c
(c)The momentum of the two-particle system, at \(t = 0\) is
\({\vec P_i} = {m_1}{\vec v_1} + {m_2}{\vec v_2}\)
Collision between the two does not affect the total momentum of the system.
A constant external force \(({m_1} + {m_2})g\) acts on the system.
The impulse given by this force, in time \(t = 0\)  to \(t = 2{t_0}\) is \(({m_1} + {m_2})g \times 2{t_0}\)
 |Change in momentum in this interval
\( = \,|{m_1}\vec v{'_1} + {m_2}\vec v{'_2} - ({m_1}{\vec v_1} + {m_2}{\vec v_2})|\, = 2({m_1} + {m_2})g{t_0}\)