Collisions of more than two bodies

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Discussion Overview

The discussion revolves around the complexities of analyzing collisions involving three or more bodies, particularly in the context of elastic and inelastic collisions in two dimensions. Participants explore the implications of conservation laws and the stability of the problem when considering virtual displacements to simplify the analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that the number of unknowns increases with each additional body in a collision, while the number of constraints remains limited, raising questions about the stability of the problem.
  • Another participant argues that the problem is not stable when considering small displacements, suggesting that material properties significantly influence the outcome if the collisions cannot be treated as independent two-body interactions.
  • A different viewpoint is presented regarding Newton's cradle, where one participant mentions that the dynamics are affected by the shape of the bodies, questioning whether the same principles apply when using cylinders instead of spheres.
  • It is suggested that treating the spheres in Newton's cradle as separated by a small distance allows for the use of two-body collision equations, although it is questioned whether this approach would yield the same results for cylinders.
  • Another participant agrees that the approach could work if the cylinders are indeed separated by a small distance.

Areas of Agreement / Disagreement

Participants express differing views on the stability of the collision problem and the applicability of two-body collision equations to different shapes of bodies. There is no consensus on whether the simplifications hold true across all scenarios discussed.

Contextual Notes

Participants highlight limitations related to the assumptions made about body shapes and separations, as well as the dependence on material properties, which remain unresolved in the discussion.

greypilgrim
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Hi.

The formulae for the velocities of two bodies after a perfectly elastic or inelastic bodies, let's say in 2D, (e.g. billiard) can be derived from three equations: conservation of energy and conservation of momentum in two dimensions.

But how do you treat collisions of three or more bodies? With each additional body the number of unknowns rises by two (velocity in x and y direction), but the number of constraints is still three.

Is the problem stable with respect to shifting the bodies by arbitrarily small (or virtual) displacements such that there are only two-body collisions to consider?
 
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greypilgrim said:
Is the problem stable with respect to shifting the bodies by arbitrarily small (or virtual) displacements such that there are only two-body collisions to consider?
It is not, and in general material properties will be highly relevant for the outcome if the overall process cannot be described as independent two-body collisions separated in time.
 
I once read that the dynamics of Newton's cradle is higly dependent on the shape of the bodies used, using cylinders instead of spheres will not show the same behaviour.

Elsewhere I read that one may treat Newton's cradle as if the spheres are separated by a small distance such that only two-body collisions occur. With this assumptions one can easily derive the observed dynamics by just using the equations for two-body collisions. So is this a coincidence and wouldn't work with cylinders?
 
Apparently not. It would work if the cylinders are actually separated a bit.
 

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