CBSE Class 11 Physics Revision Notes Chapter 3: Motion in a Plane

Motion in a Plane explains how position, velocity and acceleration are represented when an object moves in two dimensions. The chapter uses vectors to study projectile motion, relative velocity and uniform circular motion.

Motion along a straight line can be represented using positive and negative signs. However, motion in a plane may occur in several directions, so both magnitude and direction become important.

These CBSE Class 11 Physics Revision Notes Chapter 3 cover vector operations, components, projectile motion and circular motion. The Class 11 Physics Chapter 3 notes also explain how one two-dimensional problem can be divided into separate horizontal and vertical motions.

Key Takeaways

• Vectors have magnitude and direction and follow vector-addition laws.
• Vector components simplify motion by separating it along perpendicular axes.
• Projectile motion combines constant horizontal velocity with vertical acceleration.
• Centripetal acceleration acts towards the centre during circular motion.

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Access Class 11 Physics Chapter 3 Motion in a Plane Notes in 30 Minutes

The chapter on Motion in a Plane Class 11 begins with vector algebra and then applies it to two-dimensional motion.

Summary of Class 11 Physics Motion in a Plane

Motion in one dimension allows only two directions. In two dimensions, an object may change its position along both the x-axis and y-axis.

The chapter therefore introduces:

• Scalars and vectors
• Vector addition and subtraction
• Resolution of vectors
• Motion in two dimensions
• 2D relative velocity
• Projectile motion
• Uniform circular motion

The same equations of motion used in a straight line can be applied independently to horizontal and vertical components.

Scalars and Vectors

Physical quantities are classified as scalars or vectors based on whether direction is required.

Scalar Quantities

A scalar quantity has magnitude only.

It is completely described by a numerical value and unit.

Examples include:

• Mass
• Time
• Distance
• Speed
• Temperature
• Energy
• Work

Scalars follow the rules of ordinary algebra.

Vector Quantities

A vector quantity has magnitude and direction.

It must also follow the triangle or parallelogram law of addition.

Examples include:

• Displacement
• Velocity
• Acceleration
• Force
• Momentum

Difference Between Scalars and Vectors

Representation of a Vector

A vector is represented by a directed line segment.

• Its length represents magnitude.
• Its arrowhead shows direction.
• Its starting point is called the tail.
• Its ending point is called the head.

The magnitude of vector A is written as:

|A| = A

The magnitude itself is a scalar.

Position and Displacement Vectors

The position vector locates a particle relative to an origin.

For a particle at coordinates (x, y):

r = xî + yĵ

If the particle moves from r to r′, its displacement is:

Δr = r′ - r

The position and displacement vectors differ in their starting points.

The displacement vector does not depend on the actual path followed.

Unit Vectors

Unit vectors have magnitude 1 and indicate direction.

The standard unit vectors are:

• î along the x-axis
• ĵ along the y-axis
• k̂ along the z-axis

Their magnitudes are:

|î| = |ĵ| = |k̂| = 1

A unit vector has no physical unit.

The unit vector along A is:

 = A/|A|

Parallel Vectors

Two vectors are parallel when they act along parallel lines.

If:

B = λA

then B is parallel to A.

When λ is positive, both vectors have the same direction. When λ is negative, their directions are opposite.

Equal Vectors

Two vectors are equal when they have:

• Equal magnitudes
• The same direction

Their positions in the plane may be different.

Negative Vector

The vector -A has the same magnitude as A but points in the opposite direction.

Therefore:

|-A| = |A|

Zero Vector

A vector having zero magnitude is called a zero or null vector.

A + (-A) = 0

Its direction cannot be defined.

A displacement becomes a zero vector when the particle returns to its initial position.

Vector Addition and Subtraction

Vectors must be combined by considering both magnitude and direction.

Multiplication of a Vector by a Scalar

Multiplying vector A by a positive scalar λ changes its magnitude but not its direction.

|λA| = λ|A|

If λ is negative, the direction reverses.

Examples:

• 2A has twice the magnitude of A.
• -2A has twice the magnitude in the opposite direction.
• 0A gives the zero vector.

Triangle Law of Vector Addition

The triangle law uses the head-to-tail method.

To add A and B:

1. Draw A.
2. Place the tail of B at the head of A.
3. Join the tail of A to the head of B.
4. The joining vector gives the resultant.

Therefore:

R = A + B

The two vectors and their resultant form a triangle.

Parallelogram Law of Vector Addition

For the parallelogram law:

1. Draw A and B from a common origin.
2. Complete a parallelogram using them as adjacent sides.
3. Draw the diagonal from the common origin.
4. The diagonal represents the resultant.

Both graphical methods produce the same vector.

Properties of Vector Addition

Vector addition is commutative:

A + B = B + A

It is also associative:

(A + B) + C = A + (B + C)

Vector Subtraction

Vector addition and subtraction are closely related.

Subtracting B from A means adding the negative of B:

A - B = A + (-B)

The direction of B is reversed before addition.

