Average Velocity and Average Speed

Average Values Use a Complete Time Interval

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 returns toward the start, distance and displacement can have different values.

\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 route drawn in the scene. The green path shows the total distance because it follows the whole track. The purple line shows the displacement because it connects the starting position directly to the ending position.

To calculate the total distance, split the green route into straight and curved segments. A straight segment uses its line length. A curved segment uses its arc length, which is the radius multiplied by the angle in radians. Add all segments, then compare the result with the same elapsed time.

\begin{aligned} &s_{\text{total}}=\sum L_{\text{segment}} \\ &L_{\text{line}}=d \\ &L_{\text{arc}}=r|\theta| \end{aligned}

For displacement, first find the change in each coordinate:

\begin{aligned} &\Delta x=x_{\text{end}}-x_{\text{start}} \\ &\Delta z=z_{\text{end}}-z_{\text{start}} \\ &|\Delta\vec r|=\sqrt{(\Delta x)^2+(\Delta z)^2} \end{aligned}

Divide the total distance and the displacement magnitude by the same elapsed time:

\begin{aligned} &\text{average speed} \\ &=\frac{s_{\text{total}}}{\Delta t} \\ &|\bar{\vec v}|=\frac{|\Delta \vec r|}{\Delta t} \end{aligned}

In the Smooth Bend case, the ball follows the curve, so its total distance is about $16.7 \text{ m}$ . The start and finish positions, however, are only $14 \text{ m}$ apart. Because the trip takes $5 \text{ s}$ , the average speed and average velocity magnitude are:

\begin{aligned} &s_{\text{total}}\approx 16.7\text{ m} \\ &|\Delta \vec r|=14\text{ m} \\ &\Delta t=5\text{ s} \\ &\text{average speed} \\ &\approx \frac{16.7}{5} \\ &\approx3.3\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	Comparison Result
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

Both calculations use the same elapsed time, but average speed divides total distance by that time and average velocity divides displacement by it.

Average Speed and Average Velocity on a Return Route

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}

Average speed uses the entire path in the numerator, while average velocity uses the change in position.

The Sign of Average Velocity Shows Direction

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

Before using a formula, check whether the question asks for distance or displacement. This choice determines whether to calculate average speed or average velocity.