**inelastic relativistic collision**

A particle of mass *m*,
moving at speed *v* = 4*c*/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* – v*^{2})^{–1/2
} = 5/3.

The
4-momentum of the incoming particle (in the rest frame) is (γ*m*,
γ*mv*, 0, 0), while the
4-momentum of the particle at rest is (*m*, 0, 0, 0), so the total
4-momentum is

*
p _{μ}*
= (

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 _{c}*

Hence* m _{c }*= (4/√3)

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

* v _{c}*
=

Hence* v _{c }= c*/2.