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Lorentz Transformations

  1. Sep 18, 2007 #1
    1. The problem statement, all variables and given/known data

    A physics professor on Earth gives an exam to her students who are on a spaceship traveling at speed v relative to Earth. The moment the ship passes the professor she signals the start of the exam. If she wishes her students to have time To (spaceship time) to complete the exam, show that she should wait a time (Earth Time) of

    T = (To)sqrt[(1-v/c)/(1+v/c)]

    before sending a light signal telling them to stop. (Hint: Remember that it takes some time for the second light signal to travel from the professor to the students.)

    2. Relevant equations

    Lorentz Transformations :

    Time dilation: t=γt'


    3. The attempt at a solution

    Well I first I defined all the different times.

    To= total time in the spaceship frame

    t= total time in Earth's frame = γTo = To/sqrt(1-(v^2/c^2))

    tx= time it takes the professor's light signal to travel from her to the students = x/c

    T= how long the professor should wait to send the second light signal after the first one = t-tx

    x is the distance between the professor and the spaceship when the students recieve the second signal.

    I'm not really sure where to go from here. I know I've gotta use x'=γ(x-vt) where x' is the distance that the spaceship has travelled before recieving the second signal somewhere, but I'm not sure what time to use in it and stuff like that.

    Any help would be appreciated.
  2. jcsd
  3. Sep 18, 2007 #2


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    First, focus on the EVENTS.
    (Can you draw a position-vs-time graph of what is happening [a spacetime diagram]?).
    Then, it should be more clear what the various t's and x's mean.
  4. Sep 18, 2007 #3

    Earth Frame:
    at time t=0, the first signal is sent.
    at time T the second signal is sent.
    at time t the second signal is recieved by the spaceship. At this time both the spaceship and the light signal have travelled x.

    Spaceship Frame:
    at time t=0 the first signal is recieved.
    at time To the second signal is recieved. The spaceship has travelled x'.

    Am I not thinking of this correctly?
  5. Sep 18, 2007 #4
    Okay, so I've been playing around with stuff and this is what I've got:

    Earth Frame:

    tx=x/c because it's the amount of time it takes for the signal to reach the students.
    so x= c(tx)

    but x=vt also because t=x/v is the amount of time it takes the students to travel distance x.

    So I said ctx=vt, therefore, tx=vt/c

    then I filled it into

    T= t-tx = t-vt/c = (ct-vt)/c = t(c-v)/c = t(1-v/c)

    Since we already know that t=γTo I filled that in too and got

    T= To(1-v/c) / sqrt[1-(v^2/c^2)]

    which is starting to look a little bit like what I'm aiming for...

    Does what I did make any sense or have I just totally lost track?
  6. Sep 18, 2007 #5
    I don't know if I'll get in trouble for this, but I just wanted to bring this back to the top before its pushed onto page 2 and forgotten... if that's not allowed just let me know, haha. Sorry.
  7. Sep 18, 2007 #6


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    Sorry, I've been busy today. I'll try and get back on this later. But maybe someone else will chime in.
  8. Sep 18, 2007 #7
    Figured it out! Thanks anyway :).

    I was on the right track, all I did after that was square both sides of that last equation I had... and then it was clear from there!
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