Paper 4, Section II, B

(a) Write down the relativistic energy $E$ of a particle of rest mass $m$ and speed $v$. Find the approximate form for $E$ when $v$ is small compared to $c$, keeping all terms up to order $(v / c)^{2}$. What new physical idea (when compared to Newtonian Dynamics) is revealed in this approximation?

(b) A particle of rest mass $m$ is fired at an identical particle which is at rest in the laboratory frame. Let $E$ be the relativistic energy and $v$ the speed of the incident particle in this frame. After the collision, there are $N$ particles in total, each with rest mass $m$. Assuming that four-momentum is conserved, find a lower bound on $E$ and hence show that

$v \geqslant \frac{N\left(N^{2}-4\right)^{1 / 2}}{N^{2}-2} c$

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