Physics for Scientists & l l l
Engineers with Modern Physics
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(Volume 3) 5e (Global Edition)
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By Douglas C. Giancoli
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(Solutions Manual All Chapters,
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100% Original Verified, A+
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Grade) l
(Chapters 36-44) l
All Chapters Solutions Manual
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l Supplement files download link l l l
attheendofthisfile.
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,CHAPTER 36: The Special Theory of Relativity l l l l l l
Responses to Questions
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1. No. Since the windowless car in an exceptionally smooth train moving at a constant velocity is an
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inertial reference frame and the basic laws of physics are the same in all inertial reference frames, there is
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no way for you to tell if you are moving or not. The first postulate of the special theory of relativity can be
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phrased as “no experiment can tell you if an inertial reference frame is at rest or moving uniformly at
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constant velocity.”
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2. The fact that you instinctively think you are moving is consistent with the relativity principle appliedto
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mechanics. Even though you are at rest relative to the ground, when the car next to you creeps forward,
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you are moving backward relative to that car.
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3. Since the railroad car is traveling with a constant velocity, the ball will land back in his hand. Both the
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ball and the car are already moving forward (relative to the ground), so when the ball is thrown straight
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up into the air with respect to the car, it will continue to move forward at the same rate as thecar and fall
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back down to land in his hand.
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4. Whether you say the Earth goes around the Sun or the Sun goes around the Earth depends on your
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reference frame. It is valid to say either one, depending on which frame you choose. The laws of
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physics, though, won’t be the same in each of these reference frames, since the Earth is acceleratingas it
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goes around the Sun. The Sun is nearly an inertial reference frame, but the Earth is not.
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5. The starlight would pass at c, regardless of your spaceship’s speed. This is consistent with the
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second postulate of relativity, which states that the speed of light through empty space is
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independent of the speed of the source or the observer.
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6. The clocks are not at fault and they are functioning properly. Time itself is actually measured to pass
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more slowly in moving reference frames when compared to a rest frame. Any measurement of time
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(heartbeats or decay rates, for instance) would be measured as slower than normal when viewed by an
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observer outside the moving reference frame.
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7. Time actually passes more slowly in the moving reference frame, including aging and other life
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processes. It is not just that it seems this way–time has actually been measured to pass more slowlyin
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the moving reference frame, as predicted by special relativity.
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8. This situation is an example of the “twin paradox” applied to parent–child instead of to twins. This
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situation would be possible if the woman was traveling at high enough speeds during her trip. Time
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would have passed more slowly for her and she would have aged less than her son, who stayed on
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Earth. (Note that the situations of the woman and son are not symmetric; she must undergo
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acceleration during her journey.)
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9. You would not notice a change in your ownheartbeat, mass,height, or waistline. No matter howfast you are
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moving relative to Earth, you are at rest in your own reference frame. Thus, you would not notice any
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changes in your own characteristics. To observers on Earth, you are moving away at 0.6c,which gives
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l= 1.25. If we assume that you are standing up, so that your body is perpendicular to the directionof motion,
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thento theobserversonEarth,itwould appearthatyourheartbeathasslowedby afactor of 1/1.25 = 0.80 and
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that your waistline has decreased by a factor of 0.80 (due to time dilationand length contraction). Your
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height would be unchanged (since there is no relative motion between you and Earth in that direction).
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lAlso note the comments in Section 36–9 of the text on “Rest Mass
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,Physics for Scientists & Engineers with Modern Physics, 5e, Global Edition Instructor Solutions Manual
and RelativisticMass” for commentsaboutmass changeand relativity. Youractual mass hasnotchanged.
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10. Yes, they do occur. However, at a speed of only 90 km/hr, v
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very close to one, so the effects would not be noticeable.
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11. Length contraction and time dilation would not occur. If the speed of light were infinite, v c would
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be 0 for all finite values of v, and therefore γ would always be 1, resulting in t = t0 and l = l0 .
