NCERT Physics Class 11 Chapter 2 Instantaneous Acceleration Notes with Graphs

NCERT Physics Class 11 Chapter 2 Easy Line-by-Line Notes

Instantaneous Acceleration

“Instantaneous acceleration is defined in the same way as instantaneous velocity.”

• Just like instantaneous velocity gives velocity at one instant,

• Instantaneous acceleration gives acceleration at one instant.

Formula of Instantaneous Acceleration

a = lim(Δt → 0) [ Δv / Δt ]

• Take very small time interval.

• Find change in velocity.

• Divide by tiny time interval.

Calculus Form

a = dv / dt

Acceleration = rate of change of velocity with time.

“The acceleration at an instant is the slope of the tangent to the v–t curve.”

Important Graph Rule:

On velocity-time graph:

Slope of tangent = instantaneous acceleration

Velocity and Acceleration

“Velocity has both magnitude and direction.”

Velocity is a vector quantity.

It can change by:

• Changing speed
• Changing direction
• Changing both

“Acceleration may result from change in speed, direction or both.”

Situation	Type of Change
Car speeding up	Speed changes
Circular motion	Direction changes
Curved fast motion	Both change

Positive, Negative and Zero Acceleration

Positive Acceleration

• Velocity increases with time.
• Object speeds up.

Example: Bike accelerating forward.

Negative Acceleration (Retardation)

• Velocity decreases with time.
• Object slows down.

Example: Applying brakes.

Zero Acceleration

• Velocity remains constant.
• No change in motion.

Example: Car moving at constant speed.

Position-Time Graphs

“Graph curves upward for positive acceleration.”

• Slope keeps increasing.

• Object moves faster and faster.

“Graph curves downward for negative acceleration.”

• Slope decreases with time.

• Object slows down.

“Straight line for zero acceleration.”

• Constant slope.

• Constant velocity.

Constant Acceleration

“Our study will be restricted to constant acceleration.”

Acceleration remains same throughout motion.

Example: Free fall near Earth.

Equation for Constant Acceleration

a = (v - v₀) / t

Rearranging Equation

v = v₀ + at

First Equation of Motion

Meaning of Symbols

Symbol	Meaning
v₀	Initial velocity
v	Final velocity
a	Acceleration
t	Time

Velocity-Time Graph Cases

Case (a)

Object moving in positive direction with positive acceleration.

• Velocity positive
• Acceleration positive

Object moves forward faster and faster.

Graph: Upward sloping line above time axis.

Case (b)

Object moving in positive direction with negative acceleration.

• Velocity positive
• Acceleration negative

Object still moves forward but slows down.

Example: Car braking while moving forward.

Case (c)

Object moving in negative direction with negative acceleration.

• Velocity negative
• Acceleration negative

Object moves backward and speeds up backward.

Case (d)

Object moving in positive direction till t₁ and then turns back.

• Initially moving forward
• Velocity becomes zero at t₁
• Then object reverses direction

v = 0

Important Point: At turning point velocity becomes zero.

Area Under Velocity-Time Graph

“Area under the curve represents displacement.”

Very Important Concept

On velocity-time graph:

Displacement = Area under v-t graph

Why Area Gives Displacement

v = Δx / Δt

Δx = v Δt

Area = velocity × time.

Constant Velocity Case

Velocity-time graph becomes:

• Straight horizontal line

Area rectangle under graph gives displacement.

Important Graph Rules Summary

Graph	Slope Gives	Area Gives
Position-Time	Velocity	—
Velocity-Time	Acceleration	Displacement

Real-Life Examples

Situation	Acceleration Type
Bike speeding up	Positive
Car braking	Negative
Cruise control	Zero
Falling object	Constant acceleration

One-Line Summary:

Acceleration describes how velocity changes with time, and on a velocity-time graph, slope gives acceleration while area under the graph gives displacement.

- Dr.Sanjaykumar Pawar   

Instantaneous acceleration on a velocity-time graph showing tangent slope and displacement area for Class 11 Physics students.

Internal Links

Motion in a Straight Line Notes

Difference Between Speed and Velocity

Average Velocity and Instantaneous Velocity

Position-Time Graph Explained

Velocity-Time Graph Explained

Equations of Motion Derivation

Uniform and Non-Uniform Motion

Numerical Problems on Acceleration

Class 11 Physics Chapter 3 Exercise Solutions

Important NEET Kinematics Questions

Exam Preparation Links

CBSE Class 11 Physics Important Questions

NEET Physics Motion Chapter MCQs

JEE Main Kinematics Practice Problems

Previous Year Questions on Acceleration

Physics Revision Notes for Class 11 

CBSE Class 11 Physics Chapter 2 Important Questions and Answers – Instantaneous Acceleration

Very Short Answer Questions

What is instantaneous acceleration?

Instantaneous acceleration is the rate of change of velocity at a particular instant of time.

