Collisions

An instance of one moving object or person striking violently against another.

Elastic collision:

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms.

If two bodies of masses m₁ and m₂, moving with speeds u₁ and u₂ with u₁ > u₂ the same straight line, collide with the each other, their respective speeds v₁ and v₂ after collision are given by

By the law of conservation of momentum

          m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

          m₁ (u₁ – v₁) = m₂ (v₂ – u₂)        … (1)

By the law of conservation of kinetic energy
       

         m₁ (u₁² – v₁²) = m₂ (v₂² – u₂²)

         m₁ (u₁ + v_­1) (u₁ – v₁) = m₂ (v₂ + u₂) (v₂ – u₂)         … (2)

Divide the equation (2) by equation (1)

We get    

         u₁ + v₁ = v₂ + u₂        … (3)

         V₂ = u₁ + v₁ – u₂

Substitute this in the equation (1) we get

       m₁ (u₁ – v₁) = m₂ (u₁ + v₁ – u₂ – u₂)

       m₁u₁ – m₂u₁ + 2m₂u₂ = m₂v₁ + m₁v₁

     

Now from equation (3) we can write
       v₁ = u₂ + v₂ –u₁

Substitute this in the equation 1 and solve as done above then you will get the value of v₂ as

     

Inelastic collision:

An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. In collisions of macroscopic bodies, all kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.

If two bodies of masses m₁ and m₂, moving with speeds u₁ and u₂ along the same straight line, collide and stick together, the speed v of the composite body is given by

    

The kinetic energy of the system of bodies after collision is less than that before collision. The loss in kinetic energy appears in the form of heat and sound energy.

The coefficient of restitution (e):

The coefficient of restitution (COR) is a measure of the "restitution" of a collision between two objects: how much of the kinetic energy remains for the objects to rebound from one another vs. how much is lost as heat, or work done deforming the objects.

   

Suppose a body is dropped from a height h₀ and it strikes the ground with a speed v₀. Let, after the inelastic collision, the speed with which it rebounds be v₁ and h₁ the height to which it rises, then

    

If after n collisions with the ground, the speed is v_n and the height to which the body rises is h_n, then

       

Height after nth bounces is h_n = e^2π h₀.
Speed of rebound after nth bounce is . 

Total distance travelled before the body comes to rest