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{excerpt}Mass times velocity, or, alternately, a quantity whose time rate of change is proportional to the net force applied to an object.{excerpt}

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h2. Motivation for Concept

[Forces|force] are actions which cause a change in the [velocity] of an object, but a given 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.

h2. Fundamental Properties

h4. Definition

The momentum (_p_) of an object with mass _m_ and velocity _v_ is defined as:

{latex}\begin{large}\[ \vec{p} \equiv m\vec{v}\]\end{large}{latex}

h4. Definition for System

For a system composed of _N_ objects, the system momentum is defined as the vector sum of the momentum of the constituents:

{latex}\begin{large}\[ \vec{p}^{system} = \sum_{j=1}^{N} m_{j}\vec{v}_{j} \]\end{large}{latex}

h4. Law of Interaction

The rate of change of a system's momentum is equal to the vector sum of the forces applied to the object:

{latex}\begin{large}\[ \frac{d\vec{p}^{\:system}}{dt} = \sum_{k=1}^{N_{F}} \vec{F}_{k} \] \end{large}{latex}

h4. Cancellation of Internal Forces

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

{latex}\begin{large}\[ \frac{d\vec{p}^{\:system}}{dt} = \sum_{k=1}^{N_{F}} \vec{F}^{ext}_{k} \] \end{large}{latex}


h4. Law of Change

The change in momentum can be found explicitly by using the net external [impulse] (_J_^ext^):

{latex}\begin{large}\[ \vec{p}^{\:system}_{f} - \vec{p}^{\:system}_{i} = \int_{t_{i}}^{t_{f}} \sum_{k=1}^{N_{F}} \vec{F}_{k}^{ext} \:dt \equiv \sum_{k=1}^{N_{F}} \vec{J}_{k}^{ext} \]\end{large}{latex}

h2. Conservation of Momentum

h4. Conditions for True Conservation

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

{latex}\begin{large}\[ \vec{p}_{f}^{system} = \vec{p}_{i}^{system}\]\end{large}{latex}

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

{latex}\begin{large}\[ \sum_{j=1}^{N} p^{j}_{x,f} = \sum_{j=1}^{N} p^{j}_{x,i} \]
\[ \sum_{j=1}^{N} p^{j}_{y,f} = \sum_{j=1}^{N} p^{j}_{y,i} \]
\[ \sum_{j=1}^{N} p^{j}_{z,f} = \sum_{j=1}^{N} p^{j}_{z,i} \]\end{large}{latex}

h4. Approximate Conservation in Collisions

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

{latex}\begin{large}\[ \lim_{t_{f}\rightarrow t_{i}} \int_{t_{i}}^{t_{f}} F^{ext} \:dt = 0 \]\end{large}{latex}

For approximately instantaneous events such as collisions, it is often reasonable to approximate the external impulse as zero.  The key to such an assumption is if 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).

{note}Note that "dominated" and "negligible" are terms whose precise definitions depend on the accuracy desired in the results.{note}

{warning}Before the collision occurs and after the collision is complete, the collision forces will usually drop to zero.  Neglecting external impulse can only be justified _during_ the collision.{warning}