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Relativity Question with Lorentz tranformations

  1. Sep 24, 2007 #1
    1. The problem statement, all variables and given/known data

    Observer O sees a red flash of light at the origin at t=0 and a blue flash of light at x = 3.26km at a time t = 7.63 [tex]\mu[/tex] seconds. What are the distance and the time interval between the flasheds according to ovserver O', who moes relative to O in the direction of increasing x with a speed of 0.625c? Assume that the origins of the two coordinate systems line up at t = t' = 0.

    2. Relevant equations

    x' = [tex]\gamma[/tex](x - ut)

    t' = [tex]\gamma[/tex]t - (u/c^2)x



    3. The attempt at a solution

    I solved for t' and got the answer 1.73 [tex]\mu[/tex]seconds. From the back of my textbook, this seems to be the correct answer.

    When I put x = 3.26km into the first equation above, I always get the answer to be 2.34 km. The answer fromt the back of my textbook says [tex]\Delta[/tex] x' = -1.90 km.

    I'm wondering what i'm doing wrong...
     
  2. jcsd
  3. Sep 24, 2007 #2

    dynamicsolo

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    Homework Helper

    Just to make it a little clearer, this equation should be written

    t' = [tex]\gamma[/tex][t - (u/c^2)x]



    Are you sure you have the answer for the right problem? (I'm hoping they didn't change the numbers in the problem between editions and forget to change the answers -- I've seen that a little too often...)

    Something about their answer doesn't seem right because the spacetime interval should be

    (delta-s)^2 = (c delta-t)^2 - (delta-x)^2 = (c delta-t')^2 - (delta-x')^2 ,

    which doesn't happen with their values.

    I have [tex]\gamma[/tex] = 1.2810 . I guess we're agreed that x' = 0 , t' = 0 for the red flash. I also agree with you that, for the blue flash,

    x' = (1.281) [ (3.26 km) - (0.625)(300,000 km/sec)(7.63e^-6 sec) ] = 2.343 km .

    However, I get

    t' = (1.281) [ (7.63e^-6) - (0.625)(3.26 km/300,000 km/sec) ]

    = 1.074e^-6 sec or

    ct' = 0.322 km.

    In O's frame,

    (delta-s)^ 2 = [ (300,000 km/sec)(7.63e^-6 - 0 sec) ]^2 - ( 3.26 - 0 km )^2
    = -5.388 km^2

    A negative space-time separation makes sense because the light-travel time in O's frame from the location of the blue flash is 3.26 km/300,000 km/sec = 10.9 microseconds, which is longer than the time separation between the flashes.

    In O' 's frame,

    (delta-s)^ 2 = [ (300,000 km/sec)(1.074e^-6 - 0 sec) ]^2 - ( 2.343 - 0 km )^2
    = -5.386 km^2 (the difference is likely "round-off" error) .

    You do not get this value with the book's answers. Unless we're missing some point about the problem description, I find their answer suspect.
     
    Last edited: Sep 24, 2007
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