Elastic & Inelastic Collisions – Differences Explained

Introduction

Collisions are everywhere in nature and technology — from billiard balls striking each other, to car crashes, to particles colliding inside accelerators. Every time two bodies collide, forces act for a short interval, causing changes in motion.

Physics classifies collisions broadly into elastic and inelastic collisions. Both conserve momentum, but they differ in whether kinetic energy is conserved. Understanding these differences is crucial in mechanics, engineering, astrophysics, and even daily life safety.

This article explains in detail:

• What collisions are.
• Types of collisions.
• Elastic vs. inelastic collisions.
• Mathematical treatment.
• Real-life examples and applications.

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Part 1: What is a Collision?

A collision occurs when two (or more) bodies exert forces on each other for a short time, leading to changes in their velocities.

• Collisions can be head-on (linear), oblique, or even multi-particle.
• The study of collisions uses Newton’s laws, momentum conservation, and energy principles.

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Part 2: Laws Governing Collisions

1. Law of Conservation of Linear Momentum
   • In an isolated system (no external force), total momentum before collision = total momentum after collision.
   m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2m1​u1​+m2​u2​=m1​v1​+m2​v2​
2. Law of Conservation of Energy
   • Total energy is always conserved, but mechanical energy (kinetic + potential) may not always remain constant.
3. Coefficient of Restitution (e)
   • Measures “elasticity” of collision.
   e=Relative velocity after collisionRelative velocity before collisione = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}}e=Relative velocity before collisionRelative velocity after collision​
   • e=1e = 1e=1 → Perfectly elastic collision.
   • 0<e<10 < e < 10<e<1 → Inelastic collision.
   • e=0e = 0e=0 → Perfectly inelastic collision (bodies stick together).

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Part 3: Elastic Collisions

3.1 Definition

A collision is elastic if:

• Momentum is conserved.
• Kinetic energy is also conserved.

3.2 Characteristics

• No loss of kinetic energy (only redistributed).
• No permanent deformation or heat generation.
• Occurs mostly in atomic/molecular or idealized systems.

3.3 Mathematical Treatment (1D Elastic Collision)

Consider two bodies of masses m1,m2m_1, m_2m1​,m2​ with initial velocities u1,u2u_1, u_2u1​,u2​. After collision, velocities are v1,v2v_1, v_2v1​,v2​.

From momentum conservation: m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2m1​u1​+m2​u2​=m1​v1​+m2​v2​

From kinetic energy conservation: 12m1u12+12m2u22=12m1v12+12m2v22\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^221​m1​u12​+21​m2​u22​=21​m1​v12​+21​m2​v22​

Solving, we get: v1=(m1−m2)u1+2m2u2m1+m2v_1 = \frac{(m_1 – m_2)u_1 + 2m_2 u_2}{m_1 + m_2}v1​=m1​+m2​(m1​−m2​)u1​+2m2​u2​​ v2=(m2−m1)u2+2m1u1m1+m2v_2 = \frac{(m_2 – m_1)u_2 + 2m_1 u_1}{m_1 + m_2}v2​=m1​+m2​(m2​−m1​)u2​+2m1​u1​​

3.4 Examples of Elastic Collisions

• Collisions between gas molecules.
• Electron-electron scattering.
• Steel balls in Newton’s cradle (nearly elastic).
• Billiard balls (approximately elastic).

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Part 4: Inelastic Collisions

4.1 Definition

A collision is inelastic if:

• Momentum is conserved.
• Kinetic energy is not conserved (some is lost to heat, sound, deformation).

4.2 Characteristics

• Part of kinetic energy converts into other forms.
• Objects may deform permanently.
• Most real-life collisions are inelastic.

4.3 Types of Inelastic Collisions

1. Partially Inelastic Collision – Bodies do not stick but KE decreases.
2. Perfectly Inelastic Collision – Bodies stick together and move with common velocity.

