Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2010

A particle AA of rest mass mm is fired at an identical particle BB which is stationary in the laboratory. On impact, AA and BB annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle θ\theta between the photon trajectories is given by

cosθ=E3mc2E+mc2\cos \theta=\frac{E-3 m c^{2}}{E+m c^{2}}

where EE is the relativistic energy of AA.

Let vv be the speed of the incident particle AA. For what value of v/cv / c will the photons move in perpendicular directions? If vv is very small compared with cc, show that

θπv/c\theta \approx \pi-v / c

[All quantities referred to are measured in the laboratory frame.]

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