# Special Relativity - Relativistic Dynamics

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!

Thanks.

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

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