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

Explanation

In a collision, the total initial kinetic energy of the bodies (before collision) is not equal to the total final kinetic energy of the bodies (after collision) then, it is called as 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_(i)=1/2m_(1)u_(1)^(2)+1/2m_(2)u_(2)^(2)" ...(1)"`
Total kinetic energy after collision,
`KE_(f)=1/2(m_(1)+m_(2))v_(2) " ...(2)"`
Then the loss of kinetic energy is
Loss of KE, `DeltaQ=KE_(i)-KE_(f)`
`=(1/2m_(1)u_(1)^(2)+1/2m_(2)u_(2)^(2))-1/2(m_(1)+m_(2))v^(2) " ...(3)"`
Substituting equation
`" "v=(m_(1)u_(1)+m_(2)u_(2))/(m_(1)+m_(2))`
in equation (3), 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)`.