Special Relativity, light down a rocket

1. The problem statement, all variables and given/known data

A rocket of length ##L_o## flies with constant velocity ##v## (in frame S' relative to a frame S in the x direction). At time t=t'=0, the capsule on the top of the rocket passes the point ##P_o## in S. At this moment, a light signal is sent from the top of the rocket to the bottom.

1)In the rest frame of the rocket, how long does it take the light signal to reach the end of the rocket?

2)In the rest frame of the observer, S, at which time does the signal reach the end of the rocket?



2. Relevant equations

Lorentz transformations, proper length and proper time.



3. The attempt at a solution

1)The wording of the question at the beginning is confusing, but I take it that S is the rest frame of the observer, S' is the rest frame of the rocket and S' moves wrt S at velocity v. I also took ##L_o## to be the proper length of the rocket (i.e the value measured in S', the rest frame of the rocket). Are these correct interpretations?



At ##t_o' = 0## the signal leaves the top of the rocket. At ##t_1'## it arrives at the bottom in frame S'. Speed of light same in all inertial frames (S' inertial since it moves with constant velocity v wrt another inertial frame) and ##L_o## the length of the rocket in S'. So ##t_1' = L_o/c## is the time taken in S'.



2)In S, as the rocket passes, the observer would see the rocket contract so expect time for signal to reach bottom in S to be smaller than that in S'. Why is ##t_1 = \gamma t_1'## not valid here?

Thanks.