Velocity

A moving particle's position derivative, $\mathbf{v}(t)=\mathbf{r}'(t)$ — a vector with both speed and direction.

When $\mathbf{r}(t)$ is read as a particle's position, its derivative $\mathbf{v}(t)=\mathbf{r}'(t)$ is the particle's velocity — the direction it's moving in and how fast, bundled into one vector. Velocity is recovered from acceleration by integrating once, $\mathbf{v}(t) = \int \mathbf{a}(t)\,dt$ , with an initial velocity $\mathbf{v}(t_0)$ needed to fix the resulting constant vector.

Examples

A projectile with initial velocity $\mathbf{v}_0=\langle 60,0,80\rangle$ ft/s under gravity has $\mathbf{v}(t)=\langle 60,\ 0,\ 80-32t\rangle$ .