Acceleration

A moving particle's velocity derivative, $\mathbf{a}(t)=\mathbf{v}'(t)=\mathbf{r}''(t)$ .

When $\mathbf{r}(t)$ is read as a particle's position, $\mathbf{r}'(t)$ is its acceleration — the rate of change of velocity. Integrating a known acceleration twice, fixing one constant vector with an initial velocity and a second with an initial position, rebuilds the full trajectory $\mathbf{r}(t)$ .

Examples

Projectile motion assumes constant $\mathbf{a}(t) = \langle 0, 0, -g\rangle$ ; circular orbital motion has acceleration always pointed back at the center, forcing $\mathbf{r}(t)\cdot\mathbf{r}'(t)=0$ .