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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. KeysDescription toand ApplicabilityAssumptions

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

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h2. AssumedPrerequisite Knowledge


h4. Prior Models

* [Point Particle Dynamics]

h4. Vocabulary

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

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h2. Model SpecificationSystem

h4. System StructureConstituents

*[Constituents|The system constituent]:*  System ismust be effectively composed of [Pointpoint particles|point particle].
{note}Rigid, 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.{note}
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*[Interactions|interaction]:*   Only [external forces|external force] need be considered, since [internal forces|internal force] do not change the system's momentum.
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h4. Descriptors

*[State Variables|state variable]:*    and Parameters

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

*[Interaction Variables|interaction variable]:*   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~ext^~k~) or, alternately, impulses may be specified (_J{_}{~^}ext,k~ext^~k~).  

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h2. Model Equations

h4. Relationships Among State Variables

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

{latex}\begin{large}\[ \vec{p}^_{\:j} = m^m_{j}\vec{v}^_{\:j}\]\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. MathematicalLaws Statement of the ModelChange

h5. Differential Form
{latex}\begin{large}\[ \frac{d\vec{p}^_{\:\rm sys}}{dt} = \:\sum_{k=1}^{N_{F}} \vec{F}^{ext}_{ext,k} \]\end{large}{latex}
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or, alternately:

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

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h2. Relevant Examples

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