Velocity Is Change with Direction

Position says where an object is. Velocity says how its position is changing—and in which direction. The distinction between position, distance, displacement, speed, and velocity is the foundation of mechanics.

Average velocity

If an object moves from position $x(t_1)$ to $x(t_2)$, its displacement is

$$\Delta x=x(t_2)-x(t_1).$$

Its average velocity is

$$v_{\mathrm{avg}}=\frac{\Delta x}{\Delta t}.$$

This value ignores the details between the endpoints. A runner who completes a lap and returns to the start has zero displacement and zero average velocity, even though the runner covered a substantial distance.

Instantaneous velocity

To describe motion at one instant, shrink the time interval around it:

For $x(t)=3t^2$, the derivative gives $v(t)=6t$. At $t=2\,\mathrm{s}$, the velocity is $12\,\mathrm{m/s}$ if position is measured in meters.

Reading motion from graphs

A horizontal position graph means the object is at rest. A straight sloped line means constant velocity. A curving position graph means velocity is changing, so the object is accelerating.

On a velocity graph, the height gives velocity while signed area gives displacement:

$$x(b)-x(a)=\int_a^b v(t)\,dt.$$

Velocity connects mathematics to mechanics

The derivative is not merely a geometric slope here. Its sign, units, and changing value describe observable motion. Acceleration is the next derivative, $a(t)=dv/dt$, and forces explain why that acceleration occurs.

Direction and speeding up are different

For $x(t)=t^3-3t$, $v=3t^2-3$ and $a=6t$. On $0<t<1$, velocity is negative while acceleration is positive, so the object moves left but slows down. At $t=1$, velocity changes sign and the object reverses direction.

Check your understanding

Can velocity be zero while acceleration is nonzero?

Show the reasoning

Yes. At the highest point of a vertical toss, velocity is momentarily zero while gravitational acceleration remains downward.