Velocity and Speed

Measuring Motion with Distance and Displacement

When an object moves, speed states the distance traveled per unit time. Velocity states the displacement per unit time together with its direction.

Speed is a scalar quantity, so it only has a value. Velocity is a vector quantity, so it has both a value and a direction.

The symbol $s$ means distance traveled, $\Delta x$ means displacement on a straight line, and $\Delta t$ means time interval. These formulas calculate average speed and average velocity over that time interval.

Speed uses the whole distance traveled. Velocity uses net displacement with its sign.

Speed from Path Length and Velocity from Position Change

To separate the two ideas, check the total path actually traveled and the change in position from start to finish. Speed is calculated from the whole distance, while velocity is calculated from net displacement per time.

For straight motion without turning around, the magnitude of speed and velocity can be the same. But if the object turns back, distance becomes greater than displacement.

Notice the last row. The object clearly moves because its distance is $16 \text{ m}$, but its final position change is zero because it returns to the starting point.

Average Speed Uses Path Length

Suppose an object moves from position $0 \text{ m}$ to $8 \text{ m}$, then returns to position $4 \text{ m}$ in $4 \text{ s}$.

The distance traveled is:

The average speed is:

This value only tells how fast the object moved along the path. It does not describe the net direction.

Average speed is therefore always nonnegative. As long as the object keeps moving along a path, the accumulated path length increases.

Average Velocity Uses Displacement

For the same example, the displacement is:

The average velocity is:

The positive sign means the net velocity points right. If the result is negative, the net velocity points left.

Average velocity must therefore be read together with the positive direction chosen at the start. Without that reference direction, a positive or negative sign has no clear meaning.

When Speed and Velocity Have Different Values

Now imagine the object moves from $0 \text{ m}$ to $8 \text{ m}$, then returns to $0 \text{ m}$ in $4 \text{ s}$.

The distance traveled is $16 \text{ m}$, so the speed is:

The displacement is $0 \text{ m}$, so the velocity is:

So speed can be nonzero while velocity over the trip is zero. Speed uses path length, while velocity uses the change in position from start to finish.

Choosing the Right Formula

If the question asks how fast the path is traveled, use distance and speed. If the question asks how fast position changes in a direction, use displacement and velocity.