An antiproton p₀ has the same rest energy as a proton. It is created in the reaction p + p → p + p + p + p . In an experiment, protons at rest in the laboratory are bombarded with protons of kinetic energy K_L, which must be great enough so that kinetic energy equal to 2m₀c² can be converted into the rest energy of the two particles. In the frame of the laboratory, the total kinetic energy cannot be converted into rest energy because of conservation of momentum. However, in the zero-momentum reference frame in which the two initial protons are moving toward each other with equal speed u, the total kinetic energy can be converted into rest energy.

(a) Find the speed of each proton u such that the total kinetic energy in the zero-momentum frame is 2m₀c².

(b) Transform to the laboratory’s frame in which one proton is at rest, and find the speed u’ of the other proton.

(c) Show that the kinetic energy of the moving proton in the laboratory’s frame is K_L = 6m₀c².