KAHIBARO
Table of Contents
Meaning and Definition
Average velocity tells us how quickly position changes over a whole time interval. It does not describe every detail of the motion in between. Instead, it compares where an object started and where it ended.
If an object is at position $x_i$ at time $t_i$, and later at position $x_f$ at time $t_f$, then the average velocity is
$$
v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}
$$
The quantity $\Delta x$ is the displacement, not the total distance traveled. Because of that, average velocity depends on direction.
Average Velocity
Average velocity is defined by
$$
v_{\text{avg}} = \frac{\Delta x}{\Delta t}
$$
Use displacement $\Delta x$, not total distance.
Direction Matters
In one dimensional motion, direction is represented by the sign of the displacement. If motion is chosen to the right as positive, then motion to the left is negative.
If an object moves from $x = 2 \,\text{m}$ to $x = 8 \,\text{m}$ in $3 \,\text{s}$, then
$$
\Delta x = 8 - 2 = 6 \,\text{m}
$$
so
$$
v_{\text{avg}} = \frac{6}{3} = 2 \,\text{m/s}
$$
The positive sign means the motion, on average, was in the positive direction.
If instead an object moves from $x = 8 \,\text{m}$ to $x = 2 \,\text{m}$ in $3 \,\text{s}$, then
$$
\Delta x = 2 - 8 = -6 \,\text{m}
$$
and
$$
v_{\text{avg}} = \frac{-6}{3} = -2 \,\text{m/s}
$$
The negative sign shows that the average motion was in the negative direction.
Average Velocity Versus Average Speed
Average velocity and average speed are not the same. Average speed uses total distance traveled, while average velocity uses displacement.
Consider an object that moves from $x = 0 \,\text{m}$ to $x = 10 \,\text{m}$ and then back to $x = 4 \,\text{m}$ in a total time of $8 \,\text{s}$.
The displacement is
$$
\Delta x = 4 - 0 = 4 \,\text{m}
$$
so the average velocity is
$$
v_{\text{avg}} = \frac{4}{8} = 0.5 \,\text{m/s}
$$
But the total distance traveled is
$$
10 + 6 = 16 \,\text{m}
$$
so the average speed is
$$
\frac{16}{8} = 2 \,\text{m/s}
$$
These values are different because the object changed direction.
Average velocity can be zero even when an object has been moving.
If the final position equals the initial position, then $\Delta x = 0$, so
$$
v_{\text{avg}} = 0
$$
Interpreting the Formula
The formula for average velocity can be understood as a ratio of change in position to change in time. A larger magnitude of average velocity means either a bigger displacement in the same time, or the same displacement in less time.
Its SI unit is meters per second, written as $\text{m/s}$.
The sign and size both matter:
| Average velocity | Meaning |
|---|---|
| Positive | Net motion in the positive direction |
| Negative | Net motion in the negative direction |
| Zero | No net change in position |
| Larger magnitude | Greater change in position per unit time |
Simple Examples
Suppose a cyclist moves from $x = -3 \,\text{m}$ to $x = 9 \,\text{m}$ in $4 \,\text{s}$. Then
$$
\Delta x = 9 - (-3) = 12 \,\text{m}
$$
and
$$
v_{\text{avg}} = \frac{12}{4} = 3 \,\text{m/s}
$$
Now suppose a toy car moves from $x = 5 \,\text{m}$ to $x = -1 \,\text{m}$ in $2 \,\text{s}$. Then
$$
\Delta x = -1 - 5 = -6 \,\text{m}
$$
so
$$
v_{\text{avg}} = \frac{-6}{2} = -3 \,\text{m/s}
$$
Geometric Picture
On a position versus time graph, average velocity over an interval is the slope of the straight line connecting the starting point and ending point.
If the graph is curved, the motion may not be uniform, but the average velocity over the whole interval is still found from the endpoints only.
The red line is not the true path of motion in the graph. It only connects the initial and final points. Its slope gives
$$
v_{\text{avg}} = \frac{x_f - x_i}{t_f - t_i}
$$
Important Caution
A common mistake is to divide final position by final time. That is not the definition of average velocity. You must subtract the initial values first.
Do not use
$$
\frac{x_f}{t_f}
$$
for average velocity.
The correct formula is
$$
v_{\text{avg}} = \frac{x_f - x_i}{t_f - t_i}
$$
Final Idea
Average velocity is a compact way to describe overall motion during a time interval. It uses only the starting and ending positions and the time between them. Because it is based on displacement, it includes direction and can differ greatly from average speed.