Resultant of Two Vectors

If A and B make an angle θ, the magnitude of the resultant is:

R = √(A² + B² + 2AB cos θ)

The direction α of the resultant relative to A is:

tan α = B sin θ/(A + B cos θ)

Special cases are:

Resolution of Vectors

The resolution of vectors means dividing a vector into components along selected directions.

Rectangular components are usually taken along the x-axis and y-axis.

Rectangular Vector Components

Suppose vector A makes an angle θ with the positive x-axis.

Its components are:

Ax = A cos θ

Ay = A sin θ

Therefore:

A = Axî + Ayĵ

The x and y quantities are scalar components, while Axî and Ayĵ are component vectors.

Signs of Vector Components

The signs depend on the quadrant.

Finding Magnitude from Components

If Ax and Ay are known:

A = √(Ax² + Ay²)

This follows from the Pythagorean theorem.

Finding Direction from Components

The direction is given by:

tan θ = Ay/Ax

Therefore:

θ = tan⁻¹(Ay/Ax)

The quadrant must be checked before fixing the direction.

Analytical Method of Vector Addition

Suppose:

A = Axî + Ayĵ

B = Bxî + Byĵ

Then:

R = A + B

R = (Ax + Bx)î + (Ay + By)ĵ

Therefore:

Rx = Ax + Bx

Ry = Ay + By

The magnitude of the resultant is:

R = √(Rx² + Ry²)

Its direction is:

tan θ = Ry/Rx

This analytical method is more accurate than drawing vectors to scale.

Motion in a Plane

In motion in two dimensions, position, velocity and acceleration have components along both axes.

The x and y motions can be studied separately.

Position Vector in Two Dimensions

For a particle at coordinates (x, y):

r = xî + yĵ

Here, x and y may change with time.

Displacement in a Plane

If the position changes from r to r′:

Δr = r′ - r

In component form:

Δr = Δxî + Δyĵ

where:

Δx = x′ - x

Δy = y′ - y

Average Velocity

Average velocity is:

vavg = Δr/Δt

In component form:

vavg = (Δx/Δt)î + (Δy/Δt)ĵ

Its direction is the same as the displacement vector.

Instantaneous Velocity

Instantaneous velocity is:

v = dr/dt

In component form:

v = vxî + vyĵ

where:

vx = dx/dt

vy = dy/dt

Its magnitude is:

v = √(vx² + vy²)

The direction is:

tan θ = vy/vx

The instantaneous velocity is tangent to the particle’s path.

Acceleration in a Plane

Average acceleration is:

aavg = Δv/Δt

Instantaneous acceleration is:

a = dv/dt

In component form:

a = axî + ayĵ

where:

ax = dvx/dt

ay = dvy/dt

Velocity and acceleration may point in different directions.

Motion with Constant Acceleration

For constant acceleration:

v = v₀ + at

In component form:

vx = v₀x + axt

vy = v₀y + ayt

The position becomes:

r = r₀ + v₀t + ½at²

In component form:

x = x₀ + v₀xt + ½axt²

y = y₀ + v₀yt + ½ayt²

These equations form the basis of projectile motion.

2D Relative Velocity

2D relative velocity describes the velocity of one moving object as observed from another moving object.

If A and B have velocities vA and vB:

Velocity of A relative to B:

vAB = vA - vB

Velocity of B relative to A:

vBA = vB - vA

Therefore:

vAB = -vBA

Relative Velocity in Different Directions

Since velocity is a vector, subtraction must be done through components.

If:

vA = vAxî + vAyĵ

vB = vBxî + vByĵ

Then:

vAB = (vAx - vBx)î + (vAy - vBy)ĵ

The relative speed is:

|vAB| = √[(vAx - vBx)² + (vAy - vBy)²]

Rain-Man Problems

Suppose rain falls vertically with velocity vr and wind gives it a horizontal velocity vw.

The apparent velocity of rain is the vector sum of these components.

Its direction satisfies:

tan θ = Horizontal component/Vertical component

The umbrella must be tilted opposite to the apparent direction of rain.

Boat-River Problems

A boat’s velocity relative to the ground is:

vBG = vBW + vWG

where:

• vBG is boat velocity relative to ground.
• vBW is boat velocity relative to water.
• vWG is water velocity relative to ground.

The resultant depends on both the boat’s velocity and river current.

Projectile Motion

A projectile is an object projected into the air and then allowed to move under gravity.

Examples include:

• A thrown ball
• A kicked football
• A stone projected at an angle
• Water leaving a pipe

The standard projectile motion formulas assume negligible air resistance.

Horizontal and Vertical Components

Suppose the projectile is launched with speed u at angle θ.

Horizontal component:

ux = u cos θ

Vertical component:

uy = u sin θ

Acceleration components are:

ax = 0

ay = -g

The horizontal and vertical motions are independent.

Position at Time t

Taking the launch point as the origin:

x = u cos θ × t

y = u sin θ × t - ½gt²

The horizontal position changes uniformly.

The vertical position changes under constant acceleration.

Velocity at Time t

Horizontal velocity:

vx = u cos θ

Vertical velocity:

vy = u sin θ - gt

The horizontal velocity remains constant.

The vertical velocity changes because of gravity.

