NCERT Physics Class 11 Example 5.8 & 5.9 Solutions | Spring Compression Explained

 - Dr.Sanjaykumar Pawar 

A moving car compresses a spring, demonstrating conservation of mechanical energy and work-energy theorem.

Internal Links

• NCERT Class 11 Physics Chapter 5 Notes
• Work, Energy and Power Formulas
• Conservation of Mechanical Energy
• Work-Energy Theorem Explained
• Hooke's Law and Spring Force
• Potential Energy and Kinetic Energy
• NCERT Physics Example 5.1–5.7 Solutions
• NCERT Physics Chapter 5 Exercise Solutions
• JEE Physics Work Energy Questions
• NEET Physics Chapter-wise MCQs
• Energy Conservation Numerical Problems
• Class 11 Physics Formula Sheet

NCERT Physics Class 11

Chapter 5 - Work, Energy and Power

Example 5.8 & Example 5.9

Question 1

A car of mass 1000 kg is moving with a speed of 18 km h^-1 on a smooth horizontal road and collides with a spring of spring constant 5.25 × 10³ N m^-1. Find the maximum compression of the spring.

Answer

Given

• Mass (m) = 1000 kg
• Speed (v) = 18 km h^-1 = 5 m s^-1
• Spring constant (k) = 5.25 × 10³ N m^-1

Formula

At maximum compression,

Kinetic Energy = Spring Potential Energy

½mv² = ½kx²

Step 1 : Calculate Kinetic Energy

K = ½ × 1000 × 5²

K = 12500 J

Step 2 : Calculate Compression

12500 = ½ × 5.25 × 10³ × x²

12500 = 2625x²

x² = 12500 / 2625

x² = 4.76

x = √4.76

x ≈ 2.18 m

Maximum Compression = 2.0 m (Approx.)

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Question 2

Using Example 5.8, if the coefficient of friction is 0.5, calculate the maximum compression of the spring.

Answer

Given

• Mass = 1000 kg
• Speed = 5 m s^-1
• Spring constant = 5.25 × 10³ N m^-1
• Coefficient of friction = 0.5
• g = 10 m s^-2

Formula

Work-Energy Theorem

ΔK = W

½mv² = ½kx² + μmgx

Step 1

12500 = 2625x² + 5000x

Step 2

2625x² + 5000x − 12500 = 0

Step 3

Using quadratic formula,

x = 1.35 m

Maximum Compression = 1.35 m

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Practice Questions

Practice Question 1

A spring of spring constant 400 N m^-1 is compressed by 0.5 m. Calculate the elastic potential energy stored.

Formula

U = ½kx²

U = ½ × 400 × (0.5)²

U = 50 J

Answer = 50 J

Practice Question 2

A body of mass 2 kg moves with speed 10 m s^-1. Find its kinetic energy.

Formula

K = ½mv²

K = ½ × 2 × 10²

K = 100 J

Answer = 100 J

Practice Question 3

A spring stores 100 J of energy. Its spring constant is 500 N m^-1. Find the compression.

100 = ½ × 500 × x²

100 = 250x²

x² = 0.4

x = 0.63 m

Answer = 0.63 m

Practice Question 4

A 500 kg car moving at 10 m s^-1 hits a spring of spring constant 10000 N m^-1. Find the maximum compression.

K = ½mv²

K = ½ × 500 × 10²

K = 25000 J

25000 = ½ × 10000 × x²

25000 = 5000x²

x² = 5

x = 2.24 m

Answer = 2.24 m

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Important Formulae

Formula	Expression
Kinetic Energy	K = ½mv²
Spring Potential Energy	U = ½kx²
Work-Energy Theorem	ΔK = W
Mechanical Energy	KE + PE = Constant (Without Friction)

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Important Viva Questions

1. Why does the car stop at maximum compression?
   Because all kinetic energy is converted into spring potential energy.


2. Why is conservation of mechanical energy not applicable when friction is present?
   Because friction is a non-conservative force and converts mechanical energy into heat.


3. What is the formula of spring potential energy?
   U = ½kx²


4. What is the formula of kinetic energy?
   K = ½mv²

NEET Physics Notes

Chapter: Work, Energy and Power

Topics Covered

• Spring Force (Hooke's Law)
• Spring Potential Energy
• Kinetic Energy
• Conservation of Mechanical Energy
• Work-Energy Theorem
• Friction
• NCERT Example 5.8
• NCERT Example 5.9
• Important Formulae
• NEET MCQs

1. Spring Force (Hooke's Law)

When a spring is stretched or compressed, it tries to return to its original length. This restoring force is called Spring Force.

