Velocity: Average and Instantaneous

Velocity is the rate of change of position with time, a vector quantity that includes direction: average velocity is displacement divided by the time taken, and instantaneous velocity is the velocity at a single moment, found as the slope of the tangent to the position-time graph. In brief: v_avg = Δx ÷ Δt. Unlike speed, velocity can be positive, negative or zero, and it can be zero for a round trip even when the speed is not. The magnitude of the instantaneous velocity is the instantaneous speed shown on a speedometer.

Describing motion needs a way of saying how fast and in which direction. Velocity does both. This guide defines average and instantaneous velocity, explains how they are found from graphs and equations, compares velocity with speed in a table, works through examples (including a tangent calculation) and corrects common mistakes.

What is average velocity?

Average velocity = displacement ÷ time interval:

v_avg = Δx ÷ Δt = (x₂ − x₁) ÷ (t₂ − t₁)

The SI unit is m/s. In one dimension, the sign shows the direction: positive means in the positive direction of the axis, negative means the opposite. Because it depends only on the starting and finishing positions, the path between them does not matter. If an object returns to its starting point, the displacement and average velocity are zero, even if it traveled far.

What is instantaneous velocity?

The instantaneous velocity is the velocity at a particular instant. It is the limit of the average velocity as the time interval becomes very small: v = lim (Δx ÷ Δt) as Δt → 0. On a position-time graph, it is the slope of the tangent to the curve at that time. For uniform motion, the instantaneous velocity is the same at every time and equals the average velocity. For changing motion, it changes from moment to moment.

In practice, a short interval is used: a speed camera measures the time to cross two lines a short distance apart, and the speedometer of a car estimates the speed from the rotation of the wheels. The magnitude of the instantaneous velocity is the instantaneous speed.

For x = t² (in m, with t in s): the chord from t = 1 s to t = 3 s has slope (9 − 1) ÷ 2 = 4 m/s, the average velocity. The tangent (dashed) at t = 2 s has slope 2t = 4 m/s, the instantaneous velocity at that moment.

Velocity and speed compared

Velocity compared with speed
Feature	Speed	Velocity
Type	Scalar	Vector
Based on	Distance	Displacement
Can be negative?	No	Yes (direction sign)
Zero for a round trip?	No	Average velocity is zero
Example	60 km/h	60 km/h north

Worked example: average and instantaneous velocity

Part A. A runner goes 100 m east in 20 s, and then 60 m west in 20 s. Take east as positive.

Displacement: +100 m + (−60 m) = +40 m, so 40 m east.
Total time: 20 s + 20 s = 40 s.
Average velocity: 40 m ÷ 40 s = +1.0 m/s, so 1.0 m/s east.
Average speed for comparison: distance = 100 + 60 = 160 m, so speed = 160 ÷ 40 = 4.0 m/s.

Part B. A cart has a position x = 2t² (x in meters, t in seconds). Find its velocity at t = 3 s.

Average velocity from 2 s to 4 s: x(2) = 8 m and x(4) = 32 m, so v = (32 − 8) ÷ (4 − 2) = 12 m/s.
Average velocity from 3 s to 4 s: x(3) = 18 m, so v = (32 − 18) ÷ 1 = 14 m/s.
A short interval around 3 s (2.9 s to 3.1 s): x(2.9) = 16.82 m and x(3.1) = 19.22 m, so v = (19.22 − 16.82) ÷ 0.2 = 12.0 m/s.
Instantaneous velocity: the values approach 12 m/s as the interval shrinks. For x = 2t², calculus gives v = 4t, so v(3) = 12 m/s, confirming the numerical result.

How did the idea of instantaneous velocity develop?

The concept of the velocity at an instant was a difficulty in ancient times (Zeno’s paradoxes of motion). Galileo described motion with changing speed in the early 1600s, and Newton and Leibniz developed calculus in the late 1600s, which gave a precise definition of instantaneous rate of change as a derivative.

Where is velocity used?

Velocity is used in navigation (the velocity of a plane relative to the air and the ground), in weather forecasts (wind velocity), in the analysis of collisions, and in astronomy (radial velocities of stars from the Doppler shift).

Which mistakes are common?

Mistake 1: Dividing total distance by time for velocity

Average velocity uses displacement. Distance divided by time gives average speed.

Mistake 2: Saying that zero average velocity means that the object did not move

It means that the object returned to the start. Its speed could have been large.

Mistake 3: Mixing up instantaneous and average values

The speedometer shows the instantaneous speed, not the average for the trip.

Mistake 4: Ignoring direction signs

Choose a positive direction first, and keep to it for all displacements and velocities.