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h2. Description
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*Hierarchy of Models* [Model Hierarchy]*
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h2. Description and Assumptions

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This model is appliesapplicable to a single [point particle] subject to an external forceacceleration that is constrained to one dimension and which is either parallel to or antiparallelanti-parallel to the particle's initial velocity.  {excerpt}

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{warning}This model is rarely used (see "Problem Cues" below).  Before trying to work out the integrals, please check if any of the sub-models are applicable.{warning}

h2. Problem Cues

In practice, this model is only useful when a one-dimensional acceleration is given that has a _known_ time dependence that is _not_ sinusoidal.  If the acceleration is constant, the sub-model [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant Acceleration)] should be used.  If the acceleration is sinusoidal (described by a sine, cosine, or sum of the two), the sub-model [Simple Harmonic Motion] should be used.  Thus, in practice, the problem cue for this model is that the acceleration will be given as an explicit and integrable function of time, most often a polynomial (the acceleration might also be plotted as a linear function of time).  

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h2. AssumptionsPrerequisite and LimitationsKnowledge


h4. Prior Models

* Link to model pages that should be learned before this model.[1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]

h4. Vocabulary

* [frame of reference]
* [position (one-dimensional)]
* [velocity]
* [acceleration]

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h2. Model Specification


h4. *System Structure*


h4.  Constituents

*Internal Constituents:* Point particle.

*Environment:* external agents interacting with the particle which are the responsible of the *real* forces acting *on* the particle. The total external force must be parallel or anti-parallel to the paticle's initial velocityA single [point particle|point particle] (or a system treated as a point particle with position specified by the center of mass).

h4. Descriptors

*ObjectState Variables:* none

*State Variables:* Time (_t_), position  (_x_) , and velocity (_v_).

*Interaction Variables:* acceleration _a_.----
h2. Interactions

h4. LawsRelevant of InteractionTypes

AnySome time-varying external forceinfluence which magnitudethat is notconfined constant.to Possible forces are forces that depends on time, on the particle's position or on the particle's velocity.
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h4. Interaction Variables

Acceleration (_a_(_t_)).

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h2. Model

h4. Laws of Change 


Differential Forms:
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{latex}\begin{large}\[ \frac{dv}{large}$x(t_f) = x(t_i) + \int_{t_i}^{t_f} v(t)dt $ or in differential form $ v(t) = \frac{dx(t)}{dt} $ dt} = a\]\end{large}{latex}\\
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{latex}\begin{large}\[ \frac{dx}{dt} = v\]\end{large}{latex}\\
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Integral Forms:
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{latex}\begin {large}$v\[ v(t_f) = v(t_i{0}) +\int_{t_i{0}}^{t_f} a\;dt\]\end{large}{latex}\\
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{latex}\begin{large}\[ x(t)dt$ or in differential form $ a(t) =\frac{dv(t)}{dt}$ = x(t_{0})+\int_{t_{0}}^{t} v\;dt\]\end{large}{latex}
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h2. Diagrammatical Representations

* Acceleration versus time graph.
* Velocity versus time graph.
* Position versus time graph.

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h2. Relevant Examples

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