# Conservation of momentum in a closed system

• B
• bobdavis
In summary: Yes, this is correct.In summary, conservation of momentum implies that the system will expand outward forever.

#### 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.

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bobdavis said:
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
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
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)

## 1. What is the law of conservation of momentum in a closed system?

The law of conservation of momentum states that the total momentum of a closed system remains constant, meaning that the total amount of momentum before and after an interaction or event within the system is the same.

## 2. How does momentum differ from velocity?

Momentum is a measure of the quantity of motion of an object, taking into account both its mass and velocity. Velocity, on the other hand, is simply the speed and direction of an object's motion.

## 3. What is an example of conservation of momentum in a closed system?

A common example of conservation of momentum in a closed system is a collision between two objects. In this scenario, the total momentum of the system before the collision is equal to the total momentum after the collision, even though the individual momenta of the objects may change.

## 4. What happens to the momentum of an object in a closed system if there are no external forces acting on it?

In a closed system, the total momentum remains constant. This means that if there are no external forces acting on an object, its momentum will not change and it will continue to move with the same velocity.

## 5. How does the conservation of momentum relate to Newton's third law of motion?

Newton's third law states that for every action, there is an equal and opposite reaction. In terms of momentum, this means that the total momentum of a closed system is conserved because the momentum lost by one object is gained by another, resulting in a net change of zero.