Formula

F = -kx

Symbol	Meaning
F	Spring Force (N)
k	Spring Constant (N/m)
x	Compression or Extension (m)

Remember

• Negative sign shows restoring force.
• Force acts opposite to displacement.
• Larger k means stronger spring.
• Smaller k means softer spring.

2. Spring Potential Energy

Energy stored inside a compressed or stretched spring.

Formula

U = ½ kx²

Symbol	Meaning
U	Potential Energy
k	Spring Constant
x	Compression

3. Kinetic Energy

Energy possessed by a moving object.

Formula

K = ½ mv²

Symbol	Meaning
m	Mass
v	Velocity

4. Conservation of Mechanical Energy

When there is no friction, the total mechanical energy remains constant.

½mv² = ½kx²

Used to calculate maximum compression of a spring.

Example 5.8

To simulate car accidents, auto manufacturers study the collisions of moving cars with mounted springs and other constraints. Consider a typical simulation with a car of mass 1000 kg moving at a speed of 18.0 km/h on a smooth road and colliding with a horizontal mounted spring of spring constant 5.25 × 10³ N m⁻¹. What is the maximum compression of the spring?

Example 5.9

Consider Example 5.8 taking the coefficient of friction, μ = 0.5, and calculate the maximum compression of the spring.

NCERT Example 5.8

Question

Mass = 1000 kg
Speed = 18 km/h
Spring Constant = 5.25 × 10³ N/m
Find the maximum compression.

Solution

Step 1 18 km/h = 5 m/s

Kinetic Energy

K = ½ ×1000×5²

K = 12500 J

Using Conservation of Energy

½kx² = 12500

x = 2.0 m

Answer: Maximum Compression = 2.0 m

5. Work-Energy Theorem

The work done by all forces acting on a body equals the change in kinetic energy.

Formula

W = ΔK

6. Friction

Friction always opposes motion.

Formula

F = μN

On horizontal surface,

N = mg

Therefore,

F = μmg

NCERT Example 5.9

Question

Mass = 1000 kg
Speed = 5 m/s
k = 5.25 ×10³ N/m
μ = 0.5

Solution

Apply Work-Energy Theorem

−½mv² = −½kx² − μmgx

Rearranging,

kx² + 2μmgx − mv² = 0

x = 1.35 m

Answer: Maximum Compression = 1.35 m

Important Formula Sheet

Formula	Equation
Kinetic Energy	½mv²
Spring Force	F = -kx
Spring Potential Energy	½kx²
Friction	μmg
Work-Energy Theorem	W = ΔK
Conservation of Energy	½mv² = ½kx²

NEET Important Points

• Hooke's law is valid only within elastic limit.
• Spring force is a restoring force.
• Potential energy is always positive.
• Friction is a non-conservative force.
• Mechanical energy is conserved only without friction.
• Work done by friction is negative.
• Spring energy depends on x².
• Kinetic energy depends on velocity².

NEET Practice MCQs

Q1. The force exerted by a spring is

• A. Constant
• B. Proportional to displacement
• C. Inversely proportional
• D. Zero

Answer: B

Q2. Potential energy stored in a spring is

• A. kx
• B. kx²
• C. ½kx²
• D. ½kx

Answer: C

Q3. Which force is non-conservative?

• A. Gravity
• B. Spring
• C. Electrostatic
• D. Friction

Answer: D

Q4. Mechanical energy remains constant when

• A. Friction acts
• B. Air resistance acts
• C. Only conservative forces act
• D. External work is done

Answer: C

Quick Revision

• Hooke's Law → F = -kx
• Spring Energy → ½kx²
• Kinetic Energy → ½mv²
• Work-Energy Theorem → W = ΔK
• Without friction → Energy conserved.
• With friction → Use Work-Energy Theorem.
• Maximum compression occurs when kinetic energy becomes zero.
• Friction reduces spring compression.

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Prepared for NEET Aspirants

Easy Notes • NCERT Based • Beginner Friendly

Motion in One Dimension Notes for NEET 2026 | Distance & Displacement MCQs with Solutions

 Motion in One Dimension – Rest, Motion & Frame of Reference (NEET Notes) 

Motion in One Dimension – Complete NEET Physics Notes on Frame of Reference, Distance, Displacement, Rest and Motion.

