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

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


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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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
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Differential Forms
 
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\begin{large}\[ \frac{dv}{dt} = a\]\end{large}
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\begin{large}\[ \frac{dx}{dt} = v\]\end{large}
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Integral Forms:
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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}
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{latex}\


 Latex
\begin{large}\[ x(t) = x(t_{
0
i})+\int_{t_{
0
i}}^{t} v\;dt\]\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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| !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. |
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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