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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. Keys to Applicability

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

|| Page Contents ||
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h2. Assumed Knowledge


h4. Prior Models

* [Point Particle Dynamics]

h4. Vocabulary

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

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h2. Model Specification

h4. System Structure

*[Constituents|system constituent]:*  System is composed of [Point particles|point particle].
{note}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.
\\

h4. Descriptors

*[State Variables|state variable]:*   Mass (_m{_}{^}j^) and velocity (_v{_}{^}j^) for each object or momentum (_p{_}{^}j^) for each object inside the system.

*[Interaction Variables|interaction variable]:*   External forces (_F{_}{~}ext,k~) or, alternately, impulses may be specified (_J{_}{~}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}\[ \vec{p}^{\:j} = m^{j}\vec{v}^{\:j}\]\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. Mathematical Statement of the Model


{latex}
 Table Row (tr)
valigntop
 Table Cell (td)
Introduction to the Model
Description and Assumptions
 Excerpt
hiddentrue
System: Any. — Interactions: Any. — Note: Linear momentum evolves separately from angular momentum, so all system constituents are treated as point particles in this model.

 This model is generally applicable (assuming knowledge of the external forces and system constituents). The 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 such as collisions (impulse will be easier to determine than force).
Learning Objectives
Students will be assumed to understand this model who can:
Relevant Definitions

 Latex
\begin{large}\[ \vec{p} = m\vec{v}\]\end{large}
S.I.M. Structure of the Model
Compatible Systems
The system must be effectively composed of point particles, 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.
Relevant Interactions
Only external forces need be considered, since internal forces do not change the system's momentum.
Laws of Change
Mathematical Representation
 Section
 Column
 
Differential Form
 

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

{latex}\
 Column
width40px
 Column
 
Integral Form
 

 Latex
\begin{large}\[ \vec{p}
^{\:\rm sys}
_{f} = \vec{p}
^{\: \rm sys}
_{i} + \sum
_{k=1
 \vec{J}^{
N_{F}} \vec{J
\rm ext}_{
ext,k
fi} = \vec{p}
^{\:\rm sys}
_{i} + \int
 \sum
_{t_{
k=1
i}}^{
N
t_{
F
f}} \sum \vec{F}
_
^{\rm ext
,k
}\:dt  \]
\end{large}
{latex}

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h2. Relevant Examples

{contentbylabel:constant_momentum,momentum_force,impulse|showSpace=false|showLabels=true|excerpt=true|operator=OR|maxResults=50}

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{search-box}
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| !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. |
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Diagrammatic Representations
Relevant Examples
 
 Toggle Cloak
idconst
 Examples Involving Constant Momentum
 
 Cloak
idconst
 falsetruetrue50constant_momentum
 
 Toggle Cloak
idimp
 Examples Involving Impulse
 
 Cloak
idimp
 falsetruetrue50impulse
 
 Toggle Cloak
idoned
 Examples Involving 1-D Collisions
 
 Cloak
idoned
 falsetruetrue501d_collision
 
 Toggle Cloak
idtwod
 Examples Involving 2-D Collisions
 
 Cloak
idtwod
 falsetruetrue502d_collision
 
 Toggle Cloak
idelas
 Examples Involving Elastic Collisions
 
 Cloak
idelas
 falsetruetrue50elastic_collision
 
 Toggle Cloak
idinel
 Examples Involving Totally Inelastic Collisions
 
 Cloak
idinel
 falsetruetrue50totally_inelastic
 
 Toggle Cloak
idflux
 Examples Involving Continuous Momentum Flux
 
 Cloak
idflux
 falsetruetrue50momentum_force
 
 Toggle Cloak
idall
 All Examples Using this Model
 
 Cloak
idall
 falsetruetrueOR50constant_momentum,momentum_force,impulse
 
 
 Search Box

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