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HELP calculating relative v for accelerating Frame

  1. Sep 13, 2009 #1
    Hi I am trying to do a calculation for relative velocity and time dilation difference in an accelerating frame . Between the back and the front as observed from an inertial frame.
    My math is limited and rusty so I am unsure if the procedure is valid or if the results are correct.
    So any input on either question ,from someone who has the skills would be appreciated. Thanks

    This calculation assumes Born rigidity, length contraction , and Rindler relative time dilation btween the back and the front of a single system.
    Inertial frame F
    Accelerating System S'
    rest L'= 1 km
    a= 1000g= 10km /s[tex]^{2}[/tex]

    Range .6c ===> .7c
    .7c-.6c =.1c = 3 x 10[tex]^{4}[/tex]km/s

    Time dt= (3 x 10[tex]^{4}[/tex]km/s)/(10km/s) =3000 s

    Contraction v[tex]_{i}[/tex]=.6c ------- [tex]\gamma[/tex]=1.25 --- = L'[tex]_{0}[/tex]=.8 km
    v[tex]_{f}[/tex] =.7 -------- [tex]\gamma[/tex]= 1.4 --- =L'[tex]_{1}[/tex] =.71km

    Difference in length over course of acceleration = .09 km
    .09km/ 3000s = 3 x 10 [tex]^{-5}[/tex] km /s

    relative velocity between front and back v[tex]_{fb}[/tex]= (3 x 10 [tex]^{-5}[/tex] km /s) /(3 x 10 [tex]^{5}[/tex]km /s ) = 10[tex]^{-10}[/tex]c
    Additive average relative velocity between front and back = (.65+10[tex]^{-10}[/tex])+ .65c = (1.7316 e[tex]^{-10}[/tex] ) c

    average velocity difference v[tex]_{d}[/tex]= 1.7316 e[tex]^{-10}[/tex] c
    avg [tex]\gamma[/tex]= 1 +( 2.9484 x 10 [tex]^{-20}[/tex] ) between front and back

    Relative to inertial frame F ,,, S' avg v=.65c [tex]\gamma[/tex]= 1.32
    dt/1.32 = 3000/1.32 = 2,273 s = overall elapsed time on S'

    2,273 x (1 +( 2.9484 x 10 [tex]^{-20}[/tex] )) = 6.782 x 10 [tex]^{-17}[/tex] s
    elapsed time difference between back and front.
    SO I considered calculating overall distance traveled for the front and the back as a function of time , but then decided that it was equivalent to simply calculating based on the difference itself.
    I simply took the distance that the back moved toward the front ,the relative velocity as a function of total elapsed time. And then divided that by c to derive a normalized relative velocity relative to the inertial frame.
    I then assumed that a simple, median average ,overall system velocity (.6.5c) would be close enough for a ballpark given constant acceleration.
    I then added the back/front relative velocity to one end and the raw velocity for the other end and used the additive velocity formula to get an internal relative velocity between the two.
    Entered this figure into the gamma function to get a gamma factor and then applied this to the gamma corrected dt' for the difference between the back clock and the front
    over the total course of acceleration. The accrued time dilation.
    I understand that this approach may only provide an approximation as compared to a more detailed analysis but is it basically a correct approach to the problem and is the result in the ballpark???
  2. jcsd
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