Displacement

Displacement: Definition and Mathematical Representation

Introduction

Displacement is a fundamental concept in physics and kinematics. It represents the change in the position of an object from its initial point to its final point. Displacement is a vector quantity, meaning it has both magnitude and direction.

It differs from distance, which only considers how much ground an object has covered, regardless of direction.

Definition

Mathematically, displacement is defined as:

$\overrightarrow{d}={\overrightarrow{r}}_{\text{final}}-{\overrightarrow{r}}_{\text{initial}}$

Where:

• $\overrightarrow{d}$ is the displacement vector,
• ${\overrightarrow{r}}_{\text{final}}$ is the final position vector,
• ${\overrightarrow{r}}_{\text{initial}}$ is the initial position vector.

SI Unit

The SI unit of displacement is the meter (m):

$1\mathrm{m}=100\mathrm{c}\mathrm{m}=0.001\mathrm{k}\mathrm{m}$

Characteristics

• It is a vector (includes both magnitude and direction).
• Can be positive, negative, or zero depending on direction.
• Independent of the actual path taken; depends only on start and end points.
• Minimum possible distance between two positions.

One-Dimensional Example

If an object moves from position $x_1=2\mathrm{m}$ to $x_2=5\mathrm{m}$, then:

$\mathrm{\Delta }x=x_2-x_1=5\mathrm{m}-2\mathrm{m}=3\mathrm{m}$

If it moves from 5 m to 2 m:

$\mathrm{\Delta }x=2\mathrm{m}-5\mathrm{m}=-3\mathrm{m}$

The sign indicates direction.

Relation to Velocity

Velocity is the rate of change of displacement:

$\overrightarrow{v}=\frac{d\overrightarrow{d}}{dt}$

This highlights the importance of displacement in determining an object’s motion.

Graphical Representation

• A displacement–time graph shows how position changes over time.
• The slope of this graph gives the object's velocity.

Displacement vs. Distance

Quantity	Type	Directional?	Value Range
Distance	Scalar	No	Always ≥ 0
Displacement	Vector	Yes	Can be positive, negative, or zero

Example: If a person walks 4 m east and 3 m west, total distance is:

$d=4+3=7\mathrm{m}$

Displacement is:

$\overrightarrow{d}=4-3=1\mathrm{m}\text{(east)}$

Applications

• Describing motion in kinematics
• Calculating velocity and acceleration
• Physics simulations and animation
• GPS and navigation systems

See Also

• Distance
• Velocity
• Vector (physics)
• Kinematics
• Position Vector