Redshift of a reflected photon from a solar sail

Question

Can someone help me understand my mistake here? I'm thinking of a photon travelling with momentum $p=h/\lambda$ which reflects off a solar sail of mass $m$ travelling with momentum $p_0$ in the same direction. Before the reflection, the total momentum is $p+p_0$ and after it is $-p'+p_1$ where $p'$ is the magnitude of the reflected photon's momentum and $p_1$ is the sail's new momentum. Conservation of momentum and energy then requires $$p+p_0 = -p'+p_1$$ and $$pc + \frac{p_0^2}{2m} = p'c + \frac{p_1^2}{2m}$$ if the sail is travelling at much less than relativistic speeds. But then solving the resulting quadratic for $p_1$ I get the entirely unreasonable $$p_1 = 4m\left\{-c \pm \sqrt{c^2+\frac{2}{m}\left(\frac{p_0^2}{2m}+p_0c+2pc\right)}\right\}$$ What have I done wrong?

Answer 1

There is a small error in your math by the way - the factor of $4$ should be a factor of $1$ - check the denominator of the quadratic equation.

Why is this so unreasonable? At a glance, it looks like the momentum of the sail is a bit larger than what it started out with. If I make the reasonable assumptions that:

• The sail is non-relativistic $\frac{p_0}{mc} = \frac{v}{c} \ll 1$
• The energy of the photon is much less than $mc^2$

and use the approximation $\sqrt{1-x}\approx 1-\frac{x}{2}\;{\rm for}\;x\ll1$, your answer reduces to:

$$p_1 \approx p_0+2p$$

This should look familiar - it's like a ball of mass $m$ colliding elastically with a ball of mass $M$ with $m\ll M$.