In the frame of reference of a planet, two spaceships move directly towards each

Question

In the frame of reference of a planet, two spaceships move directly towards each other, each with speed c/4 relative to the planet in opposite directions.
(a) What is the speed of one ship relative to the other? Call this speed v.
(b) Draw the situation in the frame of reference of one of the spaceships, indicating the speeds of the planet and both spaceships relative to this frame by velocity magnitude, and vector where appropriate. (c) A clock on ship 1 ticks once a second as observed from ship 1. How much time does it take for the clock on ship 1 to tick once as observed from ship 2?

Explanation / Answer

Given that,

speed of the spaceships = v1 = v2 = c/4 = 0.25c

(a)The speed of one ship relative to the other will be:

v = v1 + v2 / ( 1 + v1xv2/c2) = 0.25c + 0.25c / (1 + 0.25c x 0.25 c /c2) = 0.5c / 1.0625 = 0.471 c

Hence, the speed of one ship wrt to other is v = 0.471 c.

(c)we have, t1 = 1 and velocity wrt to each other v = 0.471 c

time will be given by:

t2 = t1/ sqrt (1 - v2/c2) = 1 / sqrt [ 1 - (0.471c)2/c2] = 1.134 s

hence, t = 1.134 sec

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