NCERT Physics Class 11 Chapter 2 Notes: Instantaneous Velocity & Acceleration

NCERT Physics Class 11 Chapter 2

Continuation: Instantaneous Velocity

“The sixth column lists the difference Δx = x(t₂) − x(t₁)...”

Easy Meaning

• Find displacement between two times:

Δx = x(t₂) − x(t₁)

• This tells how much position changed.

“...the last column gives the ratio Δx/Δt...”

Meaning

Average velocity is:

v_avg = Δx / Δt

• Divide displacement by time interval.

“As we decrease Δt from 2.0 s to 0.010 s, the average velocity approaches 3.84 m/s.”

Important Idea

• Smaller time interval gives more accurate velocity.
• As:

Δt → 0

average velocity becomes instantaneous velocity.

“...which is the value of velocity at t = 4.0 s.”

Final Result

v = 3.84 m/s

“In this manner, we can calculate velocity at each instant...”

Meaning

• Using calculus or graphs:
• velocity can be found at any time.

Graphical Method Limitation

“The graphical method... is not always a convenient method.”

Meaning

Drawing tangents accurately is difficult.

“We must carefully plot the position-time graph...”

Meaning

• For accurate velocity:
• graph must be very precise.

“It is easier to calculate velocity if we have data or an equation.”

Important Point

• Equations are easier than graphs.
• Example:
• If position equation is known, velocity can be calculated directly.

Example 2.1

Given Equation

“The position of an object moving along x-axis is given by”

x = a + bt²

Where:

• a = 8.5 m
• b = 2.5 m/s²

Finding Velocity

“In differential calculus, velocity is”

v = dx / dt

Differentiate Position Equation

Given:

x = a + bt²

Differentiate with respect to time:

v = d/dt (a + bt²) = 2bt

Substitute Value of b

Since:

• b = 2.5

Then:

v = 5.0t

Unit: m/s

Velocity at t = 0 s

“At t = 0 s, v = 0 m/s”

Meaning

Object starts from rest.

Velocity at t = 2 s

Substitute (t = 2):

v = 5 × 2

v = 10 m/s

Average Velocity Between 2 s and 4 s

Formula

v_avg = [x(4.0) − x(2.0)] / (4.0 − 2.0)

Substitute Position Values

Using:

x = a + bt²

At t = 4:

x(4) = 8.5 + 2.5(16)

x(4) = 48.5 m

At t = 2:

x(2) = 8.5 + 2.5(4)

x(2) = 18.5 m

Calculate Average Velocity

v_avg = (48.5 − 18.5) / 2

v_avg = 15 m/s

Uniform Motion

“For uniform motion, velocity is same as average velocity.”

Meaning

• If velocity does not change:
• instantaneous velocity = average velocity

Instantaneous Speed

“Instantaneous speed is the magnitude of velocity.”

Definition

Speed = magnitude (numerical value) of velocity.

Example

Velocity: +24 m/s

or

Velocity: −24 m/s

Both have speed:

24 m/s

because speed has no direction.

Important Difference

Quantity	Direction Needed?
Velocity	Yes
Speed	No

2.3 ACCELERATION

“The velocity of an object changes during motion.”

Meaning

• Objects may:
• speed up
• slow down
• change direction

“How do we describe this change?”

Answer

Using acceleration.

Galileo’s Idea

“Galileo concluded that rate of change of velocity with time is constant...”

Meaning

• In free fall:
• velocity changes uniformly with time.

This led to acceleration concept.

Definition of Acceleration

“Acceleration is the rate of change of velocity with time.”

Main Definition

a = Δv / Δt

Average Acceleration Formula

ā = (v₂ − v₁) / (t₂ − t₁)

Meaning of Symbols

Symbol	Meaning
v₁	initial velocity
v₂	final velocity
t₁	initial time
t₂	final time

SI Unit of Acceleration

m/s²

Meaning:

• velocity changes by meters per second every second.

Graphical Meaning

“On velocity-time graph, acceleration is the slope.”

