Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2011

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

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

vN(N24)1/2N22cv \geqslant \frac{N\left(N^{2}-4\right)^{1 / 2}}{N^{2}-2} c

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