1. Can the Distance Between Two Places Be Fixed?

Example

Question: What is the distance between City A and City B?

Answer: This is an incomplete (incorrectly framed) question because the distance depends on the path or route taken.

For example:

• Route 1 = 1050 km
• Route 2 = 1510 km

Therefore, distance depends on the path followed.

Important: Always specify the route while asking about distance.

---

2. Frame of Reference

Definition

A frame of reference is a place or point from where we observe the position or motion of an object.

The person who makes the observation is called the observer.

Example

A student is standing on the roadside and watching a moving bus.

• Road = Frame of reference
• Student = Observer

---

3. Position

Definition

The position of an object is its location with respect to a frame of reference.

Example

A bicycle is parked 10 m from a tree.

Here,

• Tree = Frame of reference
• Bicycle = Object

---

4. Rest

Definition

An object is said to be at rest if its position does not change with time with respect to the chosen frame of reference.

Example

A school bag kept on a classroom desk is at rest with respect to the desk.

Position
  ^
  |
  |──────────────
  |
  +---------------------> Time

Position remains constant.

---

5. Motion

Definition

An object is said to be in motion if its position changes with time with respect to the chosen frame of reference.

Example

A car moving on a highway is in motion with respect to the road.

Position
  ^
  |        /
  |      /
  |    /
  +---------------------> Time

Position changes with time.

---

Relative Nature of Rest and Motion

Rest and motion are relative terms, not absolute.

They depend on the frame of reference.

Example

Suppose:

• Student A is sitting inside a moving bus.
• Student B is sitting beside Student A.
• A traffic police officer is standing on the roadside.

Then:

• Student A is at rest with respect to Student B.
• Student B is at rest with respect to Student A.
• Student A is in motion with respect to the traffic police officer.
• Student B is also in motion with respect to the traffic police officer.

Thus, the same object can be at rest for one observer and in motion for another observer.

---

Important NEET Points

• Frame of Reference: Place from where observation is made.
• Observer: Person making the observation.
• Position: Location of an object with respect to the frame of reference.
• Rest: Position does not change with time.
• Motion: Position changes with time.
• Distance depends on the path taken.
• Rest and motion are relative concepts.
• There is no absolute rest or absolute motion in the universe.

---

NEET Quick Revision

• Frame of Reference → Point from where observation is made.
• Observer → Person who observes.
• Position → Location relative to the reference point.
• Rest → Position remains unchanged with time.
• Motion → Position changes with time.
• Distance → Depends on the path followed.
• Rest and Motion → Relative, not absolute. 

Motion in One Dimension – Distance and Displacement (NEET Notes)

1. Observer Always Assumed to be at Rest

Definition

While studying motion, the observer is always assumed to be at rest with respect to the chosen frame of reference.

Example

A student is sitting inside a moving bus and asks,

"Sir, when will Delhi come?"

The teacher replies,

"Delhi is not coming. Our bus is moving towards Delhi."

Although it appears that Delhi is coming closer, actually the bus is moving, while Delhi is at rest with respect to the Earth.

Important Point

• Motion is always described with respect to an observer.
• The observer is usually considered to be at rest.

---

2. Distance

Definition

Distance is the actual length of the path travelled by an object.

OR

Distance is the total path covered by an object during its motion.

Formula

Distance = Actual Path Length

---

Characteristics of Distance

• It depends on the path taken.
• It is a Scalar Quantity (only magnitude, no direction).
• It is always positive or zero.
• It cannot be negative.
• SI Unit = metre (m).
• It is always greater than or equal to displacement.
• Scalar quantities are added by simple algebraic addition.

---

Example 1

A person walks

• 3 m forward
• then 4 m forward

Distance = 3 + 4 = 7 m

---

Example 2

A person walks along three different paths between the same two points.

Path A = 10 m

Path B = 15 m

Path C = 18 m

Distance depends on which path is chosen.

---

3. Displacement

Definition

Displacement is the shortest straight-line distance between the initial position and the final position of an object.

OR

Displacement is the change in the position of an object.

---

Formula

Displacement = Final Position − Initial Position

---

Characteristics of Displacement

• It depends only on the initial and final positions.
• It does not depend on the path taken.
• It is a Vector Quantity (has magnitude and direction).
• It can be positive, negative, or zero.
• SI Unit = metre (m).
• To calculate displacement, only the initial position and final position are required.