The speed is:

v = √(vx² + vy²)

Equation of Trajectory

From the horizontal equation:

t = x/(u cos θ)

Substituting into the vertical equation:

y = x tan θ - gx²/(2u² cos²θ)

This is the equation of trajectory.

Since it contains an x² term, the path is parabolic.

Time to Reach Maximum Height

At maximum height:

vy = 0

Therefore:

0 = u sin θ - gtₘ

Hence:

tₘ = u sin θ/g

Time of Flight

For a projectile returning to the same level:

T = 2u sin θ/g

The time of flight depends on the vertical component of initial velocity.

Maximum Height of Projectile

At the highest point, vertical velocity becomes zero.

The maximum height of projectile is:

H = u² sin²θ/2g

Horizontal Range of Projectile

The range is the total horizontal distance travelled.

R = Horizontal velocity × Time of flight

R = u cos θ × 2u sin θ/g

Therefore:

R = u² sin 2θ/g

Maximum Range

Range is maximum when:

sin 2θ = 1

Therefore:

2θ = 90°

θ = 45°

The maximum range is:

Rmax = u²/g

Complementary Angles

Two angles θ and (90° - θ) produce the same range.

This happens because:

sin 2θ = sin[2(90° - θ)]

However, their maximum heights and flight times are different.

Horizontal Projectile

A projectile launched horizontally has:

ux = u

uy = 0

Its horizontal position is:

x = ut

Its vertical displacement is:

y = ½gt²

The time of fall depends only on the vertical height.

Important Points About Projectile Motion

During projectile motion:

The standard range and flight-time formulas apply when the launch and landing levels are equal.

Uniform Circular Motion

An object performs uniform circular motion when it moves along a circular path with constant speed.

Its velocity changes continuously because its direction changes.

Direction of Linear Velocity

The linear velocity of the particle is tangent to the circular path.

At every point, the radius and velocity are perpendicular.

Angular Displacement

Angular displacement is the angle through which the radius vector turns.

For arc length s and radius R:

θ = s/R

Its SI unit is radian.

Angular Velocity

Angular velocity is the rate of change of angular displacement.

ω = Δθ/Δt

For uniform circular motion, angular velocity remains constant.

Its SI unit is rad s⁻¹.

Relation Between Linear and Angular Velocity

Since:

s = Rθ

Dividing by time:

v = Rω

Thus, linear speed depends on radius and angular velocity.

Centripetal Acceleration

The inward acceleration in circular motion is called centripetal acceleration.

It always points towards the centre.

Its magnitude is:

ac = v²/R

Using v = Rω:

ac = Rω²

Although speed remains constant, the change in direction produces acceleration.

Time Period and Frequency

Time period T is the time taken for one revolution.

Frequency ν is the number of revolutions completed per second.

ν = 1/T

For one revolution:

ω = 2π/T

Also:

ω = 2πν

Linear speed is:

v = 2πR/T

or:

v = 2πRν

Uniform and Non-Uniform Circular Motion

Motion in a Plane Formula Summary

Vector Formulae

Motion in a Plane Formulae

Projectile Motion Formulae

Circular Motion Formulae

Important Topics in Class 11 Physics Chapter 3

Students should revise the following areas carefully:

• Difference between scalars and vectors
• Position and displacement vectors
• Unit vectors
• Triangle and parallelogram laws
• Resolution of vectors
• Analytical vector addition
• Position, velocity and acceleration components
• Relative velocity in a plane
• Projectile-motion formulas
• Equation of trajectory
• Time of flight and maximum height
• Horizontal range
• Uniform circular motion
• Linear and angular velocity
• Centripetal acceleration

Tips for Learning Motion in a Plane Class 11

Understand Components Before Projectiles

Projectile motion becomes easier after learning how a vector splits along the x-axis and y-axis.

Draw the Coordinate Axes

Choose positive x and y directions before assigning signs to components.

Treat Both Directions Separately

In projectile motion, horizontal acceleration is zero while vertical acceleration is -g.

Check the Formula Conditions

The standard range and flight-time equations assume that the projectile lands at its launch level.

Track Vector Signs

A negative component indicates direction. The magnitude of a vector remains non-negative.

Use Diagrams for Relative Motion

Rain-man and boat-river problems become clearer after drawing all velocity vectors.

Connect Circular Motion with Direction

In circular motion, speed may remain constant while velocity changes because its direction changes.

Basic Questions from Motion in a Plane

Why does a projectile follow a parabolic path?

The horizontal position varies linearly with time, while the vertical position contains a time-squared term. Eliminating time produces an equation containing x², which represents a parabola.

Why is acceleration non-zero at maximum height?

Only the vertical velocity becomes zero at maximum height. Gravity continues to act downward, so acceleration remains equal to g.

How are vectors added using the parallelogram law?

The two vectors are drawn from a common origin as adjacent sides of a parallelogram. The diagonal from the common origin represents the resultant.

Why does centripetal acceleration point inward?

Velocity changes direction towards the centre as the particle follows the circular path. The resulting acceleration therefore acts radially inward.

How is relative velocity calculated in a plane?

Subtract the observer’s velocity vector from the object’s velocity vector. The subtraction should be performed separately along the x and y axes.

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