An observer in a rocket moves towards a mirror at speed v relative to the refere

Question

An observer in a rocket moves towards a mirror at speed v relative to the reference frame labeled by S in Figure P1.30. The Mirror is stationary with respect to S. A light pulse emitted by the rocket travels toward the mirror and is reflected back to the rocket. The front of the rocket is distance d from the mirror (as measured by observers in S) at the moment the light pulse leaves the rocket. What is the total travel time of the pulse as measured by observers in:
a.) the S frame and
b.) the front of the rocket
In Figure P1.30 the only other supplemental information given is that the rocket moves at v=0.8c towards the mirror.

Explanation / Answer

(a)
An observer at rest relative to the mirror sees the light travel a distance D = 2d - x , where
x = vt(s) is the distance the ship moves toward the mirror in time t(s) . Since this observer agrees that the speed of light is c, the time for it to travel distance D is

t(s) = D / c

= 2d - vt(s) / c

so, t(s) = 2d / c+v

(b)
The observer in the rocket measures a length-contracted initial distance to the mirror of

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

and the mirror moving toward the ship at speed v. Thus, he measures the distance the light travels as D = 2(L - y) where y = vt /2 is the distance the mirror moves toward the ship before the light reflects from it. This observer also measures the speed of light to be c, so the time for it to travel distance D is :

t = D / c

= 2/c * [ d*sqrt(1 - v^2/c^2) - vt/2 ]

so,

(c+v)t = 2d/c * sqrt((c+v)(c-v))

or

t = 2d/c * sqrt((c-v)/(c+v))

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