Instantaneous Velocity and Instantaneous Speed

Average velocity, as defined in Section 2.3, describes the overall trend of motion across a whole time interval, but it cannot tell us how fast an object was moving at one particular moment. To capture "speed right now," we need the idea of a limit, borrowed from elementary calculus.

Setting up the limit. Consider an object whose position at time $t$ is $x(t)$. Let it move to a new position $x(t+\Delta t)$ after a small further time $\Delta t$. Its average velocity over this small interval is

Now imagine making $\Delta t$ smaller and smaller — a fraction of a second, then a thousandth of a second, and so on — while always recomputing this ratio. As $\Delta t \to 0$, the ratio $\Delta x/\Delta t$ generally settles down to a definite limiting value. This limiting value is called the instantaneous velocity of the object at time $t$:

In the language of calculus, the instantaneous velocity is simply the derivative of position with respect to time. Geometrically, on a position-time graph, $\Delta x/\Delta t$ is the slope of the chord joining two nearby points on the curve; as $\Delta t \to 0$ this chord becomes the tangent to the curve at the point $t$, so instantaneous velocity equals the slope of the tangent to the x-t graph at that instant (this graphical picture is developed fully in Section 2.6).

Worked example of differentiation. If a particle's position is given by $x(t) = 2 + 3t - t^2$ (SI units), then applying the elementary rules of differentiation term by term,

so at $t = 2$ s the instantaneous velocity is $v = 3 - 2(2) = -1$ m/s — the negative sign showing the particle is, at that instant, moving in the negative x-direction even though it started out moving in the positive direction. (This calculation is carried out in detail in Example 3.)

Instantaneous speed. The instantaneous speed of an object at any instant is simply defined as the magnitude of its instantaneous velocity at that instant, $|v(t)|$. Unlike average speed, instantaneous speed is never larger than the magnitude of instantaneous velocity — in fact they are, by definition, always numerically identical at every single instant. (The distinction between speed and velocity that matters is between the average versions over an interval, not the instantaneous versions at a point.) …