(a) Explain perfect inelastic collision and derive an expression for loss of kinetic energy in perfect inelastic collision.

Explanation

In a perfectly inelastic or completely inelastic collision, the objects stick together permanently aft er collision such that they move with common velocity. Let the two bodies with masses `m_(1) "and" m_(2)` move with initial velocities `u_(1) "and" u_(2)` respectively before collision. Aft er perfect inelastic collision both the objects move together with a common velocity v as shown in figure.
Since, the linear momentum is conserved during collisions,

The common velocity can be computed by
`v = (m_(1)u_(1) + m_(2)u_(2))/(m_(1) + m_(2))` ...(1)
Loss of kinetic energy in perfect inelastic collision:
In perfectly inelastic collision, the loss in kinetic energy during collision is transformed to another form of energy like sound, thermal, heat, light etc. Let `KE_(i)` be the total kinetic energy before collision and `KE_(f)` be the total kinetic energy after collision.
Total kinetic energy before collision,
`KE_(e) = 1/2 m_(1)u_(1)^(2) + 1/2 m_(2)u_(2)^(2)` ...(2)
Total kinetic energy after collision,
`KE_(f) = 1/2 (m_(1) + m_(2))v^(2)` ...(3)
Then the loss of kinetic energy is Loss of `KE, DeltaQ = EK_(f) - KE_(i)`
= `1/2 (m_(1) + m_(2)) v^(2) - 1/2 m_(1)u_(1)^(2) - 1/2 m_(2)u_(2)^(2)` ...(4)
Substituting equation (1) in equation (4), and on simplifying (expand v by using the algebra `((a + b)^(2) = a^(2) + b^(2) + 2ab)`, we get
Loss of KE, `DeltaQ = 1/2 ((m_(1)m_(2))/(m_(1)+ m_(2))) (u_(1) – u_(2))^(2)` ...(5)