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Astrophysics - Special Relativity

  1. Sep 18, 2010 #1
    1. The problem statement, all variables and given/known data

    2) If a relativistic rocket has a proper acceleration alpha that
    increases with proper time tau according to:
    alpha(tau) = 2/[Cosine(tau)^2 - Sine(tau)^2]
    find its motion, r(t), from the point of view of a control tower
    for whom the rocket is motionless at r(0) = 0.
    (Hint: alpha(tau) here is the derivative with respect to tau of
    ln[tan(tau + pi/4)] .)



    2. Relevant equations

    1. R=Rapidity
    2. tanh(R)=β
    3. d/dτ(R)=α


    3. The attempt at a solution

    Using formula #3 and the hint, I have R. Using formula #2 and my TI-89, I got:

    (1-β)/2 = cos[t*(sqrt(1-β^2)+pi/4]^2

    Using a couple of trig formulas, I have

    β-1 = sin(2*t*sqrt(1-β^2))

    I'm stuck there. As far as I know, there is no way to solve for β, and thus for the velocity 'v', which means I can't integrate to find r(t).
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 19, 2010 #2

    vela

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    Try using

    [tex]\tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}}[/tex].
     
  4. Sep 19, 2010 #3
    I wind up at the same spot.
     
  5. Sep 19, 2010 #4

    vela

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    Do it by hand, and show your work here.
     
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