Center-of-mass velocity (${\mathbf{V}}_{\mathrm{CM}}={\sum }_i{m}_i{\mathbf{v}}_i/{\sum }_i{m}_i$)

For fixed classical particle masses, differentiating the center of mass position gives the displayed mass-weighted velocity. The total momentum is $M{\mathbf{V}}_{\mathrm{CM}}$. The relativistic zero-momentum frame instead has velocity $c^2\mathbf{P}/E$; treating rest mass as the only weight there generally fails.