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Composition Setup
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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}

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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

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System: One point particle constrained to move in one dimension. — Interactions: Any that respect the one-dimensional motion. 
Introduction to the Model
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.
Learning Objectives
Students will be assumed to understand this model who can:
  • Choose the one graph possible velocity or acceleration vs. time graphs which corresponds to a model position versus time graph.
  • Differentiate position given as a polynomial function of time to find the corresponding velocity and acceleration.
  • Integrate the velocity or acceleration when given as a polynomial function of time along with appropriate initial conditions to find the functional form of the position.
S.I.M. Structure of the Model
Compatible Systems
A single point particle (or a system treated as a point particle with position specified by the center of mass).
Relevant Interactions
Some time-varying external influence that is confined to one dimension.
Laws of Change
Mathematical Representation
 Section
 Column
 
Differential Forms
 
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\begin{large}\[ \frac{dv}{dt} = a\]\end{large}
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\\
{latex}\


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

h5. Integral Forms

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Integral Forms
 
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\begin{large}\[ v(t) = v(t_{i})+\int_{t_{i}}^{t} a\;dt\]\end{large}
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\\
{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*
|!Acceleration vs Time Graph.bmp!|
* *Velocity versus time graph*
|!Velocity vs Time Graph.bmp!|
* *Position versus time graph*
|!Position vs Time Graph.bmp!|

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h2. Relevant Examples

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

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Diagrammatic Representations
Image Added
Click here to run a simulation demonstrating position, 
velocity and acceleration graphs for general 1-D motion
Simulation provided by:
PhET Interative Simulations
University of Colorado
 http://phet.colorado.edu
Relevant Examples
 
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 All Examples Relevant to the Model
 
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example_problem,nonuniform_acceleration
example_problem,nonuniform_acceleration
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