Find the increment of the kinetic energy of the closed system comprising two spheres of masses `m_1` and `m_2` due to their perfectly inelastic collision, if the initial velocities of the sphere were equal to `v_1` and `v_2`.

Explanation

For perfectly inelastic collision, in the C.M. frame, final kinetic energy of the colliding system (both spheres) becomes zero. Hence initial kinetic energy of the system in C.M. frame completely turns into the internal energy (Q) of the formed body. Hence
`Q=overset~T_i=1/2mu|vecv_1-vecv_2|^2`
Now from energy conservation `DeltaT=-Q=-1/2mu|vecv_1-vecv_2|^2`,
In lab frame the same result is obtaiend as
`DeltaT=1/2((m_1vecv_1+m_2vecv_2)^2)/(m_1+m_2)-1/2m_1|vecv_1|^2+m_2|vecv_2|^2`
`=-1/2mu|vecv_1-vecv_2|^2`