Collision between two or more particles is the interaction for a short interval of time in which they apply relatively strong forces on each other.

In a collision physical contact of two bodies is not necessary.

There are two types of collisions:

1. Elastic collision

The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions.

In an elastic collision all the involved forces are conservative forces.

Total energy remains conserved.

2. Inelastic collision

The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions.

In an inelastic collision some or all the involved forces are non-conservative forces.

Total energy of the system remains conserved.

If after the collision two bodies stick to each other, then the collision is said to be perfectly

inelastic.

Coefficient of Restitution or Resilience

The ratio of relative velocity of separation after collision to the velocity of approach before Collision is called coefficient of restitution resilience.

It is represented by e and it depends upon the material of the colliding bodies.

For a perfectly elastic collision, e = 1

For a perfectly inelastic collision, e = 0

For all other collisions, 0 < e < 1

One Dimensional or Head-on Collision

If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or head-on collision.

Inelastic One Dimensional Collision

Applying Newton’s experimental law, we have

Velocities after collision

V₁ = (m₁ – m₂) u₁ + 2m₂u₂ / (m₁ + m₂)

and v₂ = (m₂ – m₁) u₂ + 2m₁ u₁ / (m₁ + m₂)

When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities.

V₁ = u₂ and v₂ = u₁

If second body of same mass (m1 = m2) is at rest, then after collision first body comes to rest and second body starts moving with the initial velocity of first body.

V₁ = 0 and v₂ = u₁

If a light body of mass m₁ collides with a very heavy body of mass m₂ at rest, then after collision.

V₁ = – u₁ and v₂ = 0

It means light body will rebound with its own velocity and heavy body will continue to be at rest.

If a very heavy body of mass m1 collides with a light body of mass m₂ (m₁ > > m₂₁) at rest, then after collision

V₁ = u1 and v2 = 2u₁

In Inelastic One Dimensional Collision

Loss of kinetic energy

ΔE = m₁ m₂ / 2(m₁ + m₂) (u₁ – u₂)² (1 – e²)

In Perfectly Inelastic One Dimensional Collision

Velocity of separation after collision = 0.

Loss of kinetic energy = m₁ m₂ (u₁ – u₂)²/ 2(m₁ + m₂)

If a body is dropped from a height h_o and it strikes the ground with velocity v_o and after inelastic collision it rebounds with velocity v1 and rises to a height h1, then

If after n collisions with the ground, the body rebounds with a velocity v_n and rises to a height h_n then

eⁿ= v_n / v_o = √hⁿ/ h^o

Two Dimensional or Oblique Collision

If the initial and final velocities of colliding bodies do not lie along the same line, then the collision is called two dimensional or oblique Collision.

In horizontal direction,

m₁u₁ cos α₁ + m₂u₂ cos α₂= m₁ v₁ cos β₁ + m₂ v₂ cos β₂

In vertical direction.

m₁u₁ sin α₁ – m₂ u₂ sin α₂ = m₁u₁ sin β₁ – m₂ u₂ sin β₂

If m₁ = m₂ and α₁ + α₂ = 90°

then β₁ + β₂ = 90°

If a particle A of mass m1 moving along z-axis with a speed u makes an elastic collision with another stationary body B of mass m₂