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Pi meson decay* A neutral pi meson \(\left(\pi^{0}\right)\), rest mass \(135 \mathrm{MeV}\), decays symmetrically into two photons while moving at high speed. The energy of each photon in the laboratory system is \(100 \mathrm{MeV}\). (a) Find the meson's speed \(V\) as a ratio \(V / c\). ( \(b\) ) Find the angle \(\theta\) in the laboratory system between the momentum of each photon and the initial line of motion.

Threshold for pi meson production A high energy photon ( \(\gamma\) -ray) collides with a proton at rest. A neutral pi meson \(\left(\pi^{0}\right)\) is produced according to the reaction \(\gamma+p \rightarrow\) \(p+\pi^{0}\) What is the minimum energy the \(\gamma\) -ray must have for this reaction to occur? The rest mass of a proton is \(938 \mathrm{MeV}\), and the rest mass of a \(\pi^{0}\) is \(135 \mathrm{MeV}\).

Particle decay A particle of rest mass \(M\) spontaneously decays from rest into two particles with rest masses \(m_{1}\) and \(m_{2}\). Show that the energies of the particles are \(E_{1}=\left(M^{2}+m_{1}^{2}-\right.\) \(\left.m_{2}^{2}\right) c^{2} / 2 M\) and \(E_{2}=\left(M^{2}-m_{1}^{2}+m_{2}^{2}\right) c^{2} / 2 M .\)

Threshold for nuclear reaction \(^{*}\) A nucleus of rest mass \(M_{1}\) moving at high speed with kinetic energy \(K_{1}\) collides with a nucleus of rest mass \(M_{2}\) at rest. A nuclear reaction occurs according to the scheme \(M_{1}+M_{2} \rightarrow M_{3}+M_{4}\) where \(M_{3}\) and \(M_{4}\) are the rest masses of the product nuclei. The rest masses are related by \(\left(M_{3}+M_{4}\right) c^{2}=\left(M_{1}+M_{2}\right) c^{2}+Q\). where \(Q>0 .\) Find the minimum value of \(K_{1}\) required to make the reaction occur, in terms of \(M_{1}, M_{2}\), and \(Q\).