun, and it is 4 light-years away.15.5 It was therefore thought that a round-trip even at the speed of light would require 8 years. The center of our galaxy is some 30,000 light-years away, so it was reasoned that a person traveling even at the speed of light would have to survive for 30,000 years to make such a voyage! But these arguments fail to take into account time dilation. Time for a person on Earth and time for a person in a high-speed spaceship are not the same.

A person’s heart beats to the rhythm of the realm of time it is in. One realm of time seems the same as any other realm of time to the person, but not to an observer who is located outside the person’s frame of reference—for she sees the difference. As an example, astro-nauts traveling at 99% the speed of light could go to the star Procyon (11.4 light-years distant) and back in 23.0 years in Earth time. It would take light itself 22.8 years in Earth time to make the same round trip. Because of time dilation, it would seem that only 3 years had gone by for the astronauts. All their clocks would indicate this, and biologically they would be only 3 years older. It would be the space officials greet-ing them on their return who would be 23 years older.

At higher speeds the results are even more impressive. At a speed of 99.99% the speed of light, travelers could travel slightly more than 70 light-years in a single year of their own time. At 99.999% the speed of light, this distance would be pushed appreciably farther than 200 years. A 5-year trip for them would take them farther than light travels in 1000 Earth-time years.

If traveling backward in time were possible, wouldn’t we have tour-ists from the future?

Relativistic Clocks

In 1971 atomic clocks were carried around Earth in jet planes. Upon landing, the traveling clocks were a few billionths of a second “younger” than twin clocks that stayed behind. Atomic clocks now cruise overhead at even greater speeds in the satellites that are part of the global positioning system (GPS). In designing this system, which can pinpoint positions on Earth to

within meters, scientists and engineers had to accommodate for relativistic time dilation. If they didn’t, GPS could not precisely locate positions on Earth. Time dilation is a fact of everyday life to scientists and engineers—especially those who design equipment for global navigation work.

Link to TECHNOLOGY


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15.5 Space and Time Travel

The topic of space and time travel often leads to good class discussions (and topics for papers!).


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Such journeys seem impossible to us today. The amounts of energy required to propel spaceships to relativistic speeds are bil-lions of times the energy used to put the space shuttles into orbit. The problems of shielding radiation induced by these high speeds seems formidable. The practicalities of such space journeys are pro-hibitive, so far. For the present, interstellar space travel must be rel-egated to science fiction. This is not because of scientific fantasy, but simply because of the impracticality of space travel. Traveling close to the speed of light in order to take advantage of time dilation is com-pletely consistent with the laws of physics.

If these problems are ever overcome and space travel becomes routine, people might have the option of taking a trip and returning in future centuries of their choosing. For example, one might depart from Earth in a high-speed ship in the year 2150, travel for 5 years or so, and return in the year 2500. One might live among Earthlings of that period for a while and depart again to try out the year 3000 for style. People could keep jumping into the future with some expense of their own time, but they could not travel into the past. They could never return to the same era on Earth that they bid farewell to.

Time, as we know it, travels only one way—forward. Here on Earth we constantly move into the future at the steady rate of 24 hours per day. An astronaut leaving on a deep-space voyage must live with the fact that, upon her return, much more time will have elapsed on Earth than she has experienced on her voyage. Star travel-ers will not bid “so long, see you later” to those they leave behind but, rather, a permanent “good-bye.”

CONCEPTCHECK ...

... Why does space travel at relativistic speeds seem impossible?

� FIGURE 15.16From Earth’s frame of reference, light takes 30,000 years to travel from the center of the Milky Way galaxy to our solar system.

You can see into the past, but you cannot go into the past. When you look at stars or galaxies at night, you’re looking at light that’s been on its way to you for doz-ens, hundreds, even mil-lions of years. You can only see the universe as it was in the past.

