If I have a probe being pulled by a reflective "light sail" and I need to derive an expression for the speed of the probe as the sail is hit by laser beams, how do I do this using relativistic mechanics?
This is what I think: as the probe speeds up, the momentum transferred by the light beam will diminish because of the doppler shift. Furthermore, the effective mass will increase because of relativistic effects.
Useful equations:
$$E=\gamma m c^2 - m_0 c^2$$
$$p=h/\lambda$$
$$\beta = v/c = pc/E = hc/(\lambda(mc^2+m_0c^2))$$
$$\lambda=c/f$$
$$f_D=(1+\Delta v/c)f_0$$
My idea was to find $\frac{dE}{dt}=\frac{d}{dt}(\gamma mc^2-m_0c^2)$ and then set that equal to $\frac{d}{dt}(hc/\lambda \beta)$.This yields:
$$\gamma ^3 mav = \frac{-ahf_0(c+\Delta v)}{v^2}+\frac{\Delta a h f_0}{v}$$
But at this point I get stuck. Could someone point my in the right direction?
