Instantaneous velocity

Concept & Definition

Average velocity describes a whole interval, but real motion varies from moment to moment — a speedometer does not report your average since breakfast. Instantaneous velocity is the velocity at one specific instant: what the average velocity becomes when the time interval shrinks toward zero around that instant.

It is a vector, directed along the motion at that moment, and its magnitude is the instantaneous speed — the number a speedometer actually shows.

Units and Dimensions

Instantaneous velocity is measured in meters per second ($\text{m/s}$), with dimensional formula $[\text{L}\,\text{T}^{-1}]$ — identical to average velocity, since a limit does not change units.

Mathematical Formulation

Taking the average velocity $\Delta x / \Delta t$ and letting the interval collapse to zero defines the instantaneous velocity as a derivative:

Definition

Instantaneous Velocity

$$v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$$

where $v$ is the instantaneous velocity at time $t$ and $x(t)$ is the position of the particle as a function of time.

In vector form, the same limiting process applies to each component of the position vector:

Vector Form

Velocity Vector

$$\vec{v} = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\,\hat{i} + \frac{dy}{dt}\,\hat{j} + \frac{dz}{dt}\,\hat{k}$$

where $\vec{r}(t) = x\,\hat{i} + y\,\hat{j} + z\,\hat{k}$ is the position vector of the particle as a function of time.

Key Intuition

On a position–time graph, the instantaneous velocity at a point is the slope of the tangent line to the curve there. As $\Delta t \to 0$, the chord used for average velocity rotates until it just touches the curve — that touching line is the tangent.

Common Pitfall

An average over a trip tells you nothing about any single moment, and vice versa. A car can average $60\ \text{km/h}$ over a journey while never once reading exactly $60\ \text{km/h}$ on its speedometer. Keep “average” and “instantaneous” as separate ideas.

Quick Check

A particle moves with position x(t) = 3t² (in meters, t in seconds). What is its instantaneous velocity at t = 2 s?

Reveal Explanation

Differentiate the position: $v = \dfrac{dx}{dt} = \dfrac{d}{dt}\left(3t^2\right) = 6t$.

Evaluate at $t = 2\ \text{s}$: $v = 6 \times 2 = 12\ \text{m/s}$.