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I've been wrestling with this problem but to no avail. I'm hoping that someone here can give me a little nod in the right direction.

Problem statement: A thrust-beam space vehicle works bearing a sort of sail which feels the push of a strong steady laser light beam directed at it from Earth. If the sail is perfectly reflected, calculate the mass of light required to accelerate a vehicle of rest mass $m_0$ up to a fixed value of $\gamma$.

Attempt at a solution: OK. Since the sail is perfectly reflective I view as if the vehicle is emitting photons. I also realize that the momentum of the light is $p = mv = mc = m$ (since I define $c=1$). This is where I get stuck. I have been using A.P. French's Special Relativity book and read his chapter on photon emission countless of times but the solution still evades me. Any pointer will be appreciated!


edit: I should add that I've calculated the relativistic mass of the ship as $m = m_0 \gamma$ and the momentum as $p = m \gamma = m_0 \gamma ^2$

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Well, probably, you should use momentum conservation for the system spacecraft+photons at two points: 1) photons fly towards the spacecraft at rest, 2) photons fly away from the accelerated spacecraft.

Remember that the spacecraft reflects, rather than emits photons.

Edit: the relation $p=\gamma^2 m_0$ for the spacecraft is incorrect, it should be $\gamma m_0$

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Thanks. I will try it and report back. –  docjay Mar 18 '12 at 20:43
Good luck in doing so! –  Alexey Bobrick Mar 18 '12 at 20:44
Well, this looks a lot better thanks to you. This is my train of thought so far: $p_{ph} = p_s - p_{ph}$ since the photons are reflected back. This is the same as: $m = m_0 \gamma - m$ so $2m = m_0 \gamma$. Now, $E=m$ so $E = \frac{m_0 \gamma}{2}$. Still on the right track? –  docjay Mar 18 '12 at 21:18
There is another point here: $p_{ph}$ is not the same before and after reflection. If you consider the energy of a photon $E=h\nu$ (which corresponds to $m=\dfrac{h\nu}{c^2}$), after the photon gets reflected from the spacecraft, it gets dopler-shifted, hence it changes $\nu$ and hence $m$. In the end, if one sends the photons of mass $m_1$ from the Earth, the photons of some mass $m_2$ (actually $m_2$ shall vary) will come back. If the problem is simpler, they ask for the mass $\Delta m$ the ship 'consumes' (more probable). If the problem is more advanced, they ask for the total $m_1$ sent. –  Alexey Bobrick Mar 18 '12 at 22:33
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