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# Conservation of momentum

## Conservation of momentum definition

The principle of conservation of momentum states that the total momentum of all objects in a closed system is constant.

A closed system is a system that cannot exchange matter with its surroundings (i.e. the total mass of the system is constant).

Take a gun and a bullet as a closed system. Before shooting, the total momentum is zero. When the bullet is shot (forward momentum), the gun needs to recoil (backward momentum) to keep the system at zero momentum.

The conservation of momentum is closely related to Newton's Third law of motion ("Forces always occur in equal and opposite pairs").

If the sum of all action and reaction forces at one point is zero, the total change of momentum of the system is also zero.

Momentum is transferred from one ball to the next when one of the outermost balls collides with an inner one. There is only enough momentum for one ball to move at any one time.

## Momentum conservation in collisions

The principle of conservation of momentum applies to collisions. The total momentum is the same before and after the impact: $$m_1\vecphy{u}_1+m_2\vecphy{u}_2=m_1\vecphy{v}_1+m_2\vecphy{v}_2$$ $\vecphy{u}_{1/2}$ and $\vecphy{v}_{1/2}$ are the initial and final momenta of a particle $1/2$.

In a collision of a stationary and a moving ball, the total momentum of the system is non-zero (because one ball is moving). It will remain that way unless an external force interferes.

Momentum is transferred when billiard balls strike each other. However, the total momentum is conserved.