Special Relativity | Part IB, 2003

A pion of rest mass MM decays at rest into a muon of rest mass m<Mm<M and a neutrino of zero rest mass. What is the speed uu of the muon?

In the pion rest frame SS, the muon moves in the yy-direction. A moving observer, in a frame SS^{\prime} with axes parallel to those in the pion rest frame, wishes to take measurements of the decay along the xx-axis, and notes that the pion has speed vv with respect to the xx-axis. Write down the four-dimensional Lorentz transformation relating SS^{\prime} to SS and determine the momentum of the muon in SS^{\prime}. Hence show that in SS^{\prime} the direction of motion of the muon makes an angle θ\theta with respect to the yy-axis, where

tanθ=M2+m2M2m2v(c2v2)1/2.\tan \theta=\frac{M^{2}+m^{2}}{M^{2}-m^{2}} \frac{v}{\left(c^{2}-v^{2}\right)^{1 / 2}} .

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