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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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\begin{large}\[ v(t) = v(t_{i})+\int_{t_{i}}^{t} a\;dt\]\end{large}


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\begin{large}\[ x(t) = x(t_{i})+\int_{t_{i}}^{t} v\;dt\]\end{large}
Diagrammatic Representations
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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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h2. Description
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*Hierarchy of Models* {td}{tr}{tr}{td}
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This model applies to a point particle subject to an external force that is either parallel or antiparallel to the particle's initial velocity.  {excerpt}

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h2. Assumptions and Limitations


h4. Prior Models

* Link to model pages that should be learned before this model.

h4. Vocabulary

* [frame of reference]
* [position (one-dimensional)]
* [velocity (one-dimensional)]
* [acceleration (one-dimensional)]

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h2. Model Specification


h4. *System Structure*


h4.  

*Internal Constituents:* Point particle.

*Environment:* external agents interacting with the particle which are the responsible of the *real* forces acting *on* the particle. The total external force must be parallel or anti-parallel to the paticle's initial velocity.

h4. Descriptors

*Object Variables:* none

*State Variables:* Time (_t_), position (_x_) , and velocity (_v_).

*Interaction Variables:* acceleration _a_.

h4. Laws of Interaction

Any external force which magnitude is not constant. Possible forces are forces that depends on time, on the particle's position or on the particle's velocity.
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h4. Laws of Change

{latex}\begin {large}$x(t_f) = x(t_i) + \int_{t_i}^{t_f} v(t)dt  v(t) = \frac{dx(t)}{dt} $ \end{large}{latex}

 
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{latex}\begin {large}$v(t_f) = v(t_i) +\int_{t_i}^{t_f} a(t)dt v(t) =\frac{dv(t)}{dt}$ \end{large}{latex}
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h2. Relevant Examples

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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/]. |
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