Conservation of Momentum

in Two-Particle Systems


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We have discussed conservation of momentum in the simplest case - a single-particle system. What happens in a two-particle system? Well, things get more interesting, but not, happily, more difficult. Here's the story.

The Theory:

Suppose we have a system consisting of object (particle) A and object B. Assume that these particles interact in some way - perhaps they collide, exert electrical , magnetic, or gravitational forces on one another, whatever - but A and B are not influenced by outside forces. Newton's Third Law says that if object A exerts a force (of any type) on object B (FBA), then B exerts an equal and opposite force on object A (FAB). So:


where the negative sign indicates that the forces are in opposite directions. Newton's Third Law also insists that these two forces occur at exactly the same time. This means that the impulse (force times time) that A exerts on B is equal and opposite to the impulse that B exerts on A.

ImpulseBA = FBA t = -FABt = -ImpulseAB

Finally, since the impulses are equal and opposite, B's change in momentum is equal and opposite to A's change in momentum:

Change in momentum of B = ImpulseBA = FBA t = -FABt = -ImpulseAB = -Change in momentum of A

This is conservation of momentum - whatever momentum is lost by A must be gained by B, and vice versa. This means that even though the momentum of A and B can increase or decrease (or change direction), the total (vector) momentum of both A and B must remain constant.

An Analogy:

Johnny and Suzie are playing marbles. Johnny starts with so many marbles, and Suzie starts with so many marbles. As they play, Johnny loses marbles and Suzie wins marbles, then Johnny wins marbles and Suzie loses marbles, and so on. As long as they don't lose their marbles (!) the total number of marbles in the game stays the same. However many marbles one player loses, the other player gains - and the total number of marbles stays the same.

Conservation of momentum is like that. One particle can lose momentum, but the other particle automatically gains that amount of momentum - what one loses, the other gains, what one gains, the other loses. One particle cannot gain momentum unless the other particle loses the same amount of momentum. There is only so much momentum in the system. The particles in the system can't create new momentum and they can't destroy momentum - they can only pass the existing momentum back and forth.

last update December 12, 2005 by JL Stanbrough