= Velocity
{title2=$\mathbf v=d\mathbf r/dt$}
{wiki}
<Velocity> is the <vector> giving the rate of change of <position> with <time>, measured in a specified <reference frame>. For a differentiable path $\mathbf r(t)$, its instantaneous <velocity> and its average <velocity> between two <times> are
$$
\mathbf v(t)=\frac{d\mathbf r}{dt},\qquad \mathbf v_{\mathrm{avg}}=\frac{\mathbf r(t_2)-\mathbf r(t_1)}{t_2-t_1},\qquad t_2>t_1.
$$
The <displacement> in the numerator retains direction. In <Cartesian coordinates>, each <vector component> is a <derivative>, for example $v_x=dx/dt$; its sign tells whether the corresponding coordinate is increasing or decreasing. The <SI> unit of <velocity> is $\mathrm{m/s}$. https://openstax.org/books/university-physics-volume-1/pages/3-2-instantaneous-velocity-and-speed[Ling, Sanny and Moebs, University Physics Volume 1, section 3.2: instantaneous velocity and speed]{ref}
For a concrete reversal, let $x(t)=(1\,\mathrm m)+(1\,\mathrm{m/s^2})t^2-(2\,\mathrm{m/s})t$ for $0\le t\le2\,\mathrm s$. Its signed <velocity> is $v_x(t)=(2\,\mathrm{m/s^2})t-2\,\mathrm{m/s}$: at $t=0$ it is $-2\,\mathrm{m/s}$, at $t=1\,\mathrm s$ it is zero, and at $t=2\,\mathrm s$ it is $+2\,\mathrm{m/s}$. The opposite endpoint <velocities> describe opposite directions, although the endpoint <speeds> agree. The <position> moves from $1\,\mathrm m$ to $0$ and back to $1\,\mathrm m$.
Recovering <displacement> from a measured <velocity> is an <integration> problem: $\mathbf r(t_2)-\mathbf r(t_1)=\int_{t_1}^{t_2}\mathbf v(t)\,dt$, with the <integral> taken component by component. In the reversal example, $\int_0^{2\,\mathrm s}v_x(t)\,dt=0$: the negative and positive contributions cancel. This distinction is useful in navigation, where the final <position> depends on direction as well as <speed>.
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