A rocket that has a proper length of 900m moves away from a space station in the

Question

A rocket that has a proper length of 900m moves away from a space station in the +x direction at 0.60c relative to an observer on the station. An astronaut stands at the rear of the rocket and fires a dart toward the front of the rocket at 0.95c relative to the rocket. How long does it take the dart to reach the front of the rocket as measured in each of the following reference frames?

1) in the frame of the rocket?

___________microseconds

2) as measured in the frame of the space station?

___________microseconds

3) as measured in the frame of the dart?     

___________microseconds

Explanation / Answer

Part 1)

Apply d = vt

900 = (.95)(3 X 10^8)(t)

t = 3.16 us

Part 2)

We need the relative velocity

v = .95c - .6c

v = .35c

d = vt

900 = (.35)(3 X 10^8)(t)

t = 8.57 us

Part 3)

The length is not 900 m to the dart

L = L'sqrt (1 - v^2/c^2)

L = 900sqrt(1 - .95^2)

L = 281 m

d = vt

281 = (.95)(3 X 10^8)(t)

t = .987 us

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