Could somebody help me check the solution please. (special relativity)

[phalanx123]

A rocket of length l0 measured in its rest frame S0 is traveling away from an observer in a frame S with a velocity u = 2/3 c. A light pulse is emitted from the nose of the rocket (x' = l0) at t0 = 0 and travels to the tail (x' = 0) where it is reflected back to the nose.

In S0, when does the light pulse reach the tail and when does it get back to the nose?

How long does the light pulse take to travel from nose to tail and back again
as determined by the observer in S?


Here is my solution

the time the pulse takes to travel from nose to tail is l0/c.
The time when it get back to the nose is t=2*l0/c.

For the second part, the total time it takes is gamma*2*l0/c where gamma=1/(1-v^2/c^2)^1/2 and v=2/3c

I think my solution is too simple, have I misunderstood the question ? Could somebody help me check it please. Thanks

 

[Doc Al]

the time the pulse takes to travel from nose to tail is l0/c.
The time when it get back to the nose is t=2*l0/c.

Correct.

For the second part, the total time it takes is gamma*2*l0/c where gamma=1/(1-v^2/c^2)^1/2 and v=2/3c

Also correct, but do you understand what allows you to apply the simple time dilation formula? Since, in the rocket frame, a single clock is used to measure the round-trip time of the light pulse, you are justified in using the time dilation formula (which describes the behavior of a single moving clock).

What if they asked for the time the pulse takes to get from nose to tail according to observers in S?

 

[phalanx123]

What if they asked for the time the pulse takes to get from nose to tail according to observers in S?

Would that not be gamma*l0/c where gamma is the same as defined above?

 

[Doc Al]

Would that not be gamma*l0/c where gamma is the same as defined above?

No, it wouldn't. That case is actually a bit more complicated to answer since two clocks are involved in the rocket frame, one at the nose and another at the tail. So you can't just apply the time dilation formula to that time interval, since no single clock directly measures it.

You can figure that one out by considering what distance the light must travel according to the S frame observers. (You'll need to take into account both length contraction and the motion of the rocket during the light's journey from nose to tail.)

 

[phalanx123]

Oh I see. so if the signal returned to the nose, only one clock is involved, so the time dilation can be used. but when measure the time interval between the emmission and reflection at the back, the events takes place at diffenrent places, so two clocks were involved, in that case the Lorentz transformation has to be used. Is my explanation/logic right?

 

[Doc Al]

Exactly right. (You could figure it out just using the known behavior of light, clocks, and measuring sticks... but that's equivalent to using the LT.)

 

[phalanx123]

Ok thank you very much. That cleared the SR up a little for me ^_^