Two spheres A and B move on a smooth horizontal surface with the same velocity V and have some separation between them. A third sphere C is moving in opposite direction on the same surface with same speed. All the spheres are of equal mass. The collisions are elastic. $V_{C M}$ represents the centre of mass velocity of all the three spheres. Column II represents the values after all the possible impacts have occurred.

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Solution

$V_{c m} = \frac{m v + m v - m v}{3 m} = \frac{V}{3}$
after all collision, $V_{c m} = \frac{- m v + m v + m v}{3 m} = \frac{V}{3}$
in figure (1) momentum of B+C is
$m v - m v = m v + m v 0 \neq 2 m v$
momentum root conserve for system A+B momentum is mv+mv=-mv+mv
2mv $\neq$ 0 momentum not conserve.
(B) Similarly as in part A.
(C) In this situation, A and B continue to change their directions in opposite each other. This is also a condition like part A.But in this case momentum of the system (B+C) is conserved for some time as fig (7) and (8). After second collision no 7 AB remains at zero momentum.
(D)This is a condition like part C. Only string converted into spring rest of all part same.

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