Obtain an expression for common velocity and loss in the kinetic energy for a moving body `m_(1)` colliding against another at rest.

Explanation

Let `v_(1i)` be the initial velocity of `m_(1)`.
Let 'v' be the common velocity for an imperfect collision
Applying the law of linear momentum,
`m_(1)v_(1i)=(m_(1)+m_(2))v`
i.e., `v=((m_(1))/(m_(1)+m_(2)))v_(1i)`

The loss in the kinetic energy on collision is
`DeltaK.E=(1)/(2)m_(1)v_(1i)^(2)(1)/(2)(m_(1)+m_(2))v^(2)=(1)/(2)m_(1)v_(1i)^(2)(1)/(2)(m_(1)+m_(2))((m_(1))/(m_(1)+m_(2)))^(2)v_(1i)^(2)`
i.e., `DeltaKE=(1)/(2)m_(1)v_(1i)^(2)(1)/(2)(m_(1)^(2)v_(1i)^(2))/(m_(1)+m_(2))=(1)/(2)m_(1)v_(1i)^(2)[1-(1)/(2).(m_(1))/(m_(1)+m_(2))]`
`=(1)/(2)m_(1)v_(1i)^(2)[(2cancelm_(1)+2m_(2)2cancelm_(1))/(2(m_(1)+m_(2)))]`
i.e., `DeltaKE=(1)/(2)((m_(1)m_(2))/(m_(1)+m_(2)))v_(1i)^(2)`.