What is the formula to prove collision is elastic or inelastic?

The formula to determine whether a collision is elastic or inelastic is the coefficient of restitution (e). This coefficient is defined as the ratio of the relative velocity of separation to the relative velocity of approach of two colliding objects.

If the coefficient of restitution is equal to 1, then the collision is considered to be perfectly elastic, meaning that kinetic energy is conserved during the collision. If the coefficient of restitution is less than 1, then the collision is considered to be inelastic, meaning that kinetic energy is not conserved and some of it is lost in the form of heat, sound, or deformation.

Elastic Collision:

- Momentum is conserved: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
- Kinetic energy is conserved: 1/2 m₁v₁² + 1/2 m₂v₂² = 1/2 m₁v₁'² + 1/2 m₂v₂'²

Inelastic Collision:

- Momentum is conserved: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
- Kinetic energy is not conserved: 1/2 m₁v₁² + 1/2 m₂v₂² > 1/2 m₁v₁'² + 1/2 m₂v₂'²

Coefficient of Restitution (e):

If the collision is perfectly elastic, e = 1. If the collision is perfectly inelastic, e = 0. For all other cases, 0 < e < 1.

e = (v₁' - v₂') / (v₁ - v₂)

where:

- v₁ is the initial velocity of the first object
- v₁' is the final velocity of the first object
- v₂ is the initial velocity of the second object
- v₂' is the final velocity of the second object