Looking in Goldstein's book, there doesn't seem to be a standard formula to calculate the COM frame velocity for two particles, from their relativistic velocities in the lab frame, although it is done for the case where one particle is initially at rest. I find this a glaring omission and would like to know if there is a general formula for two relativistic particles moving along the $x$-axis of the lab frame.
[Physics] Calculate relativistic boost to COM frame from two arbitary velocities
classical-mechanicshomework-and-exercisesinertial-framesspecial-relativityvelocity
Best Answer
The center of mass 4-momentum is the sum of the 4-momenta of the particles (no vector symbol or index, but the v's are four-component vectors) using the masses as the weights:
$$ P_\mathrm{CM} = m_1 v_1 + m_2 v_2 $$
The length of this is the mass of the combined system, (mostly minus metric)
$$ M^2 = |P|^2 = m_1^2 + m_2^2 + 2m_1 m_2 v_1 \cdot v_2 $$
The four-velocity of the center of mass is then
$$ v_\mathrm{CM} = {m_1v_1 + m_2 v_2 \over M} $$
and the three velocity is given by the ratio of the space-components of the four vector to the time component:
$$ v^0_\mathrm{CM} = {m_1\gamma_1 + m_2 \gamma_2 \over M}$$
So that the center of mass velocity is:
$$ \vec{v}_\mathrm{CM} = {m_1\gamma_1 \vec{v}_1 + m_2\gamma_2 \vec{v}_2 \over m_1\gamma_1 + m_2\gamma_2}$$
or the weighted average of the velocities using the relativistic mass (the energy). This formula usually appears with energy letters replacing mass letters:
$$ \vec{v}_\mathrm{CM} = { E_1 \vec{v}_1 + E_2 \vec{v}_2 \over E_1 + E_2}$$
Where m_1 and m_2 are the masses, $v_1$ and $v_2$ are the 4-velocities, $E_1$ and $E_2$ are the energies, $\gamma_1 = {1\over \sqrt{1-|\vec v_1|^2}}$ and similarly for $\gamma_2$.