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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. Description and Assumptions

{excerpt}This model is [generally applicable|generally applicable model] (assuming knowledge of the external forces and system constituents).{excerpt}

h2. Problem Cues

This model is especially useful when describing the momentum of systems where external forces are absent (system momentum will be constant) or estimating the force in a process that occurs in a very short time interval as in collisions (impulse will be easier to determine than force).

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h2. Prerequisite Knowledge

h4. Prior Models

* [Point Particle Dynamics]

h4. Vocabulary

* [system]
* [force]
* [impulse]
* [momentum]
* [velocity]

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h2. System

h4. Constituents

The system must be effectively composed of [point particles|point particle], though rigid bodies may be treated as point particles with positions specified by the center of mass positions of the rigid body when this model is used.

h4. Variables and Parameters

Mass (_m{_}{~}j~) and velocity (_v{_}{~}j~) for each object or momentum (_p{_}{~}j~) for each object inside the system.

h2. Interactions

h4. Relevant Types

Only [external forces|external force] need be considered, since [internal forces|internal force] do not change the system's momentum.

h4. Interaction Variables

External forces (_F{_}{^}ext^~k~) or, alternately, impulses may be specified (_J{_}{^}ext^~k~).  

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h2. Model

h4. Relationships Among State Variables

If not directly given, momenta can be obtained using the definition:

{latex}\begin{large}\[ \vec{p}_{j} = m_{j}\vec{v}_{j}\]\end{large}{latex}

The relationship implied by the model is most easily expressed in terms of the *system momentum*, which is the vector sum of the constituent momenta.  For a system composed of _N_ point particles:

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

{warning}The number of point particle constituents in the system is not necessarily fixed.  A [totally inelastic collision], for example, could be viewed as a process where two separate system constituents exist in the initial state, but only one is present in the final state.{warning}
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h4. Laws of Change

h5. Differential Form
{latex}\begin{large}\[ \frac{d\vec{p}_{\rm sys}}{dt} = \:\sum_{k=1}^{N_{F}} \vec{F}^{ext}_{k} \]\end{large}{latex}
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h5. Integral Form
{latex}\begin{large}\[ \vec{p}_{\rm sys}_{f} = \vec{p}_{\rm sys}_{i} + \sum_{k=1}^{N_{F}} \vec{J}^{ext}_{k} = \vec{p}_{\rm sys}_{i} + \int \sum_{k=1}^{N_{F}} \vec{F}^{ext}_{k}\:dt  \]
\end{large}{latex}

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h2. Relevant Examples

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