---

Example 1

A person walks

• 3 m east
• then 4 m east

Distance = 7 m

Displacement = 7 m east

Since there is no change in direction,

Distance = Displacement

---

Example 2

A person walks

• 3 m east
• then 4 m west

Distance = 3 + 4 = 7 m

Displacement = 1 m west

Here,

Distance > Displacement

---

Example 3

A person starts from point A, walks around a circular park, and returns to point A.

Distance = Circumference of the park

Displacement = 0

Because the initial and final positions are the same.

---

4. Relation Between Distance and Displacement

Case 1: Straight-Line Motion (No Change in Direction)

Example

3 m → + 4 m →

Distance = 7 m

Displacement = 7 m

Distance = Displacement

---

Case 2: Motion with Change in Direction

Example

3 m → then 4 m ←

Distance = 7 m

Displacement = 1 m

Distance > Displacement

---

Important NEET Points

Distance	Displacement
Actual path length	Shortest straight-line distance
Scalar quantity	Vector quantity
Depends on path	Does not depend on path
Always positive or zero	Can be positive, negative, or zero
Cannot be negative	Can be negative
Distance ≥ Displacement	Displacement ≤ Distance
Unit = metre (m)	Unit = metre (m)

---

NEET Quick Revision

• Observer → Usually assumed to be at rest.
• Distance → Actual path travelled.
• Displacement → Shortest straight-line distance between initial and final positions.
• Distance depends on path.
• Displacement depends only on initial and final positions.
• Distance is a scalar quantity.
• Displacement is a vector quantity.
• Distance is always greater than or equal to displacement.
• Distance = Displacement only when the object moves in a straight line without changing direction.

Motion in One Dimension - Distance & Displacement (NEET Level)

Concept Questions with Solutions

Q1. Which of the following is always true for a moving object?
A) Distance = |Displacement|
B) Distance < |Displacement|
C) Distance ≥ |Displacement|
D) Distance ≤ |Displacement|

Answer: C) Distance ≥ |Displacement|
Explanation: Distance is actual path length, while displacement is shortest straight-line distance between initial and final points. So distance is always greater than or equal to displacement.

Q2. For motion in 1-D without change in direction, which is correct?
A) Distance = |Displacement|
B) Distance > |Displacement|
C) Distance < |Displacement|
D) None

Answer: A) Distance = |Displacement|
Explanation: When motion is in a straight line without reversing direction, actual path equals shortest path, so distance equals magnitude of displacement.

Q3. A person walks 4 m east and then 3 m east. Find distance and displacement.

Answer:
Distance = 4 + 3 = 7 m
Displacement = 7 m east
Explanation: Both motions are in same direction, so distance = displacement.

Q4. A person walks 6 m east and 4 m west. Find distance and displacement.

Answer:
Distance = 6 + 4 = 10 m
Displacement = 2 m east
Explanation: Distance adds total path. Displacement = final - initial = 6 - 4 = 2 m east.

Q5. Why is displacement a vector quantity?

Answer: Because displacement has both magnitude and direction.
Explanation: It depends on initial and final positions and includes direction (positive or negative in 1D).

Q6. Can distance be negative?

Answer: No.
Explanation: Distance is scalar and represents actual path length, so it is always positive or zero.

Q7. A body returns to its starting point after moving in a circle. Find distance and displacement.

Answer:
Distance = circumference of circle
Displacement = 0
Explanation: Final position = initial position, so displacement is zero but path is not zero.

Q8. What is frame of reference?

Answer: It is the point or system from which an observer measures position or motion.
Explanation: Motion is always relative to a chosen observer.

Q9. Why is motion relative?

Answer: Because it depends on the observer.
Explanation: An object may appear at rest for one observer and in motion for another.

Q10. A car moves 15 m straight without changing direction. Find distance and displacement.

Answer:
Distance = 15 m
Displacement = 15 m
Explanation: Straight-line motion without change in direction makes distance equal to displacement.

Key Formulas

• Distance = Actual path length
• Displacement = Final position - Initial position
• |Displacement| ≤ Distance

Important Points

• Distance is scalar
• Displacement is vector
• Distance depends on path
• Displacement depends only on endpoints

Q11. A person walks 3 m north, then 4 m east. Find distance and displacement.

