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 $p = m \gamma = m_0 \gamma ^2$
Best Answer
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 $\gamma m_0$