The centre-of-mass frame

The little arrow diagram boils down to two general formulae, valid for any triple of objects $A$, $B$, $C$:

General formulae

$\vec{v}_{A(B)} = \vec{v}_{A(C)} + \vec{v}_{C(B)}$ $\vec{v}_{A(B)} = -\vec{v}_{B(A)}$

The first is composition (you pass through a third object $C$); the second says that swapping who is observing only flips the sign.

A particularly useful case is the centre-of-mass reference frame (CMRF): the observer moving together with the CM, with velocity $\vec{v}_\text{CM}$. In this frame, the velocity of each body becomes its velocity relative to the CM:

In the CM frame

$\vec{v}_{i(\text{CM})} = \vec{v}_i - \vec{v}_{\text{CM}}$

This frame has a remarkable property: the total momentum measured in the CMRF is always zero. Indeed $\sum_i m_i \vec{v}_{i(\text{CM})} = \sum_i m_i \vec{v}_i - M_\text{tot}\vec{v}_\text{CM} = \vec{p}_\text{tot} - \vec{p}_\text{tot} = \vec{0}$. This is natural: the observer moves at exactly the velocity at which the overall momentum travels, so it “sees” the system at rest as a whole. The CMRF is the privileged frame for analysing collisions, explosions and the decomposition of kinetic energy.

Links

Topics: Centre of mass Concepts: Centre of mass Skills: Changing reference frame

Related exercises: Worked exercise — The cat on the trolley · Rocket exploding mid-trajectory · Disc with a tangent hole