Express T' in terms of T, v, c for light pulse to moving rocket

[Silviu]

Homework Statement

A rocket moving with speed v passes a stationary observer. The observer waits a time T (according to his clock) after the rocket passes and send a pulse of light in the direction of the rocket. The rocket pilot notes that, according to her clocks, the time elapsed between the moment she passes the observer and the moment she receives the light pulse is T'. Express T' in terms of T, v and c.

Homework Equations

The Attempt at a Solution

I first calculated the time needed to receive the pulse from the point of view of the stationary observer (S). According to S, the light is emitted after a time T and in this time (according to S) the spaceship (S') moved a distance $d=vT$. According to S the extra time needed for the light to reach S' would be $T_1$ such that $cT_1=vT+vT_1$ from where we get $T_1=\frac{vT}{c-v}$ so the total time, according to S would be $T+T_1=\frac{c}{c-v}T$. To get the time according to S' I just used $T'=\frac{T}{\gamma}=T\frac{c\sqrt{1-\frac{v}{c}}}{c-v}=\sqrt{\frac{c}{c-v}}$. However the answer in their solution is $T'=T\sqrt{\frac{1+\beta}{1-\beta}}=T\sqrt{\frac{c+v}{c-v}}$. What is wrong with my reasoning?

 

[Orodruin]

You are missing a square in the v/c in your gamma factor.

 

[Silviu]

You are missing a square in the v/c in your gamma factor.

Wow, that was really dumb. Thank you for this!