Special relativity - spaceship speed and position

[FranzDiCoccio]

Hi,

a friend today teased me with the following SR problem. I worked out a solution
but I'm not so confident about my SR. Should I better brush up on it?

Homework Statement

So there's this spaceship traveling at some unknown constant speed, getting away from Earth. It has two mirrors on it, one at the front and one at the rear.
Earth sends two simultaneous light pulses which are reflected by the two mirrors
and get back to Earth after times 2 T₁ and 2 T₂.
Knowing these times and the length L of the spaceship we should evaluate the
speed v and the distance of the spaceship (say at the time we receive the second signal).

Homework Equations

The only tricky point is time dilation, in my opinion.

The Attempt at a Solution

So this is what I would do to determine the speed v. When this is known the distance would simply be d = T2 (c + v).

I'd say that the first pulse hits the rear mirror at time T1. At this time the second pulse is at the rear of the ship as well. This second pulse hits the front mirror at time T2, so that t = T2-T1 is the time it takes to go from the rear to the front of the ship.
All of this in the reference frame of the earth.
Now from the point of view of the spaceship the second pulse takes a time t' = L/c in covering the length of the ship. The times t and t' should be related by time dilation:

t = t' / sqrt(1-v^2/c^2).

In summary I get

(T2-T1)^2 = L^2/(c^2-v^2)

which can be solved for v.


Seems correct to me, but my friend and other people put forward tricky arguments and I'm not sure any more of my solution.

Thanks

 

[BigJDubs]

in advance.Your solution is correct! Time dilation is indeed a tricky concept, but you seem to have applied it correctly here. Since you are confident in your solution, there is no need for you to brush up on SR. However, if you would like to go over the problem with someone else to make sure, you could always ask a professor or tutor for help.