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4 Acceleration of an relativistic rocket

  1. May 23, 2012 #1
    1. The problem statement, all variables and given/known data

    A rocket of (time dependant) mass M ejects fuel such that its change in mass in the instantaneous ZMF is [tex] \frac{dM}{d\tau} = -\frac{E}{c^2} [/tex] The speed of the fuel ejected is [itex]w[/itex].
    Prove that [tex] a = \frac{Ew}{Mc^2}[/tex]
    where a is defined by [itex] -a^2 = A_\mu A^\mu [/itex]

    3. The attempt at a solution

    In the rest frame, [tex]P_{before} =
    \left(\begin{array}{c}
    M \\ 0
    \end{array}\right)
    =
    P_{rocket} + P_{fuel}
    =
    P_{rocket} + \left(\begin{array}{c}
    \delta m \gamma \\ -\delta m \gamma w
    \end{array}\right)
    [/tex]

    We are given [tex]\frac{dM}{d\tau} = -E [/tex] so [tex] \delta m = -\frac{dM}{d\tau} d\tau = E d\tau [/tex]

    [tex]
    P \equiv M U \\
    \frac{dP}{d\tau} = \frac{dM}{d\tau}U + MA =
    \left(\begin{array}{c}
    \frac{dM}{d\tau} - \gamma E \\ \gamma E w
    \end{array}\right)
    \\
    -EU + MA
    =
    \left(\begin{array}{c}
    -E - \gamma E \\ \gamma E w
    \end{array}\right)
    [/tex]

    The rocket starts stationary in the ZMF so [itex]U = \left(\begin{array}{c}
    1 \\ 0
    \end{array}\right) [/itex]

    Therefore, [tex]
    A = -\frac{E}{M} \left(\begin{array}{c}
    \gamma \\ -\gamma w
    \end{array}\right)

    [/tex]

    [tex]
    -a^2 \equiv A_\mu A^\mu = \left(\frac{E}{M}\right)^2 [\gamma^2(w^2-1)] = \left(\frac{E}{M}\right)^2 \frac{w^2-1}{1 - w^2} = -\left(\frac{E}{M}\right)^2
    [/tex]

    I've lost an [itex]w^2[/itex] somewhere! Can anyone see it?
     
  2. jcsd
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