{table:border=1|frame=void|rules=cols|cellpadding=8|cellspacing=0}
{tr:valign=top}
{td:width=350|bgcolor=#F2F2F2}
{live-template:Left Column}
{td}
{td}

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}

 

{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 

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}

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

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.