In Distance and Displacement you met the difference between how far you went and where you ended up. To work with numbers, physicists put the motion on an axis: pick an origin ($x = 0$) and a positive direction. Then every position is a signed number, like the integers in Subtracting Integers.
Key idea
Displacement is final position minus initial position: $\Delta x = x_f - x_i$. Its sign tells the direction: positive means it ended up on the $+x$ side of where it started, negative means the $-x$ side.
Why it works
Subtraction measures the gap between two numbers on a number line, including which way you step to get from the first to the second. Going from $x = 2$ to $x = 7$ is a step of $+5$; going from $7$ back to $2$ is a step of $-5$. Displacement is exactly that step, for the whole trip at once, so it ignores everything that happened in between.
Worked examples
Worked example: A move to the left
A cart starts at $x = 3\ \text{m}$ and stops at $x = -5\ \text{m}$. Find its displacement.
- $\Delta x = x_f - x_i = -5 - 3$.
- $\Delta x = -8\ \text{m}$. The minus sign says it moved 8 m in the $-x$ direction.
Answer: $\Delta x = -8\ \text{m}$
Worked example: Out and back
A runner goes from $x = 10\ \text{m}$ to $x = 70\ \text{m}$, then turns and jogs back to $x = 40\ \text{m}$. Find the displacement and the distance.
- Displacement uses only the ends: $\Delta x = 40 - 10 = 30\ \text{m}$.
- Distance adds the length of every leg: $60\ \text{m}$ out and $30\ \text{m}$ back.
- Distance $= 60 + 30 = 90\ \text{m}$.
Answer: $\Delta x = 30\ \text{m}$; distance $= 90\ \text{m}$.
Common mistake
Using the total distance as the displacement. When the motion changes direction, the two differ: the runner above traveled 90 m but is only 30 m from the start. Check: displacement can be negative or zero; distance never can. If an object ends where it began, its displacement is zero however far it went.
| Distance | Displacement | |
|---|---|---|
| What it measures | total path length | change in position |
| Sign | never negative | positive, negative or zero |
| Depends on the path? | yes | no, only on the start and end |