The velocity of the centre of mass

If the centre of mass is the weighted average of the positions, its velocity is obtained simply by differentiating the definition with respect to time. Since the masses are constant, the derivative passes inside the sum and turns each position $\vec{s}_i$ into the corresponding velocity $\vec{v}_i$:

$\vec{v}_{\text{CM}} = \frac{m_1\,\vec{v}_1 + m_2\,\vec{v}_2 + \cdots}{M_{\text{tot}}} = \frac{\vec{p}_{\text{tot}}}{M_{\text{tot}}}$

The numerator $\sum_i m_i \vec{v}_i$ is exactly the total momentum of the system. So the velocity of the centre of mass is the total momentum divided by the total mass: the CM moves like a single body carrying all the mass and all the momentum of the system.

Intuition

If the total momentum is conserved, the velocity of the CM stays constant — even during a collision or an explosion! Internal forces redistribute momentum among the parts, but do not change the sum, so $\vec{v}_\text{CM}$ notices nothing. This is the key that makes the centre of mass so powerful.

Links

Topics: Centro di massa Concepts: Centro di massa · Quantità di moto · Conservazione della quantità di moto

Related exercises: Due blocchi con molla e grafico del CM · Persona che cammina sulla barca · Razzo nello spazio profondo