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 v_c of the composite particle?

(b) What is its mass m_c?

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

        p_μ  =  (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:

        m_c²  =  p_μ²  =  m²(γ+1)² - m²γ²v²  =  (γ²(1-v²) + 2γ+1)m²  =  2(γ+1)m²  =  (16/3)m².

Hence  m_c  =  (4/√3)m.

The velocity of the composite particle is its 3-momentum divided by its energy:

        v_c  =  p_x / p_t  =  mγv / m(γ+1)  =  (γ/(γ+1))v  =  (5/8)v  =  (5/8)(4/5)  =  1/2.

Hence  v_c  =  c/2.