# 3.10: Kinetic energy

- Page ID
- 8382

We remind ourselves that we are discussing *particles*, and that all kinetic energy is translational kinetic energy.

Notation:

- \(T_{C}\) = kinetic energy with respect to the centre of mass C.
- \( T\)
**=**kinetic energy with respect to the origin O.

Theorem:

\[ T = T_{C} + \frac{1}{2}M\overline{v}^{2}\tag{3.10.1}\label{eq:3.10.1} \]

Thus:

\(T = \frac{1}{2}\sum m_{i}{v}^{2}_{i} = \frac{1}{2} \sum m_{i} ({\bf v} ^{\prime}_{i} + \overline{{\bf v} })\cdot ({\bf v} ^{\prime}_{i} + \overline{{\bf v} })\)

\(= \frac{1}{2}\sum m{v}^{\prime 2}_{i} \times \overline{{\bf v} } \sum m{{\bf v} }^{\prime}_{i} + \frac{1}{2} v^{-2} \sum m_{i}\).

\(\therefore \qquad T = T_{C} + \frac{1}{2}M\overline{v}^{2} \).

Corollary:

If \(\overline{{\bf v} } = 0, T = T_{C}\) . (Think about what this means.)

Corollary:

Corollary: For a non-rotating rigid body, \( T_{C}\) = 0, and therefore \( T = \frac{1}{2}M\overline{v}^{2}\)

(Think about what this means.)