4.4 Mathematical Treatment (Perfectly Inelastic Collision)

If two masses m1,m2m_1, m_2m1​,m2​ with initial velocities u1,u2u_1, u_2u1​,u2​ collide and stick together with final velocity vvv: m1u1+m2u2=(m1+m2)vm_1 u_1 + m_2 u_2 = (m_1 + m_2)vm1​u1​+m2​u2​=(m1​+m2​)v v=m1u1+m2u2m1+m2v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}v=m1​+m2​m1​u1​+m2​u2​​

Kinetic energy after collision: KEf=12(m1+m2)v2KE_f = \frac{1}{2}(m_1 + m_2)v^2KEf​=21​(m1​+m2​)v2

Since KEf<KEiKE_f < KE_iKEf​<KEi​, some energy is lost.

4.5 Examples of Inelastic Collisions

• Car crashes.
• Clay balls colliding and sticking together.
• Meteor hitting Earth.
• Sports: Kicking a football (deformation + heat).

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Part 5: Differences Between Elastic & Inelastic Collisions

Feature	Elastic Collision	Inelastic Collision
Momentum	Conserved	Conserved
Kinetic Energy	Conserved	Not conserved
Energy Transformation	No energy lost to heat, sound, deformation	KE partly converted to heat, sound, deformation
Coefficient of Restitution (e)	e=1e = 1e=1	0≤e<10 \leq e < 10≤e<1
Nature	Ideal, microscopic, rare in daily life	Common in real-world scenarios
Examples	Gas molecules, Newton’s cradle, billiards	Car accidents, clay collisions, meteor strikes

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Part 6: Numerical Examples

Example 1: Elastic Collision

A 2 kg ball moving at 4 m/s collides elastically with a 3 kg ball at rest. Find their velocities after collision.

Solution: u1=4,  u2=0,  m1=2,  m2=3u_1 = 4, \; u_2 = 0, \; m_1 = 2, \; m_2 = 3u1​=4,u2​=0,m1​=2,m2​=3 v1=(2−3)4+2(3)(0)5=−45=−0.8 m/sv_1 = \frac{(2-3)4 + 2(3)(0)}{5} = \frac{-4}{5} = -0.8 \, m/sv1​=5(2−3)4+2(3)(0)​=5−4​=−0.8m/s v2=(3−2)(0)+2(2)(4)5=165=3.2 m/sv_2 = \frac{(3-2)(0) + 2(2)(4)}{5} = \frac{16}{5} = 3.2 \, m/sv2​=5(3−2)(0)+2(2)(4)​=516​=3.2m/s

👉 Ball 1 rebounds at –0.8 m/s, Ball 2 moves at 3.2 m/s.

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Example 2: Perfectly Inelastic Collision

Two bodies of 5 kg and 3 kg move towards each other with velocities 6 m/s and –4 m/s. Find their common velocity after sticking. v=(5)(6)+(3)(−4)5+3=30−128=188=2.25 m/sv = \frac{(5)(6) + (3)(-4)}{5+3} = \frac{30 – 12}{8} = \frac{18}{8} = 2.25 \, m/sv=5+3(5)(6)+(3)(−4)​=830−12​=818​=2.25m/s

👉 Final velocity = 2.25 m/s in the direction of the heavier body.

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Part 7: Applications of Collision Theory

1. Engineering Safety
   • Car crash testing uses inelastic collision principles to design crumple zones that absorb energy.
2. Sports Science
   • Elasticity of balls (tennis, basketball) determines bounce behavior.
3. Astronomy
   • Meteor impacts are modeled as inelastic collisions.
   • Elastic scattering studied in nuclear/particle physics.
4. Everyday Life
   • Playing billiards or pool.
   • Throwing clay or mud.
   • Sound generation when objects strike.

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Part 8: Graphical Understanding

• Elastic Collision Graph: KE before = KE after.
• Inelastic Collision Graph: KE decreases after collision, though total energy remains constant.

Graphs of relative velocity vs. time also highlight the coefficient of restitution.

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Part 9: Key Takeaways

• All collisions conserve momentum (isolated system).
• Elastic collisions also conserve kinetic energy.
• Inelastic collisions lose some KE to heat, sound, or deformation.
• Perfectly inelastic collisions involve objects sticking together.
• Coefficient of restitution defines the elasticity of collision.