Displacement and Distance

Two Ways to Read a Trip

When an object moves from a starting point to an ending point, two quantities can sound similar but mean different things: distance and displacement.

Distance is the total length of the path traveled. If the path turns, every part of the turn is included. Displacement is the change in position from start to finish, so it only depends on the starting point, ending point, and direction.

\begin{aligned} s&=\text{path length} \\ \Delta \vec{r}&=\vec{r}_{\text{final}}-\vec{r}_{\text{initial}} \end{aligned}

Here, $s$ means the total path length, while $\Delta \vec{r}$ means the change in position from start to finish.

In two-dimensional motion, displacement is drawn as the straight line from the starting point to the ending point.

Distance adds the path actually traveled. Displacement reads the change from the start point to the end point.

Path Length and Displacement

Choose a route to compare the traveled path with the straight displacement.

Distance traveled: $s=7\text{ m}$
Displacement magnitude: $|\Delta\vec{r}|=5\text{ m}$
Displacement vector: $\Delta\vec{r}=4\hat{i}+3\hat{j}\text{ m}$
What changes: Distance is longer because the car follows the turn.

Detours Change Distance

Compare an object that travels from the same start point to the same end point in two ways. A straight route gives a distance equal to the displacement magnitude. A detour gives a longer distance, but the displacement is still read as the line from start to finish.

The detour route has a greater distance because the object travels along a longer path. But the displacement stays the same as long as the starting and ending points stay the same.

This is the main question to keep separate: distance asks how much path was traveled, while displacement asks where the object ended up compared with where it started.

Distance Has No Direction

Distance is always zero or positive. It answers the question, "how long is the path traveled?" So distance does not include direction.

If an object moves $4 \text{ m}$ east and then $3 \text{ m}$ south, the distance is:

s=4 \text{ m}+3 \text{ m}=7 \text{ m}

The symbol $s$ is often used for distance traveled.

Displacement Has Direction

Displacement is a vector quantity. That means displacement has magnitude and direction. For the example $4 \text{ m}$ east and then $3 \text{ m}$ south, the displacement magnitude is the straight line from start to finish:

|\Delta \vec{r}|=\sqrt{4^2+3^2}=5 \text{ m}

The direction of displacement does not follow the turns of the path. It follows the line from the starting point to the ending point.

Returning to the Starting Point

If an object travels around and returns to the starting point, the distance is not zero because a path was still traveled. But the displacement is zero because the initial and final positions are the same.

Trip	Distance	Displacement
From start to finish by a straight route	equal to the straight line	line from start to finish
From start to finish by a detour	longer	still the line from start to finish
Around a loop back to the start	not zero	$0$

So, distance tells the length of the actual path traveled, while displacement tells the net change in position.

Winding Route with a Straight Result

A student walks $8 \text{ m}$ east, then $6 \text{ m}$ north. The distance traveled is:

s=8 \text{ m}+6 \text{ m}=14 \text{ m}

The displacement magnitude is:

|\Delta \vec{r}|=\sqrt{8^2+6^2}=10 \text{ m}

The distance is $14 \text{ m}$ , but the displacement magnitude is $10 \text{ m}$ . The values are different because the path traveled is not the same as the straight line from start to finish.