Linear Momentum of the Center of Mass

Consider a system of $N$ point-masses $m_i$ , each traveling with vector velocity $\underline{v}_i$ , and not necessarily rigidly attached to each other. Then the total momentum of the system is

$$\underline{p}\eqsp \sum_{i=1}^N m_i \underline{v}_i \eqsp \sum_{i=1}^N m_i \dot{\underline{x}}_i \eqsp M \frac{d}{dt} \left(\frac{1}{M}\sum_{i=1}^N m_i \underline{x}_i \right) \eqsp M \frac{d}{dt} \underline{x}_c \isdef M \underline{v}_c$$

where $M=\sum m_i$ denotes the total mass, and $\underline{v}_c$ is the velocity of the center of mass.

Thus, the momentum $\underline{p}$ of any collection of masses $m_i$ (including rigid bodies) equals the total mass $M$ times the velocity of the center-of-mass.