Average Velocity and Speed

Physics Classical Mechanics • Motion

Written by STEM Calculators Team Published April 30, 2026 Updated April 30, 2026

Compute average velocity from displacement and time, compare it with average speed from total distance and time, and inspect the motion on an interactive graph with animation.

Dimension

Preset

Distance unit

Time unit

Axis label

Displacement vector

Δx

Δy

Δz

Elapsed time

Total distance traveled (optional)

The displacement vector is the net change in position. If the distance field is left blank, the calculator assumes straight-line motion, so total distance equals the displacement magnitude. Accepted numeric expressions include pi/2, sqrt(2), 1e-3, sin(0.4), and abs(-5).

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Enter the displacement data and click “Calculate”.

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Theory

Average velocity and average speed are related, but they are not the same quantity. The key difference is that average velocity is based on displacement, while average speed is based on total distance traveled. Because displacement is a vector and distance is a scalar, average velocity is a vector quantity and average speed is a scalar quantity.

Average velocity

If an object has displacement vector \(\Delta \vec r\) over an elapsed time \(t\), then the average velocity vector is

Main formula.

\[ \begin{aligned} \vec v_{\text{avg}} &= \frac{\Delta \vec r}{t} \end{aligned} \]

Its magnitude is

\[ \begin{aligned} |\vec v_{\text{avg}}| &= \frac{|\Delta \vec r|}{t} \end{aligned} \]

This quantity depends only on where the motion started and where it ended. The route taken in between does not affect the average velocity.

Average speed

If the total distance traveled is \(s\), then the average speed is

\[ \begin{aligned} \text{average speed} &= \frac{s}{t} \end{aligned} \]

This depends on the full path length, not on the net displacement alone.

Why the two are different

Suppose a person walks around a curved route and ends up only a short distance away from the starting point. The total distance traveled can be large, but the displacement can be small. In that case, the average speed is much larger than the magnitude of the average velocity.

In fact, the inequality

\[ \begin{aligned} s \ge |\Delta \vec r| \end{aligned} \]

implies that

\[ \begin{aligned} \text{average speed} \ge |\vec v_{\text{avg}}| \end{aligned} \]

Equality occurs only when the motion is straight-line motion without detours.

2D and 3D support

The formulas work in both two and three dimensions. In 2D, if \(\Delta \vec r = (\Delta x,\Delta y)\), then

\[ \begin{aligned} |\Delta \vec r| &= \sqrt{(\Delta x)^2 + (\Delta y)^2} \end{aligned} \]

In 3D, if \(\Delta \vec r = (\Delta x,\Delta y,\Delta z)\), then

\[ \begin{aligned} |\Delta \vec r| &= \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2} \end{aligned} \]

Once the displacement magnitude is known, the average velocity magnitude follows directly by dividing by time.

Worked sample

Use the sample values \(|\Delta \vec r| = 13\,\mathrm{m}\), \(t = 2\,\mathrm{s}\), and \(s = 20\,\mathrm{m}\).

Step 1. Compute the magnitude of average velocity.

\[ \begin{aligned} |\vec v_{\text{avg}}| &= \frac{|\Delta \vec r|}{t} \\ &= \frac{13}{2} \\ &= 6.5\,\mathrm{m\,s^{-1}} \end{aligned} \]

Step 2. Compute the average speed.

\[ \begin{aligned} \text{average speed} &= \frac{s}{t} \\ &= \frac{20}{2} \\ &= 10\,\mathrm{m\,s^{-1}} \end{aligned} \]

Step 3. Compare them.

\[ \begin{aligned} 10\,\mathrm{m\,s^{-1}} > 6.5\,\mathrm{m\,s^{-1}} \end{aligned} \]

The reason is that the total distance traveled is larger than the displacement magnitude.

Vector meaning of average velocity

Average velocity does not just have a size; it also has a direction. Its direction is the same as the direction of the displacement vector. That is why the graph in the calculator shows the displacement arrow and the actual traveled path separately. The path determines average speed, while the displacement arrow determines average velocity.

Important restriction

The elapsed time must be greater than zero. Otherwise, dividing by time would be undefined. Also, the total distance traveled cannot be smaller than the displacement magnitude, because no real path can be shorter than the straight-line separation between the start and end points.

Summary

Quantity	Formula	Meaning
Average velocity vector	\(\vec v_{\text{avg}} = \Delta \vec r / t\)	Net displacement per unit time
Magnitude of average velocity	\(\|\vec v_{\text{avg}}\| = \|\Delta \vec r\| / t\)	Size of the vector average velocity
Average speed	\(s / t\)	Total path length per unit time
Always true	\(\text{average speed} \ge \|\vec v_{\text{avg}}\|\)	Distance is never less than displacement magnitude
Equality case	\(s = \|\Delta \vec r\|\)	Straight-line motion without detours

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Average Velocity and Speed

Interactive motion graph

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Interactive motion graph

Calculation steps

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