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� Teaching Tip Present this interesting but fictitious example of time dilation: Suppose that one could be whirled in a giant centrifuge up to relativistic speeds without physical injury. (Of course, in reality, one would be crushed to death in such a case, but pretend that somehow one is physically unaffected by the crushing centripetal forces—the fictitiousness of this example). Explain how one taking a “ride” in such a centrifuge might be strapped in a seat and told to press a button on the seat when he or she wishes the ride terminated. And suppose that after being whirled about at rim speeds near the speed of light, the occupant decides that 10 minutes is enough so he or she presses the button, signaling those outside to bring the machine to a halt. After the machine is halted, those outside open the door, peer in, and ask, “Good gosh, what have you been doing in there for the past 3 weeks!” In the laboratory frame of reference, 3 weeks would have elapsed during a 10-min interval in the rotating centrifuge. The point: One does not have to necessarily travel through wide expanses of space for time dilation to be significant. Motion in space, rather than space itself, is the key factor.

The amounts of energy required to

propel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit.





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15.6 Length ContractionFor moving objects, space as well as time undergoes changes.

When an object moves at a very high speed relative to an observer, its measured length in the direction of motion is con-tracted. The observable shortening of objects moving at speeds approaching the speed of light is length contraction. The amount of contraction is related to the amount of time dilation. For everyday speeds, the amount of contraction is much too small to be measured. For relativistic speeds, the contraction would be noticeable. As Figure 15.17 shows, a meterstick aboard a spaceship whizzing past you at 87% the speed of light, for example, would appear to you to be only 0.5 meter long. If it whizzed past at 99.5% the speed of light, it would appear to you to be contracted to one tenth its original length. The width of the stick, perpendicular to the direction of travel, doesn’t change. As relative speed gets closer and closer to the speed of light, the measured lengths of objects contract closer and closer to zero.

Do people aboard the spaceship also see their metersticks—and everything else in their environment—contracted? The answer is no. People in the spaceship see nothing at all unusual about the lengths of things in their own reference frame. If they did, it would violate the first postulate of relativity. Recall that all the laws of physics are the same in all uniformly moving reference frames. Besides, there is no relative speed between the people on the spaceship and the things they observe in their own reference frame. However, there is a relative speed between themselves and our frame of reference, so they will see our metersticks contracted—and us as well. As Figure 15.18 shows, a rule of relativity is that changes due to alterations of space-time are always seen in the frame of reference of the “other guy.”

Muons and Mutations

When cosmic rays bombard atoms at the top of the atmosphere, new particles are made. Some are muons, radio-active particles that streak downward toward Earth’s surface. A muon’s average lifetime is only two millionths of a sec-ond, seemingly too brief to reach the ground below before decay-ing. But because muons move at nearly the speed of light, length contraction dramatically shortens their distance to Earth. You are hit by hundreds of muons every second! Muon impact, like that of all high-speed elementary particles, causes bio-logical mutations. So we see a link between the effects of relativity and the evolution of living creatures on Earth.

Link to BIOLOGY

FIGURE 15.17 �A meterstick traveling at 87% the speed of light relative to an observer would be measured as only half as long as normal.

FIGURE 15.18 �In the frame of reference of the meterstick on the spaceship, its length is 1 meter. Observers from this frame see our metersticks contracted. The effects of relativity are always attributed to “the other guy.”

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15.6 Length Contraction

Key Termlength contraction

Common Misconception Objects can go faster than the speed of light from some frames of reference.

FACT If an object could reach the speed of light, its length would contract to zero which means that such an apparent speed is impossible.

� Teaching Tip Hold up a meter stick horizontally, and state that if your students made accurate measurements of its length, their measurements would agree with yours. Everyone would measure it as 1 m long. People at the back of the room would have to compensate for its shorter appearance due to distance, but nevertheless, they would agree on its 1-m length.

� Teaching Tip Walk across the room holding the meter stick like a spear. State that your measurements and those of your students would now differ. State that if you were to travel at 87% the speed of light, relative to the class, they would measure the stick to be half as long, 0.5 m. At 99.5% the speed of light, they would see it as only 10 cm long. At greater speeds, it would be even shorter. If you were traveling at the speed of light, the length of the stick would contract to zero from the point of view of the students.

