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

h4. Description and Assumptions

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.

h4. Problem Cues

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).


h2. Model


h4. {color:red}Compatible Systems{color}

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

h4. {color:red}Relevant Interactions{color}

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

h4. {color:red}Laws of Change{color}

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h5. Differential Forms

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

h4. {color:red}Diagrammatic Representations{color}
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These graphs show the *position*, *velocity* , and *acceleration* for the motion of a particle for which the equation of motion is
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{latex}\begin{large} \[x = -0.1t^{3} -t^{2} + 30t -100 \]\end{large}{latex}
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|!Position vs Time Graph.bmp!|
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Consequently the *velocity*, which is the derivative of the position with respect to time, is given by
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{latex}\begin{large} \[x = -0.3t^{2} -2t + 30 \]\end{large}{latex}
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|!Velocity vs Time Graph.bmp!|
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and the *acceleration* is given by the second derivative of the position with respect to time:
\\
{latex}\begin{large} \[x = -0.6t -2 \]\end{large}{latex}
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|!Acceleration vs Time Graph.bmp!|
\\[position versus time graph]
* [velocity versus time graph]
* acceleration versus time graph
* [motion diagram]

h2. Relevant Examples

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

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