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}{composition-setup} {table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols} {tr:valign=top} {td:width=350px|bgcolor=#F2F2F2} {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 {cloak:id=desc} 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. {cloak} h4. {toggle-cloak:id=cues}Problem Cues {cloak:id=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). {cloak} h4. {toggle-cloak:id=pri} Prior Models {cloak:id=pri} * [1-D Motion (Constant Velocity)] * [1-D Motion (Constant Acceleration)] {cloak} h4. {toggle-cloak:id=vocab}Vocabulary {cloak:id=vocab} * [position (one-dimensional)] * [velocity] * [acceleration] {cloak} 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). {cloak} h4. {toggle-cloak:id=int} Relevant Interactions {cloak:id=int} Some time-varying external influence that is confined to one dimension. {cloak} h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color} {cloak:id=laws} {section}{column} h5. Differential Forms {latex}
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Excerpt
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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}
{latex}\\ \\ {latex}\



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\begin{large}\[ \frac{dx}{dt} = v\]\end{large}
{latex}\\ \\ {column}{column} h5. Integral Forms {latex}
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Integral Forms
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\begin{large}\[ v(t) = v(t_{
0
i})+\int_{t_{
0
i}}^{t} a\;dt\]\end{large}
{latex}\\ \\ {latex}\



Latex
\begin{large}\[ x(t) = x(t_{
0
i})+\int_{t_{
0
i}}^{t} v\;dt\]\end{large}
{latex}\\ {column}{section} {cloak} h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color} {cloak:id=diag} * Acceleration versus time graph. * Velocity versus time graph. * Position versus time graph. {cloak} h2. Relevant Examples None yet. ---- {td} {tr} {table} {live-template:RELATE license}
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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