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12. Both the length contraction and time dilation formulas include the term
l 1−v2 c2 . If c were not
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l
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the limiting speed in the universe, then it would be possible to have a situation with v c. However,this
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would result in a negative number under the square root, which gives an imaginary number as a result,
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indicating that c must be the limiting speed. Also, assuming the relativistic formulas were still correct,
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as v gets very close to c, an outside observer should be able to show that
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l = l0 1 − v c is getting smaller and smaller and that the limit as v→c is l → 0.
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show that c is a limiting speed, since nothing can get smaller than having a length of 0. A similar
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to
analysis for time dilation should show that t = is getting longer and longer and that the
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1 − v 2 c2
limit as v →c is t → . This would show that c is a limiting speed, since the slowest that time
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can pass is that it comes to a stop.
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13. If the speed of light was 25 m/s, then we would see relativistic effects all the time, something like the
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Chapter opening figure or Figure 36–16 with Question 21. Everything moving relative to us would be
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length contracted and time dilation would have to be taken into account for many events. There would be
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no “absolute time” on which we would all agree, so it would be more difficult, for instance,to plan to
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meet friends for lunch at a certain time. Many “twin paradox” kind of events would occur,and the
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momentum of moving objects would become very large, making it very difficult to change their motion.
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One of the most unusual changes for today’s modern inhabitants of Earth would be thatnothing would be
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able to move faster than 25 m/s, which is only about 56 mi/h.
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14. No. The relativistic momentum of the electron is given by
l . At low speeds
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1 − v2 c2
(compared to c) this reduces to the classical momentum, p = mv. As v approaches c, γ approaches l l l l l l l l l l l l l l l l l
infinity so there is no upper limit to the electron’s momentum. l l l l l l l l l l
15. No. To accelerate a particle with nonzero rest mass up to the speed of light would require an infinite
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amount of kinetic energy, according to Eq. 36–10a, and so is not possible.
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16. No, E = mc2 does not conflict with the conservation of energy, it actually completes it. Since this
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equation shows us that mass and energy are interconvertible, it says it is now necessary to include
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mass as a form of energy in the analysis of energy conservation in physical processes.
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17. Every observer will measure the speed of a beam of light to be c. Check it with Eq. 36–7d. “Away”from
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the Earth is taken as the positive direction, so “towards” the Earth is the negative direction.
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, Chapter 36 The Special Theory of Relativity
v + u (−c)+ 0.70c
= u= = −c .
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vu 1 + (−1)(0.70)
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l
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1+ 2 l l
c
The beam’s speed (magnitude of velocity), relative to Earth, is c.
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18. Yes. One way to describe the energy stored in the compressed spring is to say it is a mass increase
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(although it would be so small that it could not be measured). This “mass” will convert back to energy
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when the spring is uncompressed.
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19. Matter and energy are interconvertible (matter can be converted into energy and energy can be
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converted into matter). Thus we should say, “Energy can neither be created nor destroyed.”
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20. No, our intuitive notion that velocities simply add is not completely wrong. Our intuition is based onour
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everyday experiences, and at these everyday speeds our intuition is correct regarding how velocities
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add. Our intuition does break down, though, at very high speeds, where we have to take into account
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relativistic effects. Relativity does not contradict classical mechanics, but it is a more general theory
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whereas classical mechanics is a limiting case.
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21. (a) From the girlfriend’s frame of reference, she and her Vespa are at rest while the observer andthe
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streetscape are moving to the left at 70 km/h. As a result the observer and the streetscapewill be
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narrower (in the horizontal direction), and she and her Vespa appear at their original width. The
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observer and streetscape will appear unchanged in the vertical direction.
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Responses to MisConceptual Questions l l l
1. (e) Answer (e) is one of the postulates of special relativity: Light propagates through empty spacewith
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a definite speed c independent of the speed of the source or observer. The other answerscontradict
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this postulate. l l
2. (c, d)
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l frames,” and “A reference frame that moves with constant velocity with respect to an inertial frame
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l is itself also an inertial frame.” So answers (a) and (b) describe inertial frames (answer awith a
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l relative velocity of 0), and answers (c) and (d) describe noninertial frames.
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3. (a) Proper length is the length measured by a person at rest with the object measured. The ship’s
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captain is at rest with the ship, so that measurement is the proper length. l l l l l l l l l l l l l
