(a) A moving $\pi^{0}$ particle of rest-mass $m_{\pi}$ decays into two photons of zero rest-mass,

$$\pi^{0} \rightarrow \gamma+\gamma$$

Show that

$$\sin \frac{\theta}{2}=\frac{m_{\pi} c^{2}}{2 \sqrt{E_{1} E_{2}}}$$

where $\theta$ is the angle between the three-momenta of the two photons and $E_{1}, E_{2}$ are their energies.

(b) The $\pi^{-}$particle of rest-mass $m_{\pi}$ decays into an electron of rest-mass $m_{e}$ and a neutrino of zero rest mass,

$$\pi^{-} \rightarrow e^{-}+\nu .$$

Show that $v$, the speed of the electron in the rest frame of the $\pi^{-}$, is

$$v=c\left[\frac{1-\left(m_{e} / m_{\pi}\right)^{2}}{1+\left(m_{e} / m_{\pi}\right)^{2}}\right]$$