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{live-template:Left Column}{td}{td}{excerpt:hidden=true}{*}System:* One [point particle] constrained to move in one dimension. --- *Interactions:* Any that respect the one-dimensional motion. {excerpt}

h1. One-Dimensional Motion (General)


h4. {toggle-cloak:id=desc}Description and Assumptions

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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.
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h4. {toggle-cloak:id=cues}Problem Cues

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In practice, this model is only useful when a one-dimensional acceleration is given that has a _known_ time dependence. 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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h4. {toggle-cloak:id=pri} Prior Models

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

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h4. {toggle-cloak:id=vocab}Vocabulary

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* [position]
* [velocity]
* [acceleration]

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h2. Model


h4. {toggle-cloak:id=sys} {color:red}Compatible Systems{color}

{cloak:id=sys}
A single [point particle|point particle] (or a system treated as a point particle with position specified by the center of mass).
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h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}

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Some time-varying external influence that is confined to one dimension.
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h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

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h5. Differential Forms

{latex}\begin{large}\[ \frac{dv}{dt} = a\]\end{large}{latex}\\
\\
{latex}\begin{large}\[ \frac{dx}{dt} = v\]\end{large}{latex}\\
\\ {column}{column}

h5. Integral Forms

{latex}\begin{large}\[ v(t) = v(t_{i})+\int_{t_{i}}^{t} a\;dt\]\end{large}{latex}\\
\\
{latex}\begin{large}\[ x(t) = x(t_{i})+\int_{t_{i}}^{t} v\;dt\]\end{large}{latex}\\ {column}{section}
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h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

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

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h2. Relevant Examples

* [Accelerate, Decelerate]
* [An Exercise in Continuity]

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