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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

$\displaystyle \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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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