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

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

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)v_f

Example:

Two Clay balls,

m₁=2kg    v₁=3m/s

m₂=3kg   v₂=0m/s

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

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

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

Conservation of momentum:

In X: m₁v_1x+m₂v_2x = m₁v_1x'+m₂v_2x'

In Y: m₁v_1y+m₂v_2y = m₁v_1y'+m₂v_2y'

These are often solved using vector components

The kinetic energy lost:

ΔKE = KE_initial-KE_final

For perfectly inelastic collision:

ΔKE = 1/2 m₁v₁²+1/2 m₂v₂² - 1/2 (m₁+m₂)v_f²