inelastic relativistic collision
A particle of mass m, moving at speed v = 4c/5, collides inelastically with a similar particle at rest.
(a) What is the speed vc of the composite particle?
(b) What is its mass mc?
Solution by Ted Jacobson
Choose units where c = 1, v = 4/5. Let the incoming particle move in the x-direction of the rest frame, and let γ = (1 – v2)–1/2 = 5/3.
The 4-momentum of the incoming particle (in the rest frame) is (γm, γmv, 0, 0), while the 4-momentum of the particle at rest is (m, 0, 0, 0), so the total 4-momentum is
pμ = (m(γ+1), mγv, 0, 0),
which must be conserved, and is therefore the 4-momentum of the composite particle. The mass of the composite particle is the magnitude of its 4-momentum:
mc2 = pμ2 = m2(γ+1)2 - m2γ2v2 = (γ2(1-v2) + 2γ+1)m2 = 2(γ+1)m2 = (16/3)m2.
Hence mc = (4/√3)m.
The velocity of the composite particle is its 3-momentum divided by its energy:
vc = px / pt = mγv / m(γ+1) = (γ/(γ+1))v = (5/8)v = (5/8)(4/5) = 1/2.
Hence vc = c/2.