Important Rule

Slope = Δv / Δt

Summary Table

Concept	Formula
Velocity	v = dx/dt
Average velocity	v = Δx/Δt
Acceleration	a = Δv/Δt
Speed	Magnitude of velocity

One-Line Summary: Acceleration tells how quickly velocity changes with time, and it is equal to the slope of the velocity-time graph.

Instantaneous Velocity and Acceleration Class 11 Physics Explained 

CBSE Class 11 Physics - Motion in a Straight Line

Very Short Answer Questions (1 Mark)

Q1. Define instantaneous velocity.

Ans: Instantaneous velocity is the velocity of an object at a particular instant of time.

Q2. Write the formula for average velocity.

Ans: v = Δx / Δt

Q3. What is the SI unit of velocity?

Ans: m/s

Q4. What is acceleration?

Ans: Acceleration is the rate of change of velocity with time.

Q5. What is the SI unit of acceleration?

Ans: m/s²

Short Answer Questions (2-3 Marks)

Q6. Differentiate between speed and velocity.

Speed	Velocity
Scalar quantity	Vector quantity
Magnitude only	Magnitude and direction
Always positive	May be positive, negative or zero

Q7. Define average acceleration.

Ans: Average acceleration is the change in velocity divided by the time interval.
a = (v₂ − v₁)/(t₂ − t₁)

Q8. Why is instantaneous velocity more accurate than average velocity?

Ans: Instantaneous velocity is calculated over an extremely small time interval, giving the exact velocity at a particular instant.

Long Answer Questions (5 Marks)

Q9. Derive the expression for velocity when position is given by: x = a + bt²

Ans:
Given:
x = a + bt²

Velocity:
v = dx/dt

Differentiating:
v = d(a + bt²)/dt
v = 2bt

Therefore,
v = 2bt

Q10. Calculate velocity at t = 2 s if b = 2.5 m/s².

v = 2bt
v = 2 × 2.5 × 2
v = 10 m/s

Multiple Choice Questions (MCQs)

1. The slope of a velocity-time graph represents:

• A. Speed
• B. Distance
• C. Acceleration
• D. Displacement

Answer: C. Acceleration

2. Speed is:

• A. Vector quantity
• B. Scalar quantity
• C. Tensor quantity
• D. None of these

Answer: B. Scalar quantity

Assertion and Reason

Assertion (A): Acceleration is the rate of change of velocity.
Reason (R): Velocity is a vector quantity.

Answer: Both A and R are true and R correctly explains A.

Fill in the Blanks

1. Speed is the ______ of velocity.
2. The SI unit of acceleration is ______.
3. The slope of a velocity-time graph gives ______.
4. Average velocity equals ______ divided by time interval.

Answers:

1. Magnitude
2. m/s²
3. Acceleration
4. Displacement

Match the Columns

Column A	Column B
Velocity	dx/dt
Acceleration	Δv/Δt
Speed	Magnitude of Velocity
Slope of x-t graph	Velocity

Case Study Questions

A particle moves along the x-axis according to:
x = 8.5 + 2.5t²

1. Find the velocity equation.

v = dx/dt = 5t

2. Find velocity at t = 2 s.

v = 5 × 2 = 10 m/s

3. Find velocity at t = 0 s.

0 m/s

Instantaneous velocity and acceleration explained using position-time and velocity-time graphs for NCERT Class 11 Physics.

Internal Links

Velocity Section

Motion in a Straight Line Notes

Difference Between Speed and Velocity

Graphs in Kinematics Explained

Position-Time Graph Questions

Velocity-Time Graph Notes

Example 2.1 Section

NCERT Class 11 Physics Solved Examples

Numerical Problems on Motion in a Straight Line

Derivatives in Physics Made Easy

Acceleration Section

Uniform and Non-Uniform Motion

Acceleration and Retardation Notes

Free Fall Motion Explained

Galileo's Contributions to Physics

Exam Preparation

Class 11 Physics Important Questions

NEET Physics Kinematics Questions

JEE Motion in One Dimension Problems

CBSE Class 11 Physics Revision Notes