Displacement Formula

The Formula

\Delta \vec{x} = \vec{x}_{\text{final}} - \vec{x}_{\text{initial}}

When to use: How far you are from where you started, in a straight line. Not the path you took.

Quick Example

Walk 3m east, then 4m north. Displacement = 5m northeast (not 7m total distance).

Notation

\Delta\vec{r} or \Delta\vec{x} is the displacement vector in metres, \vec{r}_i and \vec{r}_f are the initial and final position vectors, and |\Delta\vec{r}| is the magnitude (scalar distance between endpoints).

What This Formula Means

The change in position of an object, measured as the straight-line distance and direction from the starting point to the ending point.

How far you are from where you started, in a straight line. Not the path you took.

Formal View

Displacement is the vector \Delta\vec{r} = \vec{r}_f - \vec{r}_i. In Cartesian coordinates, \Delta\vec{r} = (x_f - x_i)\hat{i} + (y_f - y_i)\hat{j}. Its magnitude is |\Delta\vec{r}| = \sqrt{(\Delta x)^2 + (\Delta y)^2}.

Worked Examples

Example 1

easy
A person walks 4 \text{ m} east and then 3 \text{ m} north. What is the total distance traveled and the displacement?

Solution

  1. 1
    Total distance traveled is the sum of path lengths: d = 4 + 3 = 7 \text{ m}.
  2. 2
    Displacement is the straight-line distance from start to finish: |\vec{d}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ m}
  3. 3
    Direction: \theta = \tan^{-1}\left(\frac{3}{4}\right) \approx 36.9° north of east.

Answer

\text{Distance} = 7 \text{ m}, \quad \text{Displacement} = 5 \text{ m at } 36.9° \text{ N of E}
Displacement is a vector quantity measuring the shortest path from start to end point, while distance is the total path length traveled. They differ whenever the path is not a straight line.

Example 2

medium
A runner completes one full lap around a 400 \text{ m} circular track. What is the runner's displacement and total distance?

Common Mistakes

  • Using total distance travelled instead of the straight-line change in position — a round trip of 10 km has zero displacement but 10 km of distance.
  • Forgetting that displacement is a vector — stating '5 metres' without a direction is incomplete; the answer should be '5 metres northeast' or similar.
  • Adding displacements as scalars instead of using vector addition — when directions differ, you must add components, not magnitudes.

Why This Formula Matters

Displacement distinguishes actual position change from total distance travelled. It is essential for correctly calculating velocity and for solving navigation, projectile, and orbital mechanics problems where direction matters as much as distance.

Frequently Asked Questions

What is the Displacement formula?

The change in position of an object, measured as the straight-line distance and direction from the starting point to the ending point.

How do you use the Displacement formula?

How far you are from where you started, in a straight line. Not the path you took.

What do the symbols mean in the Displacement formula?

\Delta\vec{r} or \Delta\vec{x} is the displacement vector in metres, \vec{r}_i and \vec{r}_f are the initial and final position vectors, and |\Delta\vec{r}| is the magnitude (scalar distance between endpoints).

Why is the Displacement formula important in Physics?

Displacement distinguishes actual position change from total distance travelled. It is essential for correctly calculating velocity and for solving navigation, projectile, and orbital mechanics problems where direction matters as much as distance.

What do students get wrong about Displacement?

Displacement is a vector with direction; distance is a scalar that is always positive.

What should I learn before the Displacement formula?

Before studying the Displacement formula, you should understand: position.

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Related Concepts

PositionVelocityVectors

Related Formulas

Velocity Formula