SCN3797-ABED_1: 3.5 The Law of Conservation of Momentum

For any isolated system, the total momentum does not change. In a collision, momentum is conserved, the total momentum before the collision is equal to the total momentum after the collision. Expressed as an equation, the law of conservation of momentum is:

$\sum_{} {\vec{p}}_{i n t i a l} = \sum_{} {\vec{p}}_{f i n a l}$

This reads as the sum of the initial momentum equals the sum of the final momentum.

You may see this principle expressed in other ways as well. The following shows other ways to express the conservation of momentum law:

Conservation of Momentum: the sum of the initial momentum equal the sum of the final momentum

$\sum_{} {\vec{p}}_{i n t i a l} = \sum_{} {\vec{p}}_{f i n a l}$

The sum of the initial momenta equals the sum of the final momenta

$\sum_{} {\vec{p}}_{i n t i a l} = \sum_{} {\vec{p}}_{f i n a l}$

${\vec{p}}_{1 i} + {\vec{p}}_{2 i} = {\vec{p}}_{1 f} + {\vec{p}}_{2 f}$

Or, the change in the momenta of object 1 is equal but opposite to the change in the momenta of object 2

${\vec{p}}_{1 f} - {\vec{p}}_{2 i} = - ({\vec{p}}_{2 f} - {\vec{p}}_{2 i}) ∆ {\vec{p}}_{1} = - ∆ {\vec{p}}_{2}$

The law of conservation of momentum governs all physical interactions; it is considered to be one of the fundamental laws of physics. The law has been used to investigate and analyze all types of interactions including vehicular accidents, subatomic particle interaction, and the study of planets, stars, and galaxies.

This law controls the universe by keeping the total quantity of motion in the universe fixed. That is, if one body slows down and comes to rest, another body must speed up and start moving. Newton's cradle of swinging masses is a good example of this, as you can see in the picture on the right.

The total momentum of a system is conserved, as long as there are no external forces acting on a system. The law of conservation of momentum will be used to analyze various collisions. When solving problems, it is useful to follow the GRASP model for problem-solving. (You may wish to review the GRASP model on page 867 of the textbook.)

Self-Check 1

A 5.0-kg object is travelling to the right at 10.0 m/s. It collides with a 7.0-kg object that initially is at rest. After the collision, the 5.0-kg object continues to move to the right, but at 1.17 m/s. In what direction and with what speed is the other object moving? You may verify your answers, where possible, by using the Collision 1D Simulator.