Displacement

Concept & Definition

If position answers “where is the body?”, then displacement answers the question that actually matters in physics: “how far, and in which direction, did it move?” It is defined as the change in position — a single straight arrow drawn from where the motion started to where it ended, regardless of the path taken in between.

Because it carries both a size and a direction, displacement is a vector quantity, written $\Delta \vec{r}$. The Greek letter $\Delta$ (delta) is the standard mathematical shorthand for “change in” — always computed as final minus initial.

Units and Dimensions

The SI unit of displacement is the meter ($\text{m}$), and its dimensional formula is $[\text{L}]$ — the same as position, since it is a difference of two positions.

Mathematical Formulation

For motion along a straight line, displacement is the difference between the final coordinate $x_f$ and the initial coordinate $x_i$:

Key Formula

Displacement in 1D

$$\Delta x = x_f - x_i$$

where $\Delta x$ is the displacement, $x_f$ is the final position, and $x_i$ is the initial position of the particle on the axis.

In three dimensions the same idea applies component by component, and the displacement vector is the difference of the final and initial position vectors:

Vector Form

Displacement Vector in 3D

$$\Delta \vec{r} = \vec{r}_f - \vec{r}_i = (x_f - x_i)\,\hat{i} + (y_f - y_i)\,\hat{j} + (z_f - z_i)\,\hat{k}$$

where $\vec{r}_i = (x_i, y_i, z_i)$ and $\vec{r}_f = (x_f, y_f, z_f)$ are the initial and final position vectors of the particle.

The sign of $\Delta x$ in 1D tells you the direction: a positive displacement means net motion along the $+x$ direction, and a negative displacement means net motion along $-x$. The magnitude $|\Delta \vec{r}|$ is the straight-line distance between start and finish.

Displacement vs Distance

These two words are used interchangeably in everyday speech, but in physics they are strictly different:

• Distance is a scalar — the total length of the path actually traveled.
• Displacement is a vector — the straight arrow from start to end.

Walk once around a 400 m track and stop where you started: your distance traveled is $400\ \text{m}$, but your displacement is exactly zero, because the initial and final positions coincide. Distance can never decrease along a journey; displacement can shrink, grow, or even vanish.

Common Pitfall

Never report a round trip as “zero movement”. The displacement is zero, but the distance is not — and the two answer different questions. Always ask: does the problem care about the path (distance) or only the endpoints (displacement)?

Quick Check

A particle moves from x = 2 m to x = 7 m, then back to x = 3 m. What are its displacement and total distance traveled?

Reveal Explanation

Displacement only sees the endpoints: $\Delta x = x_f - x_i = 3 - 2 = +1\ \text{m}$.

Distance counts every step: first leg $|7 - 2| = 5\ \text{m}$, second leg $|3 - 7| = 4\ \text{m}$, so the total distance is $5 + 4 = 9\ \text{m}$.