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Chapter 5: Problem 9

Suppose an astronaut is moving relative to Earth at a significant fraction of the speed of light. (a) Does he observe the rate of his clocks to have slowed? (b) What change in the rate of earthbound clocks does he see? (c) Does his ship seem to him to shorten? (d) What about the distance between two stars that lie in the direction of his motion? (e) Do he and an earthbound observer agree on his velocity relative to Earth?

Short Answer

Expert verified
In summary, (a) the astronaut does not observe the rate of his clocks to have slowed, as he is at rest in his own reference frame. (b) The astronaut observes the rate of earthbound clocks to have slowed due to time dilation. (c) The astronaut's ship does not appear to shorten since he is in the same reference frame. (d) The distance between stars in the direction of his motion appears shorter due to length contraction. (e) Both the astronaut and the earthbound observer agree on the astronaut's velocity relative to Earth.

Step by step solution

01

Understand the perspective of the astronaut

The astronaut is moving relative to Earth, so from the astronaut's perspective, he is at rest while Earth is moving backward.
02

Consider time dilation

Since the astronaut is moving at a significant fraction of the speed of light, the effect of time dilation becomes noticeable. Time dilation is a difference in the elapsed time measured by two observers, due to the relative motion between them.
03

Conclusion

From the astronaut's perspective, the rate of his clocks does not appear to have slowed down, as he is at rest in his own reference frame. Time dilation affects the relative time observed by different frames, not the time observed in one's own reference frame. b) What change in the rate of earthbound clocks does he see?
04

Consider the astronaut's perspective

The astronaut sees Earth moving backward at a significant fraction of the speed of light.
05

Apply time dilation

Due to time dilation, the rate of clocks on Earth, as observed by the astronaut, will appear to have slowed down. c) Does his ship seem to him to shorten?
06

Understand the perspective

The astronaut is inside his ship in his own reference frame, which he perceives to be at rest.
07

Consider length contraction

Length contraction occurs when an object is moving at a significant fraction of the speed of light relative to an observer and consequently appears shorter to that observer.
08

Conclusion

From the astronaut's perspective, his ship does not appear to shorten, as he is in the same reference frame as the ship and perceives it to be at rest. Length contraction affects the appearance of objects in different reference frames, not in one's own reference frame. d) What about the distance between two stars that lie in the direction of his motion?
09

Understand the astronaut's perspective

The astronaut is moving towards the two stars.
10

Apply length contraction

Due to length contraction, the distance between the two stars, as observed by the astronaut, will appear to be shorter than that observed by a stationary observer on Earth. e) Do he and an earthbound observer agree on his velocity relative to Earth?
11

Consider relative velocities

The velocity of the astronaut relative to Earth is the same for both the astronaut and the Earth-based observer, as it is the difference between their velocities.
12

Apply the speed of light

Both the astronaut and the Earth-bound observer will agree that the astronaut is moving at a significant fraction of the speed of light relative to Earth, which indicates that no object can achieve or exceed the speed of light.
13

Conclusion

Yes, both the astronaut and the Earth-bound observer will agree on the astronaut's velocity relative to Earth.

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