Acceleration

Remarks

Example: for uniform circular motion $x(t) = (R\cos(\omega t), R\sin(\omega t), 0)$ with $R > 0$, the acceleration is $a(t) = -\omega^2 x(t)$: it points to the centre and has size $R\omega^2$. Non-example: the rate of change of the speed, $\frac{d}{dt}\abs{v(t)}$, is not the acceleration; for the circular motion it is $0$, while $a(t) \ne 0$ when $\omega \ne 0$. When $v(t) \ne 0$, the acceleration splits into a tangential part $\frac{d}{dt}\abs{v}$ along the direction of motion and a normal part of size $\abs{v}^2 \kappa$, where $\kappa$ is the curvature of the path. Frames of reference that move with constant velocity relative to each other give the same acceleration.