The conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. For two objects moving in one dimension (along a straight line), the momentum of each object can be expressed as its mass times its velocity (p = mv).

A sign convention can be used to distinguish between the direction of motion of the objects. For example, we can assign a positive direction to the right and a negative direction to the left. Then, if an object is moving to the right, its velocity is positive, and if it is moving to the left, its velocity is negative.

Using this sign convention, the momentum before the collision can be expressed as the sum of the momenta of the two objects: p1 + p2 = (m1v1) + (m2v2). After the collision, the momenta of the objects change, but the total momentum remains constant: p1′ + p2′ = (m1v1′) + (m2v2′) = p1 + p2.

By substituting the expressions for the momenta of the objects before and after the collision, we can solve for their velocities after the collision and determine how they change as a result of the collision.