# Conservation of momentum in a closed system

• B
bobdavis
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.

Last edited:

Homework Helper
Gold Member
2022 Award
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?
The system is effectively a gas of particles and must exert some pressure on the walls of any container. If there is no container, then the pressure must gradually decrease and the volume increase. In other words, if the gas has internal KE then it cannot be pressureless.

That's one way to look at it.

• vanhees71 and bobdavis
bobdavis
I see. When I posted the question I was just thinking in terms of a regular mechanical system and I didn't make the connection to kinetic theory of gases but now I know where to look for more insight, thank you

• vanhees71, berkeman and PeroK
bobdavis
If the particles are all identical point particles with equal mass, can this be extended to a stronger statement that the evolution of the system is described by the rays extending from the initial velocity vectors of the particles? i.e. for each particle with initial position p and initial velocity v, at any time t > 0 there will be a particle with position p+vt and velocity v ? ex.. if the particle never collides with any other particles, then this particle will satisfy the criteria at every time t, and if the particle does collide with another particle then second particle will acquire the velocity of the first and traverse the next section with the same velocity v until the next collision and so on (the same as if the particles had just passed through each other without interacting, so the point of collision is just a point of intersection of the trajectory rays)