� Teaching Tip Point out that as far as people aboard a spaceship are concerned, they are at rest and other things move away from or toward them.



The contraction of speeding objects is the contraction of space itself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference. So if an object, like the baseball in Figure 15.19, is moving horizontally, no contraction takes place vertically.

Relativistic length contraction is stated mathematically:

v2

c2L L0 1 ( )

In this equation, v is the speed of the object relative to the observer, cis the speed of light, L is the length of the moving object as measured by the observer, and L

0 is the measured length of the object at rest.15.6

Suppose that an object is at rest, so that v � 0. When 0 is substi-tuted for v in the equation, we find L � L

0, as we would expect. It was

stated earlier that if an object were moving at 87% the speed of light, it would contract to half its length. When 0.87c is substituted for v in the equation, we find L � 0.5L

0. Or when 0.995c is substituted for v,

we find L � 0.1L0, as stated earlier. If the object could reach the speed

c, its length would contract to zero. This is one of the reasons that the speed of light is the upper limit for the speed of any object.

Einstein’s theory of relativity has raised many philosophical ques-tions. What, exactly, is time? Can we say it is nature’s way of seeing to it that everything does not all happen at once? And why does time seem to move in one direction? Has it always moved forward? Are there other parts of the universe where time moves backward? Perhaps these unanswered questions will be answered by the physi-cists of tomorrow. How exciting!

CONCEPTCHECK ...... How does the length of an object change when it is

moving at a very high speed relative to an observer?

� FIGURE 15.19As relative speed increases, contraction in the direc-tion of motion increases. Lengths in the perpendicu-lar direction do not change.

In summary—Time dila-tion: moving clocks run slowly. Length contrac-tion: moving objects are shorter (in the direction of motion).

think!A spacewoman travels by a spherical planet so fast that it appears to her to be an ellipsoid (egg shaped). If she sees the short diameter as half the long diameter, what is her speed relative to the planet?Answer: 15.6

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� Teaching Tip Write the length-contraction formula on the board. State that contraction takes place only in the direction of motion. The stick moving in spear fashion appears shorter but it doesn’t appear thinner. Contrast the students’ view of the stick with yours. Since you move with the stick you see no contraction, whatever the speed. From your frame of reference, the v in the equation is zero, and so L 5 Lo. Contraction depends on the frame of reference.

� Teaching Tip With space travel between stars, the distance as seen from our rest frame of reference is quite different than it is as seen from the frame of reference of a moving spaceship. If a distance of 20 light-years separates a pair of stars in our frame of reference, a spaceship traveling at 0.87c between them would see them as only 10 light-years apart.

Ask Consider a pair of stars, one on each “edge” of the universe. Now consider a photon traveling from one star across the entire universe to the other. From the frame of reference of the photon, what is the distance of separation between stars? Zero! From a frame of reference traveling at c, the length contraction reaches zero.

When an object moves at a very high

speed relative to an observer, its measured length in the direction of motion is contracted.


• Next-Time Question 15-2

• Problem-Solving Exercises in Physics 9-2

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15.4.1 The slowing of time in moving systems is not merely an illusion resulting from motion. Time really does pass more slowly in a moving system compared with one at relative rest.

15.4.2 There would be no relative speed between you and your own pulse, so no relativistic effects would be noticed. There would be a relativistic effect between you and people back on Earth. You would find their pulse rate slower than normal (and they would find your pulse rate slower than normal). Relativity effects are always attributed to “the other guy.”

15.4.3 When A and B have different motions relative to each other, each will observe a slowing of time in the frame of reference of the other. So they will not agree on mea-surements of time. When they are moving in unison, they share the same frame of reference and will agree on measurements of time. They will see each other’s time as passing normally, and each one will see events on Earth in the same slow motion.

15.6 The spacewoman passes the spherical planet at 87% the speed of light.

think! AnswersConcept Summary ••••••

• From the viewpoint of special relativity, you travel through a combination of space and time. You travel through space-time.

• The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference.

• The second postulate of special relativ-ity states that the speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer.

