- #1

bobdavis

- 19

- 8

In a closed system consisting of a set of particles not at rest relative to each other and acting on each other only by classical mechanical collision (i.e. billiard balls model, not including gravity or other long-range interactions), does conservation of momentum imply that the system will expand outward forever? If so how would I show this and if not can a counterexample be provided?

EDIT: In particular I mean elastic collisions, so not just momentum but also kinetic energy is conserved, and the "not at rest relative to each other" criteria is just non-zero kinetic energy, otherwise a counterexample would be two particles of opposite momentum in inelastic collision such that they both stop.

Attempt at answering: my intuition is that a component of the momentum of the center of mass might at a given time correspond to the kinetic energy of particles that are displaced from the center of mass in the opposite direction as that component, but because momentum and kinetic energy are both conserved, this kinetic energy will eventually propagate in the direction of that component until it reaches the other side, i.e. until its displacement from the center of mass is in the same direction as the component of momentum, and once this is true for all the kinetic energy in the system then you would just have a radially expanding system with particles heading away from the origin forever.

Two examples to clarify:

1) Two particles displaced from the center of mass in the same direction as their momentum: the particles simply drift away from each other forever.

2) Two particles displaced from the center of mass in the opposite direction as their momentum: the particles collide at the center of mass, but kinetic energy and momentum are conserved so the particles bounce off each other and now the situation is the same as (1) and so the particles now drift away from each other forever.

EDIT: In particular I mean elastic collisions, so not just momentum but also kinetic energy is conserved, and the "not at rest relative to each other" criteria is just non-zero kinetic energy, otherwise a counterexample would be two particles of opposite momentum in inelastic collision such that they both stop.

Attempt at answering: my intuition is that a component of the momentum of the center of mass might at a given time correspond to the kinetic energy of particles that are displaced from the center of mass in the opposite direction as that component, but because momentum and kinetic energy are both conserved, this kinetic energy will eventually propagate in the direction of that component until it reaches the other side, i.e. until its displacement from the center of mass is in the same direction as the component of momentum, and once this is true for all the kinetic energy in the system then you would just have a radially expanding system with particles heading away from the origin forever.

Two examples to clarify:

1) Two particles displaced from the center of mass in the same direction as their momentum: the particles simply drift away from each other forever.

2) Two particles displaced from the center of mass in the opposite direction as their momentum: the particles collide at the center of mass, but kinetic energy and momentum are conserved so the particles bounce off each other and now the situation is the same as (1) and so the particles now drift away from each other forever.

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