VECTORS

Displacement: Definition, Formula, Types, and Real-Life Examples

INTRODUCTION

Displacement is one of the most important concepts in physics and mathematics. It helps us understand how far an object moves from its original position to its final position in a specific direction. Unlike distance, displacement not only measures length but also considers direction.

In daily life, displacement can be observed while walking, driving, running, or even traveling by air. Understanding displacement is essential for students preparing for school exams, competitive exams, and engineering studies.

What is Displacement?

Displacement is defined as the shortest distance between the initial position and final position of an object along with its direction.

It is a vector quantity because it has both magnitude and direction.

Definition:
Displacement is the change in position of an object from one point to another in a particular direction.

Displacement Formula

The formula for displacement is:

Displacement = Final Position − Initial Position

Mathematically:

s = x₂ − x₁

• s = displacement
• x₂ = final position
• x₁ = initial position

Characteristics of Displacement

• Displacement is a vector quantity.
• It depends on the initial and final positions.
• It can be positive, negative, or zero.
• The magnitude of displacement is always less than or equal to distance.
• Its SI unit is metre (m).

Difference Between Distance and Displacement

Distance	Displacement
Scalar quantity	Vector quantity
Measures total path covered	Measures shortest path
No direction involved	Direction is important
Always positive	Can be positive, negative, or zero

Types of Displacement

1. Positive Displacement

When an object moves in the positive direction from its initial point, the displacement is positive.

2. Negative Displacement

When an object moves opposite to the positive direction, displacement becomes negative.

3. Zero Displacement

If an object returns to its starting point, displacement becomes zero because the initial and final positions are the same.

Real-Life Examples of Displacement

1. A student walks 5 meters east from home to school. The displacement is 5 meters east.
2. A runner completes one full lap of a circular track and returns to the starting point. The displacement is zero.
3. A car travels 10 km north and then 4 km south. The displacement is 6 km north.

Importance of Displacement in Physics

Displacement plays a major role in understanding motion and mechanics. It is used in:

• Velocity calculations
• Motion analysis
• Projectile motion
• Navigation systems
• Engineering and robotics

Displacement and velocity

If the mathematical relation between displacement and time of a particle be given thenvelocity of the particle can be calculated.For calculating velocity, displacement is differentiated with respect to time. It means the time rate of change of displacement is called velocity.In term of calculus ,

 v = ds/dt

Example: Displacement of a particle is given as
s = 2t³ − 4t² + 3
where displacement s is in metre and time t is in second, then its velocity will be
v = ds/dt
= d(2t³ − 4t² + 3 )/dt
= 6t² - 8t
To get velocity at t= 5 seconds we have to put t =5 in the last equation, we get
v = 6(5)² - 8(5)
= 150 - 40
= 110 m

Displacement and Acceleration

If the mathematical relation between displacement and time of a particle be given then acceleration of the particle can be calculated.For calculating acceleration, displacement is differentiated with respect to time two times. It means the second order derivative of displacement with respect to time is called acceleration.In term of calculus ,

a =dv/dt= d²s/dt²

Example: Displacement of a particle is given as
s = 10t³ − 5t² + 20
where displacement s is in metre and time t is in second, then its velocity will be
v = ds/dt
= d(10t³ − 5t² + 20 )/dt
= 30t² - 10t
Now to get acceleration, the last expression is differetiated with respect to time .
a = dv/dt
= d(30t² - 10t )/dt
= 60t - 10
To get acceleration at t= 1 second we have to put t =1 in the last equation, we get
a = 60(1) - 10
= 60 - 10
= 50 m/s²

Solved Example

Question:
A person moves from 3 m to 15 m on a straight road. Find the displacement.

Solution:
Displacement = Final Position − Initial Position
= 15 − 3
= 12 m

Answer: 12 meters

Key Points to Remember

• Displacement measures change in position.
• It always includes direction.
• Shortest path is considered in displacement.
• Displacement can be zero even when distance is not zero.

Conclusion

Displacement is a fundamental concept in physics that describes the shortest distance between the starting and ending positions of an object in a particular direction. It differs from distance because it includes direction and focuses only on the shortest path. Learning displacement helps students build a strong foundation in motion, mechanics, and advanced physics concepts.

Whether in academics or real-world applications, displacement is essential for understanding how objects move and interact in space.