• Time dilation occurs ever so slightly for everyday speeds, but significantly for speeds approaching the speed of light.

• The amounts of energy required to pro-pel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit.

• When an object moves at a very high speed relative to an observer, its mea-sured length in the direction of motion is contracted.

Key Terms ••••••

space-time (p. 283)

special theory of relativity (p. 283)

postulate (p. 284)

first postulate of special relativity (p. 285)

second postulate of special relativity (p. 286)

time dilation (p. 287)

length contrac-tion (p. 294)

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CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 297 CHAPTER 15 SPECIAL RELATIVITY—SPACE AND TIME 297

15 ASSESS

Check Concepts ••••••

Section 15.1 1. What is space-time?

2. Can you travel while remaining in one place in space? Explain.

3. Does light travel through space? Through time? Through both space and time?

4. What is time dilation?

Section 15.2 5. What is the first postulate of special

relativity?

Section 15.3 6. What is the second postulate of special

relativity?

7. The ratio of velocity gain to time for a freely falling body is g. Similarly, what is the ratio of distance to time for light waves?

Section 15.4 8. The path of light in a vertical “light clock” in

a high-speed spaceship is seen to be longer when viewed from a stationary frame of ref-erence. Why, then, does the light not appear to be moving faster?

9. If we view a passing spaceship and see that the inhabitants’ time is running slow, how do they see our time running?

10. If you were traveling in a high-speed rocket ship, would clocks on board appear to you to be running slow? Defend your answer.

11. Is it possible for a person with a 70-year life span to travel farther than light travels in 70 years? Explain.

Section 15.5 12. What are the present-day obstacles to

interstellar space travel?

Section 15.6 13. How long would a meterstick appear if it

were thrown like a spear at 99.5% the speed of light?

14. How long would a meterstick appear if it were traveling at 99.5% the speed of light, but with its length perpendicular to its direction of motion? (Why are your answers to this question and the last question different?)

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ASSESS

Check Concepts 1. Space and time are two parts

of one whole.

2. Yes, in time

3. yes; no; no

4. Stretching of time due to motion in space

5. The laws of nature are the same in all uniformly moving frames.

6. c is always the same whether the source, the receiver, or both move.

7. c, the speed of light

8. It takes correspondingly more time.

9. Slow also; each sees the same effect in the other.

10. No. In your frame, there is no time dilation. Relative speed of you and clocks is zero.

11. Yes, a person moving at speeds close to c lives longer.

12. Fuel energy and radiation

13. One-tenth as long

14. No change; only change is parallel to motion, not perpendicular.


ASSESS

Concept Summary ••••••

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15 15. If you were traveling in a high-speed space-

ship, would metersticks on board appear contracted to you? Defend your answer.

Think and Rank ••••••

Rank each of the following sets of scenarios in order of the quantity or property involved. List them from left to right. If scenarios have equal rankings, then separate them with an equal sign. (e.g., A � B)

16. A spaceship emits brief flashes of light at 1-second intervals. The circles represent light already emitted by the spaceship.

A B C

Rank the following quantities from greatest to least.

a. the speeds at which the flashes reach an observer to the right, in front of the approaching spaceship

b. how frequently the flashes reach the same observer

c. the speeds of the spaceship as seen by you, an Earth observer

17. Three spaceships shoot space probes at the speeds shown.

A

B

C

Rank the following quantities from greatest to least.

a. the speeds of the probes as seen by an Earth observer

b. the speed of light reflected from the de-parting probes as seen by the spaceship

Plug and Chug ••••••

18. If a spaceship moves away from you at half the speed of light and fires a rocket away from you at half the speed of light relative to the spaceship, common sense may tell you the rocket moves at the speed of light relative to you. But it doesn’t! The relativistic addi-tion of velocities (not covered in the chapter) is given by

v1 v2

1v v1v2

c2

Substitute 0.5c for both v1 and v

2 and show

that the velocity, v, of the rocket relative to you is 0.8c.


