Momentum

Mass times velocity, or, equivalently, a quantity whose time rate of change is equal to the net force applied to a system.

Motivation for Concept

Forces are actions which cause a change in the velocity of an object, but a given application of force will have very different results when applied to objects of very different mass. Consider the force imparted by a baseball player swinging a bat. When delivered to a baseball, the change in velocity is dramatic. A 95 mph fasball might be completely reversed and exit the bat moving 110 mph in the other direction. When delivered to a car, however, the change in velocity is miniscule. A car moving 95 mph will not be slowed noticeably by the action of a bat. Thus, although the change in velocity of a system is proportional to the force applied, it is not equal to the force applied. To define a quantity whose rate of change is equal to the force applied, we must include both the mass and velocity of the system subject to the force.

Mathematical Definition

Momentum of a Point Particle

The momentum (p) of a point particle with mass m and velocity v is defined as:

Momentum of a System

For a system composed of N objects which are approximated as point particles with their position specified by the objects' centers of mass, the system momentum is defined as the vector sum of the momentum of the constituents:

This definition is completely equivalent to

where M^sys is the total mass of the system and v^CM is the velocity of the system's center of mass.

Momentum and Newton's Laws

Momentum and Newton's Second Law

One way of stating Newton's Second Law is that the rate of change of a system's momentum is equal to the vector sum of the forces applied to the object:

Momentum and Newton's Third Law

By Newton's 3rd Law, internal forces cancel from the vector sum above, leaving only the contribution of external forces:

Momentum and Impulse

The integrated change in momentum can be found explicitly by using the net external impulse (J^ext):

Conservation of Momentum

Conditions for True Conservation

In the absence of a net external force, the momentum of a system is constant:

This equation is normally broken up to explicitly show the system constituents and the vector components:

Approximate Conservation in Collisions

Because the change in momentum is proportional to the impulse, which involves a time integral, for instantaneous events:

For approximately instantaneous events such as collisions, it is often reasonable to approximate the external impulse as zero by considering a system composed of all the objects involved in the collision. The key to the utility of this assumption is that often during collisions the change in momentum of any individual system constituent being analyzed is dominated by the internal collision forces (the external forces make a negligible contribution to that constituent's change in momentum during the collision).