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Relativistic Photon Rocket

  1. May 22, 2015 #1
    1. The problem statement, all variables and given/known data
    (a) Show ##a = \frac{a_0}{\gamma^3}##.
    (b) Find proper acceleration of rocket
    (c) Find speed as a function of time.
    (d) Find acceleration of second rocket.

    2011_B1_Q5.png
    2. Relevant equations


    3. The attempt at a solution
    Part(a)
    4-vector acceleration is given by ##\gamma^2 \left[ \frac{\gamma^2}{c}(\vec u \cdot \vec a), \frac{\gamma^2}{c^2}(\vec u \cdot \vec a)\vec u + \vec a \right]##.
    For acceleration parallel to velocity, using invariance we have ##a_0^2 = a^2\gamma^6##. Thus we show that
    [tex]a = \frac{a_0}{\gamma^3}[/tex]

    Part(b)
    We know that ##\frac{M(\tau)}{d\tau} = -\alpha M(\tau)##, so solving we have ##M(\tau) = M_0 e^{-\alpha \tau}##. Considering the change in energy of the photon in time ##d\tau##, we have ##\frac{dE}{d\tau} = c \frac{dp}{d\tau} = cm_0 a_0##. Also in time ##d\tau##, the change in mass converted to energy is ##dE=c^2dM = -\alpha M_0 e^{-\alpha \tau} c^2 d\tau##. Thus we have the acceleration as
    [tex]a_0 (\tau) = \alpha c e^{-\alpha \tau}[/tex]

    Part(c)
    I'm not sure how to do this part. I have ##v = c \tanh \left( \frac{a_0 \tau}{c} \right)##.

    Part(d)
    Same concept as part(b). In time ##dt##, energy of photons produced is ##\alpha M_0 c^2 dt = p c##. Thus upon reflection, the change in momentum in time ##dt## is ##\Delta p = 2\alpha m_0 c dt##. Acceleration is ##a = 2\alpha c##. Finally, we have ##a_0 = \gamma^3 a##:
    [tex]a_0 = \gamma^3 (2 \alpha c) = \frac{2\alpha a}{\left( 1 - \frac{v^2}{c^2} \right)^{-\frac{3}{2}}}[/tex]

    How do I do part (c)?
     
  2. jcsd
  3. May 22, 2015 #2

    TSny

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    Homework Helper
    Gold Member

    In the right hand side of the last equation, shouldn't ##m_0## be replaced by ##M(\tau)##?

    Due to the motion of the rocket relative to the launch pad, the rate at which photons arrive at the rocket according to launch-pad time is not the same as the rate at which photons are produced according to lauch-pad time. Also in the launch-pad frame, there is a doppler shift in the frequency (or energy) of the photons as they reflect off of the moving rocket.
     
    Last edited: May 22, 2015
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