Answer:
Distance = 3 + 4 = 7 m
Displacement = √(3² + 4²) = √25 = 5 m (north-east direction)

Explanation:
Distance is total path length (scalar), so we simply add.
Displacement is shortest straight-line distance, so we use Pythagoras theorem because motion is at right angle.

Q12. A body moves 10 m forward and then 10 m backward. Find distance and displacement.

Answer:
Distance = 10 + 10 = 20 m
Displacement = 0 m

Explanation:
The object returns to its starting point, so final position = initial position.
Hence displacement is zero but distance is non-zero.

Q13. Which quantity can be zero even if an object moves? A) Distance B) Speed C) Displacement D) Time

Answer: C) Displacement

Explanation: If an object returns to its starting point, displacement becomes zero, but it still covers some distance.

Q14. A student says "Delhi is coming closer while we are in a train". Is this correct?

Answer: No, it is incorrect.

Explanation: Delhi is at rest with respect to Earth. The train is moving towards Delhi. Motion depends on frame of reference.

Q15. A particle moves in a straight line 8 m east and then 3 m west. Find displacement.

Answer:
Displacement = 8 - 3 = 5 m east

Explanation: Direction matters in displacement. East is positive, west is negative in 1D motion.

Q16. Why is distance always greater than or equal to displacement?

Answer:
Because distance is actual path and displacement is shortest path.

Explanation: Any curved or back-and-forth motion increases path length, so distance ≥ displacement always.

Q17. A car moves in a circular path and completes half a circle. Radius = 7 m. Find displacement.

Answer:
Displacement = diameter = 2 × 7 = 14 m

Explanation: In half circle, initial and final points are opposite ends of diameter.

Q18. Can displacement be greater than distance? A) Yes B) No

Answer: B) No

Explanation: Displacement is shortest distance, so it can never exceed actual path length.

Q19. A boy runs 5 m east, 5 m west, 5 m east again. Find distance and displacement.

Answer:
Distance = 5 + 5 + 5 = 15 m
Displacement = 5 m east

Explanation: Total path is added for distance, but final position is 5 m east of start.

Q20. Define observer in motion study.

Answer: The person who measures or observes the motion of an object is called an observer.

Explanation: All motion is described relative to the observer (frame of reference).

Q21. A particle moves 12 m north, then 5 m south. Find distance and displacement.

Answer:
Distance = 12 + 5 = 17 m
Displacement = 12 - 5 = 7 m north

Explanation:
Distance is total path length (scalar).
Displacement depends on direction, so we subtract opposite directions.

Q22. A body moves 6 m east, then 8 m north. Find displacement.

Answer:
Displacement = √(6² + 8²) = √(36 + 64) = √100 = 10 m (north-east)

Explanation:
Motion is at right angle, so we apply Pythagoras theorem.

Q23. If displacement is zero, what can we say about motion?

Answer: The object has returned to its initial position.

Explanation:
Even if the object moves, if final position = initial position, displacement becomes zero.

Q24. A car travels 20 m in a straight line. What is the relationship between distance and displacement?

Answer:
Distance = Displacement = 20 m

Explanation:
In straight-line motion without changing direction, path length equals shortest distance.

Q25. Which quantity gives only magnitude but no direction? A) Displacement B) Distance C) Velocity D) Acceleration

Answer: B) Distance

Explanation:
Distance is scalar, so it has only magnitude and no direction.

Q26. Can distance ever be negative?

Answer: No.

Explanation:
Distance is actual path length, so it is always positive or zero.

Q27. A person walks 10 m east and 10 m west. What is displacement?

Answer: 0 m

Explanation:
Final position is same as initial position, so displacement is zero.

Q28. What is the SI unit of displacement?

Answer: metre (m)

Explanation:
Displacement is a length quantity, so its SI unit is metre.

Q29. A student says “distance and displacement are always equal”. Is this correct?

Answer: No.

Explanation:
They are equal only in straight-line motion without change in direction.

Q30. A particle moves in a square path of side 4 m and returns to starting point. Find distance and displacement.

Answer:
Distance = 4 + 4 + 4 + 4 = 16 m
Displacement = 0 m

Explanation:
Distance is total perimeter, but displacement is zero because initial and final positions are same.

Q31. A car moves 18 m east and then 24 m north. Find displacement.

Answer:
Displacement = √(18² + 24²) = √(324 + 576) = √900 = 30 m (north-east)

Explanation:
Since motion is in two perpendicular directions, we use Pythagoras theorem to find shortest distance.

