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There are three blocks A,B and C, with B and C at rest connected by a light spring and A moving with a velocity v, all three lying on the same horizontal frictionless surface. After some time A collides elastically with B and we are required to find the maximum compression in the spring.

The problem is that in the solution they first consider the elastic collision of A and B separately and apply the momentum conservation with A and B as the system and form an equation using coefficient of restitution. Then using that velocity they do the further calculation.

I am unable to understand that how can conservation of momentum be used here as the instant A and B collide B will start accelerating and start compressing the spring therefore an external force i.e. the spring force will come into play. And can we use the coefficient of restitution as wouldn't there be change in the velocity of separation due to spring force.

Similarly when a third ball collides elastically with two balls in contact,all three of the same mass, then the ball at the extreme end gets all the velocity and in the justification the collisions are considered separately.

Conclusion: is it justified to use momentum conservation in the case of those three blocks? Can we consider the colliding bodies separately neglecting all the other bodies and their effect that they are in contact with? Can we use the coefficient of restitution to formulate equations without considering the effect of the rest of the bodies in the system?

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  • $\begingroup$ Spring force is non impulsive. $\endgroup$ – RunMachine_Kohli Oct 5 '19 at 14:16
  • $\begingroup$ Can you please elaborate. $\endgroup$ – Utkarsh Jha Oct 5 '19 at 14:30
  • $\begingroup$ And if it is not impulsive, yet won't it start acting the moment it is compressed? And what about the balls? $\endgroup$ – Utkarsh Jha Oct 5 '19 at 14:31
  • $\begingroup$ Spring never reacts instantaneously $\endgroup$ – RunMachine_Kohli Oct 5 '19 at 14:53
  • $\begingroup$ So can we use the analogy of springs in case of the collision between the balls? $\endgroup$ – Utkarsh Jha Oct 5 '19 at 15:54
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The collision between A and B is instantaneous. When it has finished B has moved only an infinitesimal distance towards C so the spring is not yet compressed and there is no spring force acting on B.

The coefficient of restitution between A and B can be used for the same reason : there is no spring force at this instant.

When a ball A collides with 2 other balls B, C which are already in contact we have a 3 body problem. These are notoriously more difficult to solve because B collides with C while A is still in contact with B.

If the balls are assumed to be rigid bodies then they do not deform at all and have an infinite coefficient of elasticity (eg Young's Modulus). Such problems are ambiguous or indeterminate in the general case. In real life the balls have a finite coefficient of elasticity and if this is known the final velocities after the 3-body collision can be calculated.

If there is little or no loss of kinetic energy during the collision and the balls are identical, the outcome is that ball A comes to rest and ball C moves off with the initial velocity of A.

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