Average Velocity and Average Speed

An Average Is Read from One Time Interval

When we talk about an average value, we are not reading one small moment like a speedometer. We take the starting state, the ending state, and the whole observation time, then ask, "overall, what motion would this trip be equivalent to?"

Average velocity uses displacement. Since displacement has direction, average velocity keeps direction too. Average speed uses the total distance actually traveled, so it only tells how quickly the path was covered without direction.

If the path bends or comes back, distance and displacement no longer tell the same story.

\bar{v}=\frac{\Delta x}{\Delta t}

The symbol $\Delta x$ or $\Delta \vec{r}$ means the change in position from start to finish. The symbol $s_{\text{total}}$ means the length of the path traveled during the trip.

Rolling Ball Average Motion

Watch the ball follow one measured track, then compare the traveled path with the start-to-finish displacement.

Total distance: $s_{\text{total}}=16.7\text{ m}$
Displacement magnitude: $|\Delta \vec r|=14\text{ m}$
Elapsed time: $\Delta t=5\text{ s}$
Average speed: $\frac{s_{\text{total}}}{\Delta t}=3.3\text{ m/s}$
Average velocity magnitude: $\frac{|\Delta \vec r|}{\Delta t}=2.8\text{ m/s}$

The numbers in the card come from the same route drawn in the scene. The green path is read as total distance because it follows the whole track. The purple line is read as displacement because it only connects the starting position to the ending position.

\begin{aligned} s_{\text{total}}&=\sum L_{\text{segment}} \\ L_{\text{line}}&=d,\qquad L_{\text{arc}}=r|\theta| \\ |\Delta \vec r|&=\sqrt{(x_{\text{end}}-x_{\text{start}})^2+(z_{\text{end}}-z_{\text{start}})^2} \\ \text{average speed}&=\frac{s_{\text{total}}}{\Delta t} \\ |\bar{\vec v}|&=\frac{|\Delta \vec r|}{\Delta t} \end{aligned}

In the Smooth Bend tab shown first, the green path is longer because the ball follows the bend. The purple line stays shorter because it only measures the start-to-finish position change.

\begin{aligned} s_{\text{total}}&\approx 16.7\text{ m},\qquad |\Delta \vec r|=14\text{ m},\qquad \Delta t=5\text{ s} \\ \text{average speed}&\approx \frac{16.7}{5}=3.3\text{ m/s} \\ |\bar{\vec v}|&=\frac{14}{5}=2.8\text{ m/s} \end{aligned}

The Same Time Interval Can Describe Different Trips

Suppose two trips both take $3 \text{ s}$ . The first trip goes straight $6 \text{ m}$ to the right. The second trip goes $6 \text{ m}$ to the right and then $3 \text{ m}$ back to the left. The time is the same, but the distance and displacement are different.

Trip	Total Distance	Displacement	What the Average Tells Us
Straight to the right	$6 \text{ m}$	$6 \text{ m}$ to the right	Average speed and average velocity have the same magnitude
Right then back	$9 \text{ m}$	$3 \text{ m}$ to the right	Average speed is larger than average velocity magnitude

This table shows where the difference comes from. The time does not change, but the numerator in the formula changes.

A Return Route Separates the Numbers

On the return route, the object moves $6 \text{ m}$ to the right and then $3 \text{ m}$ to the left in $3 \text{ s}$ .

The total distance is $9 \text{ m}$ , but the displacement is only $3 \text{ m}$ to the right.

\text{average speed}=\frac{9}{3}=3\text{ m/s}

Notice that both calculations use the same time interval. The difference appears because the numerator is different: average speed uses the whole path, while average velocity uses the change in position.

The Sign Shows the Direction of Average Velocity

If we choose the rightward direction as positive, displacement to the right is positive and displacement to the left is negative. That sign carries into average velocity.

For example, an object starts at $x=8 \text{ m}$ and ends at $x=2 \text{ m}$ in $2 \text{ s}$ .

\Delta x=2-8=-6\text{ m}

A negative value does not mean the motion is "slower than zero." It means the average velocity points to the left relative to the axis we chose.

A Loop Can Have Zero Average Velocity

If an object returns to its starting point, its displacement is $0 \text{ m}$ . As a result, its average velocity is $0 \text{ m/s}$ even though its average speed can still be greater than zero.

For a loop trip, the path still has length. But the start-to-finish change in position becomes zero because the starting point and ending point are the same.

Question	Quantity Used	Value Being Asked
How much path is covered each second?	Total distance	Average speed
How much does position change each second?	Displacement	Average velocity

So, before using a formula, first check whether the question asks for distance or displacement. That is the fastest way to avoid mixing up average speed and average velocity.