(a) Write down expressions for the relativistic 3 -momentum $\mathbf{p}$ and energy $E$ of a particle of rest mass $m$ and velocity $\mathbf{v}$. Show that these expressions are consistent with

$$E^{2}=\mathbf{p} \cdot \mathbf{p} c^{2}+m^{2} c^{4}$$

Define the 4-momentum $\mathbf{P}$ for such a particle and obtain $(*)$ by considering the invariance properties of $\mathbf{P}$.

(b) Two particles, each with rest mass $m$ and energy $E$, moving in opposite directions, collide head on. Show that it is consistent with the conservation of 4 -momentum for the collision to result in a set of $n$ particles of rest masses $\mu_{i}$ (for $\left.1 \leqslant i \leqslant n\right)$ only if

$$E \geqslant \frac{1}{2}\left(\sum_{i=1}^{n} \mu_{i}\right) c^{2} .$$

(c) A particle of rest mass $m_{1}$ and energy $E_{1}$ is fired at a stationary particle of rest mass $m_{2}$. Show that it is consistent with the conservation of 4 -momentum for the collision to result in a set of $n$ particles of rest masses $\mu_{i}$ (for $1 \leqslant i \leqslant n$ ) only if

$$E_{1} \geqslant \frac{\left(\sum_{i=1}^{n} \mu_{i}\right)^{2}-m_{1}^{2}-m_{2}^{2}}{2 m_{2}} c^{2}$$

Deduce the minimum frequency required for a photon fired at a stationary particle of rest mass $m_{2}$ to result in the same set of $n$ particles, assuming that the conservation of 4 -momentum is the only relevant constraint.