Q32. A body moves 15 m south and 15 m north. Find distance and displacement.

Answer:
Distance = 15 + 15 = 30 m
Displacement = 0 m

Explanation:
The object returns to its starting point, so displacement is zero.

Q33. Which statement is correct? A) Distance can be negative B) Displacement is always positive C) Distance is always ≥ |Displacement| D) Displacement is scalar

Answer: C) Distance is always ≥ |Displacement|

Explanation:
Distance is actual path length and cannot be less than shortest straight-line distance.

Q34. A student walks 7 m east, 7 m east, and 7 m west. Find displacement.

Answer:
Displacement = (7 + 7 - 7) = 7 m east

Explanation:
Take east as positive and west as negative, then add algebraically.

Q35. What is a frame of reference?

Answer: A system or point from which an observer measures position and motion.

Explanation:
Motion is always relative to a chosen reference point or observer.

Q36. A bus moves forward and a passenger feels stationary inside. Why?

Answer: Because both passenger and bus are in same frame of reference.

Explanation:
Relative motion inside the bus is zero, so passenger feels at rest.

Q37. A person completes one full circular path. What is displacement?

Answer: 0 m

Explanation:
Initial and final positions are same, so displacement is zero.

Q38. Which is vector quantity? A) Distance B) Speed C) Displacement D) Time

Answer: C) Displacement

Explanation:
Displacement has both magnitude and direction.

Q39. A body moves 9 m east and 12 m west. Find distance.

Answer: 21 m

Explanation:
Distance is total path length, so we add magnitudes.

Q40. Find displacement for Q39.

Answer: 3 m west

Explanation:
Net movement = 12 - 9 = 3 m towards west.

Q41. A particle moves 10 m east, 6 m north, and 10 m west. Find distance and displacement.

Answer:
Distance = 10 + 6 + 10 = 26 m
Displacement = 6 m north

Explanation:
Distance is total path length (scalar), so we add all parts.
East and west cancel each other (10 m east and 10 m west), leaving only 6 m north as net displacement.

Q42. A body moves in a straight line 25 m forward and then 25 m backward. Find displacement.

Answer: 0 m

Explanation:
Final position is same as initial position, so displacement becomes zero.

Q43. Which quantity depends on path taken? A) Displacement B) Position C) Distance D) Velocity

Answer: C) Distance

Explanation:
Distance depends on actual route followed, so different paths give different values.

Q44. A student says “displacement is always positive”. Is this correct?

Answer: No.

Explanation:
Displacement can be positive, negative, or zero depending on direction in 1-D motion.

Q45. A car moves 50 m straight without changing direction. What is true? A) Distance > Displacement B) Distance < Displacement C) Distance = Displacement D) Distance = 0

Answer: C) Distance = Displacement

Explanation:
In straight-line motion without reversing direction, path length equals shortest distance. 

Internal Links

• Units and Measurements Notes 

• Physical Quantities: Scalars and Vectors

 • Kinematics Complete Notes

 • Position, Distance and Displacement

 • Speed, Velocity and Acceleration

 • Graphs in Motion (Position-Time & Velocity-Time) 

• Relative Motion Explained

 • NCERT Class 11 Physics Chapter 3 Notes 

• NEET Physics Mock Tests

 • NEET Physics Previous Year Questions 

• Motion in a Straight Line Numericals 

• Laws of Motion Notes

 • Projectile Motion Notes

 • Circular Motion Basics

 • Physics Formula Sheet for NEET

Work, Energy and Power Notes for NEET: Kinetic Energy & Variable Force

-  Dr.Sanjaykumar Pawar 

Kinetic Energy and Work Done by Variable Force explained with formulas, examples, and graphical interpretation for NEET aspirants.

INTERNAL LINK SUGGESTIONS

6. Laws of Motion Complete Notes

7. Motion in One Dimension Notes

8. Motion in a Plane Notes

9. Work Energy Theorem Explained

10. Potential Energy Notes

11. Conservation of Energy Notes

12. Power Formula and Applications

13. Mechanical Energy Examples

14. NCERT Physics Class 11 Solutions

15. NEET Physics Formula Sheet

16. Rotational Motion Notes

17. Gravitation Complete Notes

18. Oscillations and SHM Notes

19. Units and Measurements Notes

20. NEET Physics MCQs with Answers 

FAQ QUESTIONS

Q1. What is kinetic energy?

A. Kinetic energy is the energy possessed by a body due to its motion.

Q2. What is the formula of kinetic energy?

