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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 *by considering a system composed of all the objects involved in the collision*.  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.  It is also completely incorrect to say that the momentum of each _object_ is conserved.  Only the _system_ momentum is (approximately) conserved.{warning}