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Chapter 2: Q20E (page 62)
Through a window in Carl's spaceship, passing al 0.5c, 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?
The time spends by Carl in spaceship for calculation is.
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Exercise 117 gives the speed u of an object accelerated under a constant force. Show that the distance it travels is given by
In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about are fatal, and there are serious concerns about the effects of prolonged accelerations greater than . Here we see how far a person could go under a constant acceleration of , producing a comfortable artificial gravity.
(a) Though traveller Anna accelerates, Bob, being on near-inertial Earth, is a reliable observer and will see less time go by on Anna's clock than on his own . Thus, , where is Anna's instantaneous speed relative to Bob. Using the result of Exercise 117(c), with replacing , substitute for , then integrate to show that .
(b) How much time goes by for observers on Earth as they “see” Anna age 20 years?
(c) Using the result of Exercise 119, show that when Anna has aged a time , she is a distance from Earth (according to Earth observers) of .
(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20. How far away from Earth will she end up and how much time will have passed on Earth?
Classically, the net work done on an initially stationary object equals the final kinetic energy of the object. Verify that this also holds relativistically. Consider only one-dimension motion. It will be helpful to use the expression for p as a function of u in the following:
In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than 1g. Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity.
(a) Though traveller Anna accelerates, Bob, being on near-inertial Earth, is a reliable observer and will see less time go by on Anna's clock (dt') than on his own (dt). Thus,, whereuis Anna's instantaneous speed relative to Bob. Using the result of Exercise 117(c), withgreplacingF/m, substitute for u, then integrate to show that
(b) How much time goes by for observers on Earth as they “see” Anna age 20 years?
(c) Using the result of Exercise 119, show that when Anna has aged a timet’, she is a distance from Earth (according to Earth observers) of
(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20. How far away from Earth will she end up and how much time will have passed on Earth?
Is it possible for the momentum of an object to be mc. If not. why not? If so, under what condition?
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