A. K = 1/2 mv².

Q3. What is work done by a variable force?

A. Work done by a variable force equals the area under the force-displacement graph.

Q4. Why is speed not reduced by 90% when kinetic energy becomes 10%?

A. Because kinetic energy is proportional to the square of speed.

Q5. What is the SI unit of work?

A. Joule (J).

WORK, ENERGY AND POWER

1. Kinetic Energy (K)

Kinetic energy is the energy possessed by a body due to its motion.

K = ½ mv²

Where:

• m = mass of the object
• v = speed of the object
• K = kinetic energy

2. Typical Kinetic Energies

Object	Mass (kg)	Speed (m/s)	Kinetic Energy (J)
Car	2000	25	6.3 × 10⁵
Running Athlete	70	10	3.5 × 10³
Bullet	0.05	200	10³
Stone Dropped from 10 m	-	14	10²
Rain Drop	3.5 × 10⁻⁵	9	1.4 × 10⁻³
Air Molecule	≈ 10⁻²⁶	500	≈ 10⁻²¹

Heavy objects moving fast possess very large kinetic energy.

3. Example: Bullet Passing Through Plywood

Given:

• Mass of bullet = 50 g = 0.05 kg
• Initial speed = 200 m/s
• Final kinetic energy = 10% of initial kinetic energy

Step 1: Initial Kinetic Energy

Ki = ½mv²

Ki = ½ × 0.05 × (200)²

Ki = 1000 J

Step 2: Final Kinetic Energy

Kf = 10% of 1000

Kf = 100 J

Step 3: Calculate Final Speed

½mvf² = 100

vf = √(2 × 100 / 0.05)

vf = 63.2 m/s

Answer: Emergent speed = 63.2 m/s

4. Important Concept for NEET

Kinetic Energy is proportional to the square of speed.

K ∝ v²

If kinetic energy becomes 10%:

vf = √0.1 × vi

vf = 0.316 × vi

Speed becomes 31.6% of original speed.

Reduction in speed:

100 − 31.6 = 68.4%

Speed is reduced by approximately 68%, not 90%.

5. Work Done by a Variable Force

Constant Force

A constant force is a force whose magnitude and direction remain unchanged.

Examples:

• Weight of a body near Earth
• Pushing a box with constant force

W = F × s

Where:

• W = Work Done
• F = Force
• s = Displacement

Variable Force

A variable force changes with position, time, or direction.

Examples:

• Spring Force
• Gravitational Force
• Electric Force

For variable force, W = Fs cannot be directly used.

6. Small Displacement Method

Consider a very small displacement Δx.

Over this tiny distance, force can be treated as constant.

ΔW = F(x)Δx

Where:

• ΔW = Small work done
• F(x) = Force at position x
• Δx = Small displacement

7. Total Work Done

Adding work done over many small intervals:

W = Σ F(x)Δx

This is called summation.

Meaning:

• Divide motion into many small parts.
• Find work done in each part.
• Add all the small works.

8. Graphical Interpretation

In a Force vs Position graph:

• Y-axis → Force F(x)
• X-axis → Position x

Area of one small rectangle:

ΔA = F(x)Δx

Since:

ΔW = F(x)Δx

Therefore:

ΔW = ΔA

Small Work Done = Area of Small Rectangle

9. Integral Form of Work Done

When Δx becomes extremely small:

W = ∫ F(x) dx

Limits:

W = ∫_xi^xf F(x) dx

Where:

• xi = Initial Position
• xf = Final Position

Work Done by Variable Force = Area Under Force-Position Curve

10. NEET Quick Revision

Kinetic Energy

K = ½mv²

• KE ∝ Mass
• KE ∝ Speed²

If Speed Doubles

K → 4K

If Speed Triples

K → 9K

Variable Force

ΔW = F(x)Δx

W = ΣF(x)Δx

W = ∫F(x)dx

Most Important NEET Statement

✔ Work Done = Area under Force-Displacement Graph

Frequently Asked Relation

If KE becomes n times:

v = √n × Initial Speed

If speed becomes n times:

K = n² × Initial Kinetic Energy

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CBSE Class 11 Physics Question Bank

Chapter: Work, Energy and Power

Section A: Very Short Answer Questions (1 Mark)

1. Define kinetic energy.

Kinetic energy is the energy possessed by a body due to its motion.

