Velocity and Speed

Two Quantities for Motion Rate

When an object moves, we can ask two different questions. First, how fast is the path being traveled? The answer is speed. Second, how fast is position changing in a direction? The answer is velocity.

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. When we discuss one whole trip, these formulas read speed and velocity over that time interval.

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

Path and Position Change Are Not the Same

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.

Speed Reads 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 speed is:

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

Velocity Reads Direction

For the same example, the displacement is:

The velocity is:

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

This is why velocity is more than speed with a sign attached. The sign only makes sense after the positive direction and the displacement have been defined.

When Speed and Velocity Look Different

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. That is not a contradiction. Speed reads path length, while velocity reads position change 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.