Proton–Proton Collision

A proton with  collides elastically with a proton a rest. If the two protons rebound with equal energies, what is the angle θ between them?

SOLUTION

Since the energies of the protons after the collision are equal, they will rebound at the same angle θ/2 relative to the initial momentum of the proton (see Figure 1.2). Again we use c = 1. In these units before the

Figure 1.2

collision

Using momentum conservation, we have

(1)

or

(2)

where  and  stand for β and γ after the collision. Energy conservation yields

(3)

or

(4)

Now,

(5)

So, we obtain by using (4)

(6)

Substituting (6) into (2) and using (5) gives

For γ ≈ 1, i.e., in the classical limit of low velocity, cos θ = 0, and we obtain the familiar result that the angle between billiard balls rebounding with equal energy is 90°. If γ >> 1 (extremely relativistic case), then cos θ ≈ 1 and θ → 0.