Average velocity and average speed

Concept & Definition

Knowing that a body moved is rarely enough — we want to know how fast it moved. Over any interval of time, physics offers two answers, and they differ in exactly the way displacement and distance do. Average velocity divides the displacement by the elapsed time; average speed divides the total path length by the elapsed time.

One is a vector with a direction and a sign; the other is a plain, always-positive scalar. Choosing the right one for a problem is half the battle.

Units and Dimensions

Both average velocity and average speed are measured in meters per second ($\text{m/s}$), with dimensional formula $[\text{L}\,\text{T}^{-1}]$.

Mathematical Formulation

If a particle undergoes displacement $\Delta x$ during the time interval $\Delta t = t_f - t_i$, its average velocity over that interval is:

Key Formula

Average Velocity

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

where $\bar{v}$ is the average velocity, $\Delta x$ is the displacement, and $\Delta t$ is the elapsed time between the initial and final instants.

The bar over $v$ is the standard notation for an average over an interval. Because $\Delta x$ can be positive, negative, or zero, average velocity inherits all three possibilities. Average speed, by contrast, never turns negative:

Scalar Form

Average Speed

$$\bar{s} = \frac{\text{total distance traveled}}{\Delta t}$$

where $\bar{s}$ is the average speed and the numerator is the full path length, not the displacement.

Key Intuition

On a position–time graph, average velocity over an interval is the slope of the straight line (the chord) joining the initial and final points of the curve.

Common Pitfall

For a round trip the displacement is zero, so the average velocity is zero — but the average speed is not. A runner who completes one lap of a track in $60\ \text{s}$ has $\bar{v} = 0$ yet $\bar{s} = 400 / 60 \approx 6.7\ \text{m/s}$. Never use displacement in the speed formula or distance in the velocity formula.

Quick Check

A car drives 60 km east in 1 hour, then 60 km west in 1 hour. Find its average velocity and average speed over the 2-hour trip.

Reveal Explanation

Average velocity uses displacement: the car ends where it started, so $\Delta x = 0$ and $\bar{v} = 0 / 2 = 0\ \text{km/h}$.

Average speed uses distance: total path is $60 + 60 = 120\ \text{km}$, so $\bar{s} = 120 / 2 = 60\ \text{km/h}$.