Paper 2, Section II, C

Dynamics and Relativity | Part IA, 2020

(a) A moving particle with rest mass MM decays into two particles (photons) with zero rest mass. Derive an expression for sinθ2\sin \frac{\theta}{2}, where θ\theta is the angle between the spatial momenta of the final state particles, and show that it depends only on Mc2M c^{2} and the energies of the massless particles. ( cc is the speed of light in vacuum.)

(b) A particle PP with rest mass MM decays into two particles: a particle RR with rest mass 0<m<M0<m<M and another particle with zero rest mass. Using dimensional analysis explain why the speed vv of RR in the rest frame of PP can be expressed as

v=cf(r), with r=mMv=c f(r), \quad \text { with } \quad r=\frac{m}{M}

and ff a dimensionless function of rr. Determine the function f(r)f(r).

Choose coordinates in the rest frame of PP such that RR is emitted at t=0t=0 from the origin in the xx-direction. The particle RR decays after a time τ\tau, measured in its own rest frame. Determine the spacetime coordinates (ct,x)(c t, x), in the rest frame of PP, corresponding to this event.

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