Conservation of Linear Momentum

Newton's second law says force is the rate of change of momentum. Apply that to two objects pushing on each other, a rifle and a bullet, two skaters, a car and a wall. Newton's third law guarantees the force each exerts on the other is equal and opposite at every instant. Equal and opposite forces, applied over the same time interval, deliver equal and opposite impulses. Since impulse is change in momentum, one object's momentum gain is the other's exact loss. Add them up and the total never moves. That is conservation of momentum: for an isolated system, with no net external force, total momentum before equals total momentum after, even though the individual pieces can end up flying in wildly different directions at wildly different speeds.

The catch is the word isolated. The law only holds for the whole system of interacting objects, not for one piece of it. A rifle recoils because the only horizontal forces are the rifle on the bullet and the bullet on the rifle, an internal pair that cancels. A bouncing ball looks like it violates the law because its own momentum clearly reverses, but the ball alone is not isolated: the earth pushes on it during the bounce. Include the earth in the system and the books balance again, the earth just recoils by an immeasurably tiny amount.

This is why momentum problems get solved with a before-and-after sum rather than tracking forces through time, and it sets up collisions and explosions, where kinetic energy is free to change but total momentum is not negotiable.