Prove that there is a loss of KE during one dimensional inelastic collision.

Explanation

In a 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 enery before collision and `KE_(f)` be the total kinetic energy after collision.
Total kinetic energy before collision,
`KE_(i)=(1)/(2)m_(1)u_(1)^(2)+(1)/(2)m_(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_(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)" "...(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)" "...(4)`