Displacement

Motion
definition

Also known as: change in position, ฮ”x

Grade 9-12

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

Definition

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

๐Ÿ’ก Intuition

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

๐ŸŽฏ Core Idea

Displacement only depends on start and end points โ€” the path taken doesn't matter.

Example

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

Formula

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

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).

๐ŸŒŸ Why It 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.

๐Ÿ’ญ Hint When Stuck

When solving a displacement problem, identify the initial and final positions. Subtract the initial position vector from the final: \Delta \vec{x} = \vec{x}_f - \vec{x}_i. In 2-D, find each component separately and then use Pythagoras for the magnitude and trigonometry for the direction.

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}.

Related Concepts

PositionVelocityVectors

Compare With Similar Concepts

Speed vs Velocity

๐Ÿšง Common Stuck Point

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

โš ๏ธ Common Mistakes

  • Using total distance travelled instead of the straight-line change in position โ€” a round trip of 10 km has zero displacement but 10 km of distance.
  • Forgetting that displacement is a vector โ€” 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 โ€” when directions differ, you must add components, not magnitudes.

Frequently Asked Questions

What is Displacement in Physics?

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

What is the Displacement formula?

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

When do you use Displacement?

When solving a displacement problem, identify the initial and final positions. Subtract the initial position vector from the final: \Delta \vec{x} = \vec{x}_f - \vec{x}_i. In 2-D, find each component separately and then use Pythagoras for the magnitude and trigonometry for the direction.

Prerequisites

position

Next Steps

velocityvectors

How Displacement Connects to Other Ideas

To understand displacement, you should first be comfortable with position. Once you have a solid grasp of displacement, you can move on to velocity and vectors.

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