In the collision of two protons, the final state contains a particle of mass m besides the protons.
(a) Give an expression for the minimum (threshold) energy E_p for the process to happen and for the corresponding momentum p_p if the target proton is at rest.
(b) Give the expression of the minimum energy E^*_p for the process to happen and of the corresponding momentum p^*_p if the two protons collide with equal and opposite velocities.
(c) How large are the threshold energies in cases (a) and (b) if the produced particle is a pion? How large is the kinetic energy in the first case?