Center of Mass Velocity & Momentum

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:
$v_{c m} = \frac{\sum_{i} m_{i} \vec{v_{i}}}{\sum_{i} m_{i}}$

Velocity center of mass in terms of momentum:
$v_{c m} = \frac{p_{t o t}}{M}$

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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. By definition of velocity is the instantaneous change of the center of mass position.

Derivation

Assumptions

To derive the center of mass velocity and momentum, assume the following:

Velocity is the derivative of displacement
The center of mass position is defined as $\vec{r_{c m}} = \frac{\sum_{i} m_{i} \vec{r_{i}}}{\sum_{i} m_{i}}$
Linear momentum is defined as $\vec{p} = m \vec{v}$

\begin{aligned} v_{c m} & = \frac{d r_{c m}}{d t} \\ = \frac{\sum_{i} \overset{p_{i}}{\overset{⏞}{m_{i} \vec{v_{i}}}}}{\sum_{i} m_{i}} \\ = \frac{\sum_{i} p_{i}}{\sum_{i} m_{i}} \\ = \frac{p_{t o t}}{M} \end{aligned}

Note

In the final line of the derivation we let $\sum_{i} p_{i} = p_{t o t}$ and $\sum_{i} m_{i} = M$