Kinetic Energy in CM Frame

The decomposition

The total kinetic energy of a system can always be split into two parts:

$KE_{total} = \underbrace{\frac{1}{2}Mv_{cm}^2}_{\text{CM translational KE}} + \underbrace{KE_{cm}}_{\text{internal KE}}$

where $KE_{cm} = \sum \frac{1}{2}m_i (v_i^{cm})^2$ is the KE of motion relative to the CM.

Derivation

$KE_{total} = \sum \frac{1}{2}m_i v_i^2 = \sum \frac{1}{2}m_i |\vec{v}_{cm} + \vec{v}_i^{cm}|^2$

$= \sum \frac{1}{2}m_i(v_{cm}^2 + 2\vec{v}_{cm}\cdot\vec{v}_i^{cm} + (v_i^{cm})^2)$

$= \frac{1}{2}v_{cm}^2\sum m_i + \vec{v}_{cm}\cdot\sum m_i\vec{v}_i^{cm} + \sum\frac{1}{2}m_i(v_i^{cm})^2$

The middle term: $\sum m_i \vec{v}_i^{cm} = 0$ (total momentum in CM frame is zero)

$KE_{total} = \frac{1}{2}Mv_{cm}^2 + \sum\frac{1}{2}m_i(v_i^{cm})^2$

$\boxed{KE_{total} = \frac{1}{2}Mv_{cm}^2 + KE_{cm}}$

Physical meaning

$\frac{1}{2}Mv_{cm}^2$: Energy of the CM moving through space. This cannot be changed by internal forces — only external forces change the CM velocity.

$KE_{cm}$: Energy of internal motions (rotation, vibration, relative motion of parts). This can be changed by internal forces. In a collision, this is what gets converted to heat, sound, or deformation.

Application to collisions

In a collision with no external forces, $v_{cm}$ is constant. Therefore $\frac{1}{2}Mv_{cm}^2$ is constant.

The only KE that can change is $KE_{cm}$:

$\Delta KE_{total} = \Delta KE_{cm}$

Elastic collision: $KE_{cm}$ is conserved (no internal energy change).

Perfectly inelastic collision: $KE_{cm}$ is completely destroyed (bodies stick together and move with CM — no relative motion, so $KE_{cm} = 0$ after).

Maximum KE loss in a collision: $\Delta KE_{max} = KE_{cm,initial}$ (all internal KE destroyed).

Minimum KE of a system

The minimum possible KE of a system with fixed total momentum $\vec{p}$ is $\frac{p^2}{2M} = \frac{1}{2}Mv_{cm}^2$.

This minimum is achieved when all particles move together (i.e., in the CM frame, all are at rest: $KE_{cm} = 0$). This is exactly the perfectly inelastic collision state.