2. Write the SI unit of kinetic energy.

Joule (J)

3. Write the formula for kinetic energy.

K = ½mv²

4. On what factors does kinetic energy depend?

Mass and square of velocity.

5. What is the kinetic energy of a body at rest?

Zero.

6. What is a variable force?

A force whose magnitude changes with position is called a variable force.

7. Write the expression for small work done by a variable force.

ΔW = F(x)Δx

8. What does the area under an F-x graph represent?

Work done by the force.

9. What is the SI unit of work?

Joule (J)

10. Write the integral form of work done.

W = ∫F(x)dx

Section B: Short Answer Questions (2 Marks)

1. Why does a bullet possess large kinetic energy despite having small mass?

A bullet has very high speed. Since kinetic energy depends on the square of velocity, it possesses large kinetic energy.

2. Calculate the kinetic energy of a body of mass 4 kg moving with speed 5 m/s.

K = ½mv²
= ½ × 4 × 25
= 50 J

3. Distinguish between constant force and variable force.

Constant Force	Variable Force
Remains unchanged	Changes with position
W = Fs	W = ∫Fdx
Example: Weight	Example: Spring Force

4. State the graphical interpretation of work done.

Work done by a force is equal to the area under the force-displacement graph.

5. If velocity doubles, how does kinetic energy change?

K ∝ v²
Therefore kinetic energy becomes four times.

Section C: Long Answer Questions (5 Marks)

1. Derive the expression for kinetic energy.

Work done W = Fs

Using F = ma
W = mas

Using equation of motion:
v² − u² = 2as
a = (v² − u²)/2s

Substituting:
W = m(v² − u²)/2

For u = 0:
K = ½mv²

2. Explain work done by a variable force with graphical interpretation.

For a small displacement Δx:
ΔW = F(x)Δx

Adding all small works:
W = ΣF(x)Δx

As Δx approaches zero:
W = ∫F(x)dx

Thus, work done by a variable force equals the area under the force-displacement graph.

Section D: Multiple Choice Questions

1. Kinetic energy depends on:

• A. Mass only
• B. Velocity only
• C. Mass and velocity
• D. Mass and square of velocity

Answer: D

2. SI unit of kinetic energy is:

• A. Newton
• B. Joule
• C. Watt
• D. Pascal

Answer: B

3. If speed triples, kinetic energy becomes:

• A. 3 times
• B. 6 times
• C. 9 times
• D. 27 times

Answer: C

4. Area under force-displacement graph represents:

• A. Velocity
• B. Momentum
• C. Work done
• D. Acceleration

Answer: C

Section E: Assertion and Reason

Assertion (A): Kinetic energy is proportional to square of velocity.
Reason (R): K = ½mv².

Both Assertion and Reason are true and Reason correctly explains Assertion.

Assertion (A): Area under force-displacement graph gives work done.
Reason (R): ΔW = FΔx.

Both Assertion and Reason are true and Reason correctly explains Assertion.

Section F: Fill in the Blanks

1. Kinetic energy is measured in Joule.
2. The formula of kinetic energy is ½mv².
3. Work done by variable force equals the area under F-x graph.
4. Kinetic energy is proportional to the square of velocity.
5. The SI unit of work is Joule.

Section G: Match the Columns

Column A	Column B
Kinetic Energy	Energy due to motion
Joule	Unit of work
Variable Force	Spring force
Area under F-x graph	Work done

Correct Matching:
A → Energy due to motion
B → Unit of work
C → Spring force
D → Work done

Section H: Statement Based Questions

Statement I: A bullet moving at high speed possesses kinetic energy.
Statement II: Kinetic energy depends on square of velocity.

Both statements are true.

Section I: Case Study Questions

A bullet of mass 0.05 kg is fired with speed 200 m/s. After passing through a wooden block, it retains only 10% of its original kinetic energy.

1. What is the initial kinetic energy?

1000 J

2. What is the final kinetic energy?

100 J

3. What is the final speed?

63.2 m/s

4. Is speed reduced by 90%?

No.

5. Why?

Because kinetic energy is proportional to velocity squared.

Section J: Competency Based Questions

1. A car moving at 20 m/s has kinetic energy K. If its speed becomes 40 m/s, what will be its kinetic energy?

K' = (40/20)² K = 4K

2. Why does a fast-moving cricket ball hurt more than a slow-moving ball?

A fast-moving ball possesses greater kinetic energy because kinetic energy depends on the square of speed.

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