Notes : Define Collision , Elastic and Inelastic Collision with Equation - Class 11 Physics

5.11 COLLISIONS

Introduction

In Physics, some quantities remain conserved during interactions between bodies. Two important conserved quantities are:

• Linear Momentum
• Energy

Collisions provide practical applications of these conservation laws.

Examples: Billiards, Carrom, Marbles, Cricket Ball striking a Bat.

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Collision

A collision is a short-duration interaction between two bodies during which they exert large forces on each other, resulting in changes in their velocities.

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Collision of Two Particles

Consider two particles:

• Mass $m_{1}$ moving with initial velocity $v_{1i}$
• Mass $m_{2}$ initially at rest

Before Collision

$$u_1=v_{1i}$$

$$u_2=0$$

Particle m₁ moves along the positive x-axis towards particle m₂.

After Collision

• m₁ moves with velocity v₁f making angle θ₁ with the x-axis.
• m₂ moves with velocity v₂f making angle θ₂ with the x-axis.

The collision changes both speed and direction of motion.

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Conservation of Linear Momentum

The total linear momentum of an isolated system remains constant during collision.

Law:

$$m_1v_{1i}=m_1v_{1f}+m_2v_{2f}$$

Momentum Conservation Along x-axis

$$m_1v_{1i}=m_1v_{1f}\cos\theta_1+m_2v_{2f}\cos\theta_2$$

Momentum Conservation Along y-axis

$$0=m_1v_{1f}\sin\theta_1-m_2v_{2f}\sin\theta_2$$

or

$$m_1v_{1f}\sin\theta_1=m_2v_{2f}\sin\theta_2$$

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Proof of Momentum Conservation

$$\Delta p_1=F_{12}\Delta t$$

$$\Delta p_2=F_{21}\Delta t$$

According to Newton's Third Law:

$$F_{12}=-F_{21}$$

Therefore,

$$\Delta p_1+\Delta p_2=0$$

Hence,

$$p_{\text{initial}}=p_{\text{final}}$$

Thus, total momentum remains conserved during collision.

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Kinetic Energy in Collisions

Kinetic energy may or may not be conserved.

Some kinetic energy may be converted into:

• Heat
• Sound
• Internal Energy
• Deformation Energy

Kinetic Energy:

$$KE=\frac{1}{2}mv^2$$

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Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved.

$$p_i=p_f$$

$$KE_i=KE_f$$

Characteristics:

• No loss of kinetic energy.
• No permanent deformation.
• Bodies regain their original shape.

Examples: Billiard balls, Molecular collisions.

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Inelastic Collision

An inelastic collision is one in which momentum is conserved but kinetic energy is not conserved.

$$p_i=p_f$$

$$KE_i>KE_f$$

Characteristics:

• Partial loss of kinetic energy.
• Energy converts into heat, sound and deformation.

Examples: Car accidents, Hammer striking a nail.

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Completely Inelastic Collision

In a completely inelastic collision, the colliding bodies stick together and move with a common velocity.

Common Velocity:

$$v=\frac{m_1u_1+m_2u_2}{m_1+m_2}$$

Characteristics:

• Momentum is conserved.
• Maximum loss of kinetic energy.
• Bodies move together after collision.

Example: Bullet embedded in a wooden block.

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Comparison of Collisions

Property	Elastic	Inelastic	Completely Inelastic
Momentum Conserved	Yes	Yes	Yes
Kinetic Energy Conserved	Yes	No	No
Bodies Stick Together	No	No	Yes
Loss of KE	Zero	Partial	Maximum

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FAQ

Q1. What is a collision?
A collision is a short-duration interaction between two bodies resulting in a change in velocity.

Q2. Is momentum conserved in all collisions?
Yes, momentum is conserved in every collision.

Q3. Is kinetic energy conserved in all collisions?
No, only in elastic collisions.

Q4. What is an elastic collision?
A collision in which both momentum and kinetic energy are conserved.

Q5. What is an inelastic collision?
A collision in which momentum is conserved but kinetic energy is not conserved.

Q6. What is a completely inelastic collision?
A collision in which colliding bodies stick together after collision.

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Quiz

1. Which quantity is always conserved during collision?
   A) Kinetic Energy
   B) Potential Energy
   C) Linear Momentum ✔
   D) Mechanical Energy


2. In an elastic collision:
   A) Only Momentum
   B) Only Kinetic Energy
   C) Both Momentum and Kinetic Energy ✔
   D) Neither


3. In a completely inelastic collision:
   A) Bodies Rebound
   B) Bodies Stop
   C) Bodies Stick Together ✔
   D) Momentum Lost


4. Momentum conservation is based on:
   A) Newton's First Law
   B) Newton's Second Law
   C) Newton's Third Law ✔
   D) Gravitation


5. In an inelastic collision, kinetic energy converts into:
   A) Heat and Sound ✔
   B) Momentum
   C) Mass
   D) Gravity

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Key Points

• Momentum is conserved in all collisions.
• Kinetic energy is conserved only in elastic collisions.
• Bodies stick together in completely inelastic collisions.
• Newton's Third Law explains conservation of momentum.
• Heat, sound and deformation may be produced during inelastic collisions.

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Important Formulae

Momentum:

$$p=mv$$

General Conservation of Momentum:

$$\sum \vec{p}_{initial}=\sum \vec{p}_{final}$$

One-Dimensional Collision:

$$m_1u_1+m_2u_2=m_1v_1+m_2v_2$$

Coefficient of Restitution:

$$e=\frac{v_2-v_1}{u_1-u_2}$$

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One-Line Definitions

Collision: A short-duration interaction between two bodies causing changes in velocity.

Elastic Collision: A collision in which both momentum and kinetic energy are conserved.

Inelastic Collision: A collision in which momentum is conserved but kinetic energy is not conserved.

Completely Inelastic Collision: A collision in which colliding bodies stick together and move with a common velocity.