1. The problem statement, all variables and given/known data A high speed proton of rest mass M collides with a proton at rest in such a way that not only do the two protons emerge from the collision, but also a pion of mass m. Find the threshold kinetic energy of the incoming proton for such a pion to be generated. Show that the efficiency of the interaction is never better than a half. 3. The attempt at a solution I've done very similar questions before but usually it would be the minimum kinetic energy required to produce one particle from two colliding ones, so there is only one variable on the RHS of the equation. Quite simply, can I just take the centre of mass of the two protons and the pion on the RHS so I don't have to have three separate velocities? If this is wrong what should I do instead, also how do I deal with the efficiency bit? Mgamma(u)[c,u]+M[c,0]=(2M+m)gamma(w)[c,w] This leads to an answer of c^2(2m+m^2/2M) which looks neat enough to be reasonable. mc^2/(c^2+2m+m^2/2M) can never be more than a half, so is the efficiency the rest energy of the new particle over the energy used.