What's the Difference Between Instantaneous and Average Velocity?

Average velocity is total displacement divided by the total time interval, whereas instantaneous velocity is velocity at one particular moment. In IB Physics A.1 Kinematics, both are vector quantities, so their signs indicate direction.

The Reasoning

For one-dimensional motion, average velocity is

$$\bar{v}=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}.$$

It describes the overall rate of change of position during a finite interval. It does not show how velocity varied within that interval; an object can have zero average velocity while moving throughout the journey if it returns to its starting position.

Instantaneous velocity is the rate of change of position at a specific time. Mathematically, it is the limit of average velocity as the time interval approaches zero:

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

On a position versus time graph, average velocity is the gradient of a straight line joining two points, while instantaneous velocity is the gradient of the tangent at one point.

For example, a runner travels $100\text{ m}$ east in $20\text{ s}$, then $40\text{ m}$ west in $10\text{ s}$. Taking east as positive, the displacement is $+60\text{ m}$, so

$$\bar{v}=\frac{60}{30}=+2.0\text{ m s}^{-1}.$$

The runner's instantaneous velocity could have many different values, including negative values while moving west.

A common misconception is that average velocity equals average speed. Here, average speed is $140/30=4.7\text{ m s}^{-1}$, because speed uses total distance, whereas velocity uses displacement. Average speed is scalar and cannot be negative.

Exam Technique

This is an SL and HL concept. For Calculate, show the equation, substitution, and answer. On graph questions, use a chord for average velocity and draw a tangent at the specified point for instantaneous velocity; then calculate its gradient using a large triangle.