3.6 Linear Momentum

Linear momentum of an object is defined as the product of the object’s mass and its velocity. Momentum is generally represented by the letter "p." Since velocity is a vector quantity, momentum is also a vector, and has the same direction as velocity. The SI unit for momentum is kilograms times meters per second, or k× m/s.

The concept of momentum lets us generalize Newton’s second law of motion. Recall Newton’s second law: an object’s change of motion is proportional to both the magnitude of the applied force and the object’s mass. Equation 2.1, F = ma, was a translation of this law that was only valid for special cases: those in which the object’s mass was constant. However, utilizing the definition of momentum, the general equation is below:

For a constant mass, D p = mD v. Thus,

The Law of Conservation of Momentum states that, in a closed system, momentum is neither lost nor gained, but can be transfered. This law is derived from Newton’s second and third laws of motion. Suppose that there are two interacting particles, A and B. The second law states that their interacting forces are equal to D p/D t. According to Newton’s third law, the forces between them are equal and opposite, or

Adding the changes in momentum of A and B together would give us the change in momentum of the system, which, in this case, is 0. Say we add a third interactive particle. The change in momentum of the system would remain 0, because an increase in momentum in one particle would be compensated by an equal decrease in another (or in the other two). Thus, the Law of Conservation of Momentum is true in a closed system.