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15. No. In your frame, there is no length contraction. The relative speed of you and sticks is zero.

Think and Rank 16. a. A 5 B 5 C

b. C, A, Bc. C, A, B

17. a. A, C, Bb. A 5 B 5 C

Plug and Chug 18. v 5 (0.5c 1 0.5c) 4

[1 1 (0.5c)2/c2] 5 c/1.25 5 0.8c


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15 ASSESS


19. If the spaceship in question 18 somehow travels at c relative to you, and it somehow fires its rocket at c relative to itself, use the equation to show that the speed of the rocket relative to you is still c!

20. Substitute small values of v1 and v

2 in

the preceding equation and show for everyday speeds that v is practically equal to v

1 + v

2.

Think and Explain ••••••

21. If you were in a smooth-riding train with no windows, could you sense the difference between uniform motion and rest? Between accelerated motion and rest? Explain how you could do this with a bowl filled with water.

22. Suppose you’re playing catch with a friend in a moving train. When you toss the ball in the direction the train is moving, how does the speed of the ball appear to an observer standing at rest outside the train? (Does it increase or appear the same as if the observer were riding on the train?)

23. Suppose you’re shining a light while riding on a train. When you shine the light in the direction the train is moving, how would the speed of light appear to an observer standing at rest outside the train? (Does it increase or appear the same as if the observer were riding with the train?)

24. People who ride in a bus all know they’re moving through space. But you know that they’re also moving through something else. What else are they moving through?

25. Light travels a certain distance in, say, 10,000 years. Can an astronaut travel more slowly than the speed of light and yet travel the same distance in a 10-year trip? Explain.

26. Can you get younger by traveling at speeds near the speed of light? Explain.

27. Explain why it is that when we look out into the universe, we see into the past.

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19. v 5 (c 1 c)/[1 1 (c2/c2)] 5 2c/(1 1 1) 5 c

20. For small speeds, v1v2/c2 < 0, so v 5 (v1 1 v2)/(1 1 0) 5 v1 1 v2.

Think and Explain 21. No; yes; by observing that the

surface of water in a bowl is not horizontal

22. It appears to travel faster.

23. Same; the speed of light is invariant.

24. Time

25. Yes, because of time dilation

26. No, the effects of relativity always are attributed to the “other guy.”

27. It takes time for light to travel from one place to another, and so we always see distant places as they were when the light left—in the past.


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Concept Summary ••••••

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15 28. One of the fads of the future might be

“century hopping,” where occupants of high-speed spaceships would depart from Earth for a few years and return centuries later. What are the present-day obstacles to such a practice?

29. If you were in a high-speed spaceship traveling away from Earth at a speed close to that of light, would you measure your nor-mal pulse to be slower, the same, or faster? How would your measurements of pulses of friends back on Earth be if you could moni-tor them from your ship? Explain.

30. Is it possible for a person to be biologi-cally older than his or her parents? Explain.

31. If stationary observers measure the shape of an emblem on fast-moving rocket ship as exactly circular, then what is the shape according to observers on the rocket ship?

32. The two-mile-long linear accelerator at Stanford University in California is less than a meter long to the electrons that travel in it. Explain.

Think and Solve ••••••

33. Joe Burpy is 30 years old and has a daughter who is 6 years old. Joe leaves on a space bus and takes a 5-year (space-bus time) round-trip at 0.99c. How old will he and his daughter be when he returns?

34. Assume that your heart normally beats once every second, and that you are in a space-ship that moves past Earth at 0.6c.

a. What time do you measure between your own heartbeats?

b. Show that your heartbeats are measured by someone on Earth to be 1.25 s apart.

c. As you and the spaceship whiz past Earth, you make similar measurements on Earthlings who measure their own heart-beats to be 1 s apart. How much time do you measure between their heartbeats?

35. Thomas, a rhino, is 2.5 meters long when at rest.

a. How long will you measure him to be when he’s running by at 0.80c?

b. Show that you would measure a time of 6.3 ns for the length of his body to pass you.


