Collision

## a collision is an event in which two or more bodies exert forces on each other for a relatively short time. Collisions can occur between objects in motion (e.g., two billiard balls hitting each other) or between a moving object and a stationary one (e.g., a car hitting a wall).

# Basic Concepts in Collisions

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a.Conservation Laws

Two important physical quantities are often conserved during collisions:

- Momentum (p): Always conserved in all types of collisions (in an isolated system).

p=mv

Total momentum before = Total momentum after

m1 v1+m2 v2=m1 v1′+m2v2′​

- Kinetic Energy (KE): Conserved only in elastic collisions.

KE=1/2mv2

Types of Collisions

## There are three main types of collisions:

1. Elastic Collision

- Both momentum and kinetic energy are conserved.
- Common in atomic and molecular physics.
- Example: Collisions between gas molecules, or two steel balls bouncing off each other.

Formulae:

In one dimension(1D):

v1'=( (m1-m2)v1+2m2v2 ) / (m1+m2)

v2'=( (m2-m1)v2+2m1v1 ) / (m1+m2)

Example:

Two balls

m1=1kg v1=2m/s

m2=1kg v2=-1m/s

v1'=( (1-1)(2)+2(1)(-1) )/(1+1) =-2/2=-1m/s

v2'=( (1-1)(-1)+2(1)(2) )/(1+1) =4/2=2m/s

2. Inelastic Collision

- Momentum is conserved, but kinetic energy is not.
- Some energy is lost to heat, sound, or deformation.
- Objects do not stick together.
- Example: A car crash where the vehicles dent but don’t stick together.

Formula:

m1v1+m2v2=m1v1'+m2v2'

But:

1/2 m1v12+1/2 m2v22 ≠ 1/2 m1v1' 2+1/2 m2v2' 2

3. Perfectly In-Elastic Collision

- A special case of inelastic collision where the objects stick together after the collision.
- Momentum is conserved, but kinetic energy is lost.
- Maximum possible loss of kinetic energy.

Formula:

m1v1+m2v2=(m1+m2)vf

Example:

Two Clay balls,

m1=2kg v1=3m/s

m2=3kg v2=0m/s

vf= ( 2(3)+3(0) ) / (2+3) =6/5=1.2m/s

Coefficient of Restitution (e)

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Defines how bouncy a collision is.

e=Relative Velocity After Collision / Relative Velocity before Collision =v2'-v1' / v1-v2

- 𝑒=1 : Elastic
- 0 < e < 1 : Inelastic
- e=0 : Perfectly In-elastic

Dimensional Collisions (2D Collisions)

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When two bodies collide in 2D or 3D, momentum is conserved in all directions (x and y axes).

Conservation of momentum:

In X: m1v1x+m2v2x = m1v1x'+m2v2x'

In Y: m1v1y+m2v2y = m1v1y'+m2v2y'

These are often solved using vector components

Energy Loss in Inelastic Collisions

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The kinetic energy lost:

ΔKE = KEinitial-KEfinal

For perfectly inelastic collision:

ΔKE = 1/2 m1v12+1/2 m2v22 - 1/2 (m1+m2)vf2

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