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

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

{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. Prerequisite Knowledge

h4. Prior Models

* [1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]

h4. Vocabulary

* [position (one-dimensional)]
* [velocity]
* [acceleration]

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h2. System

h4. Constituents

A single [point particle|point particle] (or a system treated as a point particle with position specified by the center of mass).

h4. State Variables

Time (_t_), position (_x_) , and velocity (_v_).

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h2. Interactions

h4. Relevant Types

Some time-varying external influence that is confined to one dimension.

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}\[ \frac{dv}{dt} = a\]\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}\[ v(t) = v(t_{0})+\int_{t_{0}}^{t} a\;dt\]\end{large}{latex}\\
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{latex}\begin{large}\[ x(t) = x(t_{0})+\int_{t_{0}}^{t} v\;dt\]\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

None yet.
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