Velocity

Velocity ($\vec{v}$) measures how quickly position changes and in what direction. Speed is the magnitude of velocity: speed has no direction, velocity does.

Average velocity over a time interval is

$${\vec{v}}_{avg}=\frac{\Delta \vec{r}}{\Delta t}$$

Instantaneous velocity is the limiting case as the time interval becomes tiny:

$$\vec{v}=\frac{d\vec{r}}{dt}=\dot{\vec{r}}$$

This is a Derivative: velocity is the slope of the position-time graph. If the slope is constant, the object has constant velocity and zero acceleration.

For one-dimensional constant-velocity motion, with initial position $x_0$,

$$x(t)=x_0+vt$$

Common checks

• Positive/negative velocity depends on your chosen coordinate direction.
• A negative velocity does not mean “slowing down”; it means motion in the negative direction.
• Zero velocity at an instant does not mean zero acceleration. At the top of a vertical throw, $v=0$ but $a=-g$.

Units: $\mathrm{m}{\mathrm{s}}^{-1}$ = Metre per Second. See also Kinematic.

Graph view

On a position-time graph, velocity is the slope. A steeper slope means larger speed; a horizontal tangent means zero instantaneous velocity. Curved position-time graphs indicate changing velocity, which means acceleration is present. For constant acceleration, the velocity-time graph is a straight line.