Chapter I - Special Relativity

1.2 Through a window in Carl's spaceplane, passing at 0.5 c, you watch Carl doing an important physics calculation. By your watch it takes him 1 min. How much time did Carl spend on his calculation?

1.4 According to Bob on Earth, Planet Y (uninhabited) is 5 l-y away. Anna is in a spaceship moving away from Earth at 0.8 c. She is bound for Planet Y to study its geology. Unfortunately, Planet Y explodes. According to Bob, this occurred two years after Anna passed Earth. (Bob, of course, has to wait for a while for the light from the explosion to arrive, but reaches his conclusion by "working backward.") Call the passing of Anna and Bob time zero for both. (a) According to Anna, how far away is Planet Y when it explodes? (b) At what time does it explode?

1.9 Bob and Bob Jr. stand at open doorways at opposite ends of an airplane hangar 25 m long. Anna owns a spaceplane, 40 m long as it sits on the runway. Anna takes off in her spaceplane, then swoops through the hangar at constant velocity. At precisely zero time on both Bob's clock and Anna's, Bob sees the nose of Anna's spaceplane reach his doorway. At zero time on his clock, Bob Jr. sees the tail of Anna's spaceplane at his doorway. (a) How fast is Anna's spaceplane moving? (b) What will Anna's clock read when she sees the tail of her spaceplane at the doorway where Bob Jr. is standing? (c) How far will Anna say the nose of her spaceplane is from Bob at this time?

1.18 A plank is stationary in frame S. It is of length L0 and makes an angle with the x-axis. It is then caused to move relative to frame S at a constant speed v parallel to the x-axis. Show that according to an observer who remains at rest in frame S, the length of the plank is now

and the angle it makes with the x-axis is .

1.19 Bob, in frame S, is observing the moving plank of exercise 1.18. He quickly fabricates a wall, fixed in his frame, that has a hole of length L and is slanted at angle , such that the plank will completely fill the hole as it passes through. This occurs at the instant t=0. According to Anna, moving with the plank, the plank is of course not of length L, but of length L0. Moreover, because Bob's wall moves relative to her, Anna sees a hole that is less than L in length; a plank longer than L is headed toward a hole shorter than L. Can the plank pass through the hole according to Anna? If so, at what time(s)? Explain.

1.25 Planet W is 12 l-y from Earth. Anna and Bob are both 20 years old. Anna travels to Planet W at 0.6 c, quickly turns around, and returns to Earth at 0.6 c. How old will Anna and Bob be when Anna gets back?

1.29 Show that for a source moving toward an observer, the equation for the relativistic Doppler shift becomes

1.31 At rest, a light source emits 532-nm light. (a) As it moves along the line connecting it and Earth, observers on Earth see 412 nm. What is the source's velocity (magnitude and direction)? (b) Were it to move in the opposite direction at the same speed, what wavelength would be seen? (c) Were it to circle Earth at the same speed, what wavelength would be seen?

1.33 A space probe has a powerful light beacon that emits 500-nm light in its own rest frame. Relative to Earth, the space probe is moving at 0.8 c. An observer on Earth is viewing the light arrive from the distant beacon and detects a wavelength of 500 nm. Is this possible? Explain.

1.35 For reasons having to do with quantum mechanics, a given kind of atom can emit only certain wavelengths of light. These "spectral lines" serve as a "fingerprint." For instance, hydrogen's only visible spectral lines are 656, 486, 434, and 410 nm. Were spectral lines of absolutely precise wavelength, they would be very difficult to discern. Fortunately, two factors broaden them: the uncertainty principle (discussed in quantum physics) and Doppler broadening. Atoms in a gas are in motion so some light will arrive having been emitted by atoms moving toward the observer and some from atoms moving away. Thus, the light reaching the observer will cover a range of wavelengths. (a) Making the assumption that atoms move no faster than their rms (root mean square) speed, where kB is the Boltzmann constant, obtain a formula in terms of the wavelength of the spectral line, atomic mass m, and temperature T for the range of wavelengths. (Note: ) (b) Evaluate this range for the 656 nm hydrogen spectral line, assuming a temperature of 105 K.

1.37 Bob is on Earth. Anna is on a spacecraft moving away from Earth at 0.6 c. At some point in Anna's outward travel, Bob fires a projectile loaded with supplies at 0.8 c. (a) How fast does the projectile move relative to Anna? (b) Bob also sends out a light signal, "Greetings from Earth," out to Anna's ship. How fast does the light signal move relative to Anna?

1.39 Prove that if v and u' (along the direction of v) are less than c, it is impossible for a speed u greater than c to result from the addition of velocities equation. [Hint: the product (c - u')(c - v) is positive.]

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