> [!summary] The center of mass in terms of velocity is the velocity of the point where all the mass of a rigid body or collection of particles lies. > **Key Equations:** > Velocity center of mass: $v_{cm}=\frac{\sum_{i} m_{i} \vec{v_{i}}}{\sum_{i}m_{i}}$ > Velocity center of mass in terms of momentum: $v_{cm}=\frac{p_{tot}}{M}$ >[!info]+ Read Time **⏱ 1 min** # Definition The center of mass in terms of velocity describes the velocity as if all the mass were concentrated at that point. If a rigid body or system of particles has a velocity, then it also has [[Linear Momentum|linear momentum]]. By definition of [[Velocity|velocity]] is the [[Instantaneous|instantaneous change]] of the [[Center of Mass Displacement|center of mass position]]. ## Derivation > [!warning] Assumptions To derive the center of mass velocity and momentum, assume the following: > - [[Velocity|Velocity]] is the [[Derivative|derivative]] of [[Displacement|displacement]] > - The [[Center of Mass Displacement|center of mass position]] is defined as $\vec{r_{cm}} = \frac{\displaystyle \sum_{i}m_{i}\vec{r_{i}}}{\displaystyle \sum_{i} m_{i}}$ > - [[Linear Momentum|Linear momentum]] is defined as $\vec{p} =m\vec{v}$ $ \begin{align*} v_{cm}&= \frac{dr_{cm}}{dt} \\ &= \frac{\sum_{i} \overbrace{m_{i} \vec{v_{i}}}^{p_{i}}}{\sum_{i}m_{i}} \\ &= \frac{\sum_{i} p_{i}}{\sum_{i}m_{i}} \\ &= \frac{p_{tot}}{M} \end{align*} $ > [!note] In the final line of the derivation we let $\displaystyle\sum_{i}p_{i}=p_{tot}$ and $\displaystyle\sum_{i}m_{i}=M$