While calculating the scattering cross section for one particle scattering we use the formula $$J_i=\rho v$$ where $\rho$ is the density of the incoming beam and $v$ is the velocity of the incoming beam.
But for two particle scattering we use instead of $v$, a relative velocity for the incoming flux: $$v_{rel}=\frac{\sqrt{(p_1.p_2)^2-m_1^2.m_2^2}}{E_1E_2}$$ where $p_i$ and $E_i$ denote, for $i=1,2$, the $3-$momenta and energies of the particle whose masses are $m_1$ and $m_2$. In a collinear frame, i.e., if $p_1$ and $p_2$ are along the same line, this reduces to$$v_{rel}=\Bigg|\frac{p_1}{E_1}-\frac{p_2}{E_2}\Bigg|$$ where this coincides for relative velocities in case for non-relativistic speeds. Is there any derivation for the relative velocity available anywhere? I have tried finding it but have been unable to do so.
