Rest mass of a composite particle A particle of rest mass m0

Rest mass of a composite particle* A particle of rest mass m_0 and kinetic energy xm_0c^2, where x is some number, strikes an identical particle at rest and sticks to it. What is the rest mass of the resultant particle?

Solution

Total energy of the particle is

E=E₀+K

Here E₀ is the rest mass energy and k is the kinetic energy.

Energy and momentum are related by the equation:

E² =E²₀+P²C²

(E₀+K)²= E²₀+P²C²

E²₀+ K²+2 E₀ K=  E²₀+P²C²

K²+2 E₀ K= P²C²

2 E₀ K = P²C² -K²

E₀ = (P²C² -K²)/2K

_M0C²= (P²C² -K²)/2K

_M0= (P²C² -K²)/2 C² K

Here K=xm0C²

_M0= (P²C² -(xm0C²)²)/2 (xm0C² ) C²

_M0= ( P² /2 xm0C² ) - (xm02)