Particle physics and conservation of momentum. You fire high intensity laser light (many photons) at a stationary block of iron, looking for an iron nuclei to fragment. Specifically, you are looking for the following interaction, or channel:

⁵⁶Fe + y → 13 ⁴He + 4n

That is, the photon (gamma) hits an iron nucleus, which has 56 nucleons (protons plus neutrons). The iron nucleus is smashed apart, decaying into 13 helium nuclei, and 4 individual neutrons. Let m = the mass of a proton or a neutron. The mass then, of the iron nucleus is 56 m, while the mass of a neutron is just m. The photon has no mass at all. However, photons are special in that the momentum of a photon is equal to the energy of a photon.

All velocities of the particles are relative to the speed of light, c. Let the photon have momentum q.

1. Prove that if the final velocity vectors of all the helium add up to zero, that the sum of the speeds of the neutrons Is equal to q/m.

2. Show then that q!4m gives the average speed of the neutrons.

3. What is the average kinetic energy of the neutrons, in terms of q and m?