A particle $A$ of rest mass $m$ is fired at an identical particle $B$ which is stationary in the laboratory. On impact, $A$ and $B$ annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle $\theta$ between the photon trajectories is given by

$$\cos \theta=\frac{E-3 m c^{2}}{E+m c^{2}}$$

where $E$ is the relativistic energy of $A$.

Let $v$ be the speed of the incident particle $A$. For what value of $v / c$ will the photons move in perpendicular directions? If $v$ is very small compared with $c$, show that

$$\theta \approx \pi-v / c$$

[All quantities referred to are measured in the laboratory frame.]