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28. No means of propelling such a massive body to such speeds; no way of effectively shielding the occupants from the radiation resulting from the high-speed collisions with interstellar matter.

29. Same; theirs would appear to be slower. No relativistic changes occur in your own frame of reference.

30. Yes, if the person stays behind while the parents take a relativistic trip.

31. The shape is elliptical, with the long axis in the direction of motion.

32. To the electrons, length contraction shortens their journey.

Think and Solve 33. Joe will be 30 1 5 5 35 years

old when he returns. Due to time dilation, his stay-at-home daughter experiences

t 5 (5 yr)/√1 2(0.99c)2/c2 5 35.4 yr. Her age will be 6 1 35.4 5 41.4 yr.

34. a. You measure 1 s, because you’re at rest in your reference frame.b. t 5 t0/√1 2 v2 / c2 5

1 s/ √1 2 (0.6c)2 / c2 5 1.25 sc. Same as above, t 5 1.25 s.

35. a. L 5 L0√1 2 (v/c)2 5 2.5 m 3

√1 2 (0.8c)2 / c2 5 1.5 mb. From v 5 d/t, t 5 Lyou measure/v 5 1.5 m/[0.8 3

(3.00 3 108 m/s)] 5 6.3 3 1029 s 5 6.3 ns


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15 ASSESS


36. A ship whizzes by you at 0.60c. Someone aboard is making a 3-minute egg for breakfast.

a. What cooking time will you measure for the egg?

b. Why should you not be surprised when the egg turns out to be perfectly cooked, rather than overcooked?

37. Before leaving the planet Hislaurels for a starship voyage, you pack a meterstick in your luggage. After the ship has settled down to a steady speed of 0.50c, you take the meterstick out of your bag.

a. How long will you measure the meterstick to be?

b. If the meterstick is moving parallel to an observer resting on Hislaurels, how long will the observer measure the meterstick?

38. You are standing facing forward on the floor of your starship, which is moving at 0.80crelative to Earth. Before you left Earth, you measured your feet to be 25 cm long.

a. People on Earth will now measure your feet to be how long?

b. Do you need to be concerned now that the shoes that you packed for the trip will be too big?

39. Pinocchio is concerned that Gepetto will see his long nose and realize that he has been lying. So Pinocchio decides to run past Ge-petto fast enough that his 10-inch long nose will be seen by Gepetto to be only 2 inches long.

a. How fast must Pinocchio run?b. When running past Gepetto, will Pinoc-

chio see Gepetto’s nose shortened?

40. Lizzie is scooting down the Interstate at 17 percent of the speed of light and mea-sures the distance between mileposts to be less than 5,280 feet.

a. What distance does she measure?b. What distance would she measure at

32 percent the speed of light?

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36. a. tyou 5 tship/ √1 2 v2 / c2 5

3 min/ √1 2 (0.6c)2 / c2 5 3 min/0.8 5 3.8 minb. The cook and the egg both age 3 min in their own frames of reference. Times differ in differently moving frames of reference.

37. a. No relative motion between you and stick, so you measure it to be 1 m long.b. An observer at rest sees the meterstick at v 5 0.50c, so length measures L 5

L0√1 2 (v / c)2 5 1.00 m 3 √1 2 (0.5c)2 / c2 5 0.87 m

38. a. L 5 L0√1 2 (v / c)2 5 25 cm

√1 2 (0.8c)2 / c2 5 25 cm(0.60) 5 15 cmb. Your feet and shoes are in the same frame, so no relativistic changes. Earth observers say feet and shoes contract by same amount so shoes should still fit—moving or not.

39. a. From L 5 L0√1 2 (v / c)2,

v 5 c √1 2 (L / L0)2, so v 5

c√1 2 (2 / 10)2 5 0.98cb. Yes, both see the other’s nose shortened.

40. a. L 5 L0√1 2 (v/c)2 5 5280 ft 3

√1 2 (0.17c / c)2 5 5200 ft

b. L 5 L0√1 2 (v/c)2 5 5280 ft 3

√1 2 (0.32c / c)2 5 5000 ft


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