Two particles of masses m1 and m2 in projectile motion have velocities v→1 and v→2 respectively at time t = 0. They collide at time t0. Their velocities become v→1′ and v→2' at time 2t0 while still moving in air. The value of m1v→1′+m2v2→′−m1v1→+m2v2→ is

The momentum of the two-particle system, at t = 0 isP→i=m1v→1+m2v→2Collision between the two does not affect the total momentum of the system.A constant external force m1+m2g acts on the system. The impulse given by this force, in time t = 0  to t=2t0 is m1+m2g×2t0∴ |Change in momentum in this interval =m1v→1+m2v→2−m1v→1+m2v2→∣=2m1+m2gt0