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h1. Momentum

{excerpt}[Mass|mass] times [velocity], or, equivalently, a quantity whose time rate of change is equal to the net [force] applied to a [system].{excerpt}

h3. Motivation for Concept

[Forces|force] 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].

h3. Mathematical Definition

h4. Momentum of a Point Particle

The momentum (_p_) of a [point particle] with [mass] _m_ and [velocity] _v_ is defined as:

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

h4. Momentum of a System

For a [system] composed of _N_ objects which are approximated as [point particles|point particle] with their position specified by the objects' [centers of mass|center of mass], the [system] momentum is defined as the [vector] sum of the momentum of the [constituents|system constituent]:

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

This definition is completely equivalent to 

{latex}\begin{large}\[ \vec{p}^{\rm \: sys} = M^{\rm sys} \vec{v}^{\rm \: CM} \]\end{large}{latex}

where _M_^sys^ is the total mass of the [system] and _v_^CM^ is the [velocity] of the [system's|system] [center of mass].

h3. Momentum and Newton's Laws

h4. Momentum and Newton's Second Law

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

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

h4. Momentum and Newton's Third Law

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

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

h4. Momentum and Impulse

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

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

h3. Conservation of Momentum

h4. Conditions for True Conservation

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

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

This equation is normally broken up to explicitly show the [system constituents|system constituent] 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}\[ \lim_{t_{f}\rightarrow t_{i}} \int_{t_{i}}^{t_{f}} \vec{F}^{\rm ext} \:dt = 0 \]\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 the utility of this assumption is that often during collisions the change in momentum of any individual [system] [constituent|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_).

{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}