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h1. Momentum

{excerpt}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 any 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}

{info}When physicists discuss the "law" or "principle" of [conservation] of momentum, they are _assuming_ (or defining?) that the universe is an _isolated system_ (it cannot be subject to external forces).{info}

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}\[ \lim_{t_{f}\rightarrow t_{i}} \int_{t_{i}}^{t_{f}} F^{\rm ext} \:dt = 0 \]\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 such an assumption is if 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_).

{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}

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