Time dilation and photon detector

[Eric_meyers]

Homework Statement

Relativistic protons that have a certain speed "v" are selected by measuring the time it takes the proton to travel between two detectors separated by a distance "L". Each detector produces an electronic pulse of very short duration ($$\Delta$$t << L/v) when a proton passes through it. A coincidence circuit is made by delaying the pulse from the first detector by an amount L/v. The signals from the two detectors are fed into a logic circuit that produces an output pulse if the pulses arrive at the same time. For input pulses that arrive at the same time as measured in the laboratory frame, calculate the time difference between arrival of the input pulses as measured in the rest frame of the proton.

Homework Equations

$$\Delta$$t = $$\Delta$$t'/ (1-v^2/c^2)^1/2 where $$\Delta$$t' is the time elapsed in reference frame of moving particle.

The Attempt at a Solution

This problem seems too easy and so I'm wondering if I'm getting tricked somehow.

I take the $$\Delta$$t elapsed in the reference frame of the lab to = L/v and then I use the time dilation equation to solve for $$\Delta$$t'

$$\Delta$$t' = L/v * (1 - v^2/c^2)^1/2

and then take the difference between the two (as asked in the problem)

L/v - L/v * (1-v^2/c^2)^1/2

Now, I'm a little iffy if I'm taking the reference frame of the proton correctly, because in the proton's frame the detectors are moving towards it, or with a velocity of -v... however since the equations don't use vectors I take it I'm dealing with scalar speed instead of v.

 

[BigJDubs]

The answer I get is - L/v * (1-v^2/c^2)^1/2which seems correct, but I'm still not sure if I'm missing something.