Exercise 23 discusses the threshold energy for two particles of mass in in a colliding-beam accelerator to produce a final stationary mass M. If the accelerator is instead a stationary-target type, more initial kinetic energy is needed to produce the same final mass. Show that the threshold energy is (M2l2m - 2m)c2. The calculation is greatly streamlined by using the momentum-energy invariant, discussed in Section 2.10 and Exercise 2.112. The main point is that the quantity (Etotal/c)2 – P2total must be the same not only before and after the collision but also in any reference frame. The quick approach is to equate this quantity after the collision in the center-of-mass frame, where the final mass is stationary, to its value before the collision in the lab frame, where the target is stationaiy. You will need to use E2 = p2c2 + m2c4 to eliminate the moving particle's momentum in the lab frame.
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