Define the 4-momentum $P$ of a particle of rest mass $m$ and velocity $\mathbf{u}$. Calculate the power series expansion of the component $P^{0}$ for small $|\mathbf{u}| / c$ (where $c$ is the speed of light in vacuo) up to and including terms of order $|\mathbf{u}|^{4}$, and interpret the first two terms.

(a) At CERN, anti-protons are made by colliding a moving proton with another proton at rest in a fixed target. The collision in question produces three protons and an anti-proton. Assume that the rest mass of a proton is identical to the rest mass of an anti-proton. What is the smallest possible speed of the incoming proton (measured in the laboratory frame)?

(b) A moving particle of rest mass $M$ decays into $N$ particles with 4 -momenta $Q_{i}$, and rest masses $m_{i}$, where $i=1,2, \ldots, N$. Show that

$$M=\frac{1}{c} \sqrt{\left(\sum_{i=1}^{N} Q_{i}\right) \cdot\left(\sum_{j=1}^{N} Q_{j}\right)}$$

Thus, show that

$$M \geqslant \sum_{i=1}^{N} m_{i}$$

(c) A particle $A$ decays into particle $B$ and a massless particle 1 . Particle $B$ subsequently decays into particle $C$ and a massless particle 2 . Show that

$$0 \leqslant\left(Q_{1}+Q_{2}\right) \cdot\left(Q_{1}+Q_{2}\right) \leqslant \frac{\left(m_{A}^{2}-m_{B}^{2}\right)\left(m_{B}^{2}-m_{C}^{2}\right) c^{2}}{m_{B}^{2}}$$

where $Q_{1}$ and $Q_{2}$ are the 4-momenta of particles 1 and 2 respectively and $m_{A}, m_{B}, m_{C}$ are the masses of particles $A, B$ and $C$ respectively.