All answers have been checked against the answer key, and should be presumed to be correct. You should ask for help in the recitations if you are unable to obtain these results.
6. v = 0.773c = 2.32 x 108 m/s
10. (a) 78.9 years on Earth, (b) 24.6 years on the spaceship, (c) 23.4 light-years on the spaceship, (d) 0.950 c = 2.85 x 108 m/s
16. The fractional change in the mass is 6.8 x 10-10 = 6.8 x 10-8 %
24. 9.0 x 1013 J
26. (a) W = (m - m0)c2 = 13.9 GeV = 2.23 x 10-9 J, (b) The momentum is 7.91 x 10-18 kg m/s,
32. m = 0.536 MeV/c2 = 1.05 m0, v = 0.302 c = 9.06 x 107 m/s
34. m = 2m0 = 3.34 x 10-27 kg, v = 0.866c = 2.60 x 108 m/s
38. The Np mass is 237.04832 u
42. 234 MeV
48. (a) 0.98c, (b) -0.64c
2. Since the car next to you is moving forward, but you are still, your speed relative to the car next to you is negative, so it looks like you are going backwards, although your absolute speed with respect to the ground is zero.
3. The ball will land on the car. Ignoring air resistance, it will come down where it went up.
4. The earth is undergoing centripetal acceleration, while the sun is not. Therefore, the Earth is going around the Sun. The principle of relativity refers to intertial frames. The Sun is an inertial frame, but the Earth is not, due to the centripetal acceleration. Inertial frames are not equivalent to accelerated frames. (General relativity makes this more subtle, but we haven't discussed that. See Section 33-3 if interested.)
5. The starlight would pass at a speed of c, as it does no matter how fast you move. The speed of light is the same to all observers.
6. If the events occur at the same place and the same time, then they will be simultaneous for all observers. Only for events separated bysoome distance is simultaneity dependent on the observer. 7. From O1's point of view, car 2 is moving to the left with velocity v and car 1 is at rest. This is exactly the same situation as in section 26-4, except with the moving cars interchanged. Note that Fig. 26-6 is somewhat misleading, since the the ends of the moving car cannot simultaneously line up with the ends of the stationary car. It is shorter from the point of view of a ground-based observer. For this reason, I did not quite follow section 26-4 in the lectures.
8. Time dilation refers to the fact that times take longer for any process to happen on a moving platform, compared to a stationary one. In particular, clocks would run slower, but in general, this is just one instance of all times taking longer than they would at rest. It is not just that the time is measured differently, it is actually longer.
9. Time actually does pass more slowly in a moving reference frame. It is not just an apparent effect.
10. Due to relativistic time dilation, someone on a spaceship moving very close to the speed of light could age much less than people left behind on Earth. In particular, an astronaut could return home to find that she was younger than her son.
11. You would not notice any change in your heartbeat, mass, height, or waistline if you were in a spaceship moving at half the speed of light, because you would be at rest in the reference frame of the spaceship. These things would all look different to an external observer not moving with the spaceship, however. Someone observing from earth would notice that your heartbeat was slower, you mass greater, your height the same, and your waistline thinner.
12. If the speed of light were only 25 m/s, we would notice relativistic effects all the time. In particular, we would have to give up on keeping clocks synchronized, and it would be harder to agree on what events happened in what order. Also, our mass would increase rapidly as we moved things, so that nothing would ever go faster than 25 m/s.
13. Yes, they occur at any speed, but are extremely hard to measure or observe at a speed as slow as 90 km/h.
14. If the speed of light were infinite, there would be no relativistic length contraction, time dilation, or mass increase.
More generally, physics would be more like it was thought to be in the time of Galileo. This would also mean that information would propagate infinitely fast, and we could see the present at all points in the universe. There would be no speed limit, and any change in a charge distribution would instantly affect electric fields at all distances. There would be no magnetism, since its strength is proportional to m0 = 1/(e0c2). Magnetism is a relativistic effect.
15. The length goes to 0 and the time goes to infinity as v approaches c. The length cannot contract, or the time dilate, any more than that, so the limiting speed of the universe must be c. Actually, this argument is a little misleading, since it is possible to define tachyons consistently, at least in non-quantum physics. Tachyons always move faster than light, but never slower. Their mass is imaginary.
16. When the object is moving slowly, the mass stays constant, but the velocity increases at a constant rate. As the object's speed approaches the speed of light, the mass begins to increase, and the rate of increase of the velocity slows down. When the velocity gets very close to the speed of light, the velocity stops changing, and the force goes mostly into increasing the mass.
17. Yes, because its energy decreases, and its energy is proportional to its relativistic mass, since E = mc2.
18. No. There is no conflict. E = mc2 is what is conserved. Kinetic energy and mass are not separately conserved, just the total relativistic energy. In the limit of small velocities, the mass is constant, and kinetic energy is again separately conserved. E = mc2 generalizes the law of energy conservation.
19. Objects that move at the speed of light do not have a rest mass. However, they can be defined to have a relativistic mass through E = mc2 , since they have an energy. E = mc2 applies to all objects.
20. There is no upper limit on the momentum of an electron, because the mass can increase without bound, although the velocity is limited. The limiting momentum is mc, but m is not bounded.
21. Yes. Anything that contributes to energy effectively contributes to mass (relativistic inertia).
22. We can say instead that energy can be neither created nor destroyed.
23. No. Velocities add normally as long as they are small compared to the speed of light. As the speed of light is approached, this rule must be modified. For most everyday purposes, we never approach the speed of light, so we don't have to consider this.
| Physics 222 | Department of Physics | University of Tennessee |