Write the formula for instantaneous acceleration.

a = lim (Δt → 0) (Δv / Δt)

What is the SI unit of acceleration?

The SI unit of acceleration is metre per second squared (m/s²).

Is acceleration a scalar or vector quantity?

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

What does the slope of a velocity-time graph represent?

The slope of a velocity-time graph represents acceleration.

What does the area under a velocity-time graph represent?

The area under a velocity-time graph represents displacement.

Short Answer Questions

Define instantaneous acceleration.

Instantaneous acceleration is the acceleration of an object at a particular instant of time. Mathematically, it is expressed as:

a = dv/dt

It measures how quickly velocity changes at a given moment.

How can velocity change?

Velocity can change in three ways:

• Change in speed
• Change in direction
• Change in both speed and direction

Any of these changes produces acceleration.

Differentiate between positive and negative acceleration.

Positive Acceleration	Negative Acceleration
Velocity increases with time.	Velocity decreases with time.
Object speeds up.	Object slows down.
Slope of v-t graph is positive.	Slope of v-t graph is negative.

What is zero acceleration?

Zero acceleration occurs when velocity remains constant. There is no change in speed or direction.

Example: A car moving at constant speed on a straight road.

Long Answer Questions

Explain instantaneous acceleration using a velocity-time graph.

Instantaneous acceleration is the rate of change of velocity at a particular instant.

a = dv/dt

On a velocity-time graph, the instantaneous acceleration at any point is obtained by drawing a tangent at that point. The slope of the tangent gives the instantaneous acceleration.

Therefore:

Instantaneous Acceleration = Slope of Tangent to the v-t Graph

Explain positive, negative and zero acceleration with examples.

Positive Acceleration: Velocity increases with time. Example: A bike speeding up.

Negative Acceleration: Velocity decreases with time. Example: A car slowing down after brakes are applied.

Zero Acceleration: Velocity remains constant. Example: A train moving uniformly on a straight track.

Derive the first equation of motion.

For constant acceleration:

a = (v − v₀)/t

Multiplying both sides by t:

at = v − v₀

Rearranging:

v = v₀ + at

This is known as the first equation of motion.

Multiple Choice Questions (MCQs)

Instantaneous acceleration is represented by:

A. Δx/Δt
B. Δv/Δt
C. dv/dt
D. Both B and C

Answer: D. Both B and C

Area under a velocity-time graph gives:

A. Velocity
B. Acceleration
C. Displacement
D. Speed

Answer: C. Displacement

Slope of a velocity-time graph gives:

A. Velocity
B. Distance
C. Acceleration
D. Time

Answer: C. Acceleration

Assertion and Reason Questions

Assertion:

Acceleration can occur even when speed remains constant.

Reason:

Velocity changes when direction changes.

Answer: Both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Assertion:

The area under a velocity-time graph gives displacement.

Reason:

Velocity is displacement per unit time.

Answer: Both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Fill in the Blanks

Instantaneous acceleration is represented by dv/dt.

The slope of a velocity-time graph gives acceleration.

The area under a velocity-time graph gives displacement.

Negative acceleration is also called retardation.

At the turning point, velocity becomes zero.

Match the Following

Column A	Column B
Slope of v-t graph	Acceleration
Area under v-t graph	Displacement
Constant velocity	Zero acceleration
Retardation	Negative acceleration

Case Study Questions

A car starts moving with an initial velocity of 10 m/s and accelerates uniformly at 2 m/s² for 5 seconds.

What is the initial velocity of the car?

Answer: 10 m/s

What is the acceleration of the car?

Answer: 2 m/s²

Calculate the final velocity.

Using:

v = v₀ + at

v = 10 + (2 × 5) = 20 m/s

Answer: 20 m/s

Is the acceleration positive or negative?

Answer: Positive acceleration.

Which equation of motion is used?

Answer: v = v₀ + at

Instantaneous Acceleration

Definition

• Acceleration at a particular instant.
• Rate of change of velocity with time.

a = dv/dt

Formula

• Very small time interval considered.
• Change in velocity divided by time interval.

a = lim(Δt→0) (Δv/Δt)

Velocity-Time Graph

• Slope of tangent gives acceleration.
• Positive slope → Positive acceleration.
• Negative slope → Negative acceleration.

Types of Acceleration

• Positive Acceleration
• Negative Acceleration (Retardation)
• Zero Acceleration

Velocity Changes By

• Changing Speed
• Changing Direction
• Changing Both

Position-Time Graph

• Upward Curve → Positive Acceleration
• Downward Curve → Negative Acceleration
• Straight Line → Zero Acceleration

Constant Acceleration

• Acceleration remains constant.
• Example: Free Fall.

v = v₀ + at

Area Under v-t Graph

• Area under curve = Displacement
• Rectangle area for constant velocity.

Real-Life Examples

• Bike speeding up
• Car braking
• Cruise control
• Falling object