MOTION IN A PLANE

│

├── Introduction

│   ├── Motion in two dimensions

│   ├── Object moves in x-y plane

│   └── Vectors are used

│

├── Vectors

│   ├── Have magnitude

│   ├── Have direction

│   ├── Examples

│   │   ├── Displacement

│   │   ├── Velocity

│   │   └── Acceleration

│

├── Coordinate System

│   ├── x-axis → Horizontal direction

│   ├── y-axis → Vertical direction

│   └── Origin (O)

│

├── Position Vector

│   ├── Gives location of particle

│   ├── Starts from origin

│   ├── Ends at particle P

│   ├── Coordinates

│   │   ├── x-coordinate

│   │   └── y-coordinate

│   └── Formula

│       └── r = x î + y ĵ

│

├── Unit Vectors

│   ├── î → Along x-axis

│   └── ĵ → Along y-axis

│

├── Displacement

│   ├── Change in position

│   ├── Initial point → P

│   ├── Final point → P′

│   ├── Directed from P to P′

│   ├── Independent of path

│   └── Formula

│       └── Δr = r′ − r

│

├── Component Form of Displacement

│   ├── Initial Position

│   │   └── r = x î + y ĵ

│   ├── Final Position

│   │   └── r′ = x′ î + y′ ĵ

│   └── Formula

│       └── Δr = (x′−x)î + (y′−y)ĵ

│

├── Change in Coordinates

│   ├── Along x-axis

│   │   └── Δx = x′ − x

│   └── Along y-axis

│       └── Δy = y′ − y

│

├── Final Displacement Vector

│   └── Δr = Δx î + Δy ĵ

│

├── Numerical Example

│   ├── Initial Position = (2,3)

│   ├── Final Position = (7,8)

│   ├── Δx = 5

│   ├── Δy = 5

│   └── Δr = 5î + 5ĵ

│

└── Important NEET Points

    ├── Motion in plane is 2D motion

    ├── Position vector gives location

    ├── Displacement is vector quantity

    ├── î along x-direction

    ├── ĵ along y-direction

    └── Displacement depends only on

        initial and final positions

Motion in a Plane (2D Motion)

Introduction

In one-dimensional motion, an object moves only in a straight line. In two-dimensional motion, an object moves in a plane. This type of motion is called Motion in a Plane.

To study motion in two dimensions easily, we use vectors.

• A vector has magnitude and direction.
• Examples: displacement, velocity, acceleration.

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Coordinate System

• We use an x-y coordinate system.
• The horizontal direction is called the x-axis.
• The vertical direction is called the y-axis.
• The point where both axes meet is called the origin.

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Position Vector

Consider a particle located at point P in the x-y plane.

Its coordinates are:

• x along x-axis
• y along y-axis

The position vector gives the location of the particle from the origin.

Formula of Position Vector

r = x î + y ĵ

Meaning of Symbols

• r = position vector
• x = x-coordinate
• y = y-coordinate
• î = unit vector along x-axis
• ĵ = unit vector along y-axis

Important Point:
The position vector always starts from the origin and ends at the particle.

Example

If a particle is at point (3,4), then:

r = 3î + 4ĵ

This means:

• 3 units in x-direction
• 4 units in y-direction

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Displacement in a Plane

Suppose the particle moves:

• from point P at time t
• to point P′ at time t′

The change in position is called displacement.

Formula of Displacement Vector

Δr = r′ − r

Meaning

• Δr = displacement vector
• r = initial position vector
• r′ = final position vector

Important Point:
Displacement is directed from initial point P to final point P′. It does not depend on the path followed. It depends only on initial and final positions.

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Component Form of Displacement

If:

r = x î + y ĵ

and

r′ = x′ î + y′ ĵ

then displacement becomes:

Δr = (x′ − x)î + (y′ − y)ĵ

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Change in Coordinates

Change along x-axis

Δx = x′ − x

Δx is the displacement in x-direction.

Change along y-axis

Δy = y′ − y

Δy is the displacement in y-direction.

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Final Form of Displacement Vector

Δr = Δx î + Δy ĵ

The total displacement has:

• x-component
• y-component

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Simple Numerical Example

Given:

• Initial position = (2,3)
• Final position = (7,8)

Step 1: Find change in x

Δx = 7 − 2 = 5

Step 2: Find change in y

Δy = 8 − 3 = 5

Step 3: Write displacement vector

Δr = 5î + 5ĵ

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Important NEET Points

• Motion in a plane is two-dimensional motion.
• Position vector gives location of particle from origin.
• Displacement is change in position.
• Displacement is a vector quantity.
• î represents x-direction.
• ĵ represents y-direction.
• Displacement depends only on initial and final positions.

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Quick Revision

Position Vector

r = x î + y ĵ

Displacement Vector

Δr = r′ − r

Component Form

Δr = Δx î + Δy ĵ