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

{excerpt}ThisTechnically, this model is applicable to a single [point particle] moving with constant [velocity|velocity] subject to a constant acceleration that is either parallel to or anti-parallel to the particle's initial velocity, but its real usefulness lies in the fact that it can describe mutli-dimensional motion with constant acceleration by separate application to orthogonal directions.  Thus, it can be used describe the system's motion in any situation where the net [force] on the system is constant (a point particle subject only to near-earth [gravity] is a common example).  It is a subclass of the model [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant AccelerationGeneral)] model defined by the constraint _a_da/dt = 0. {excerpt}

h2. Problem Cues

For pure kinematics problems, the problem will often explicitly state that the acceleration is constant, or else some quantitative information will be given (e.g. a linear velocity versus time plot) that implies the acceleration is constant.  This model is always applicable to the vertical direction in a problem that specified gravitational [freefall].  The model is also sometimes useful (in conjunction with [Point Particle Dynamics]) in dynamics problems when it is clear that the net force is constant.

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h2. AssumptionsPrerequisite and LimitationsKnowledge

h4. Prior Models

* None assumed. [1-D Motion (Constant Velocity)]

h4. Vocabulary

* [position (one-dimensional)]
* [velocity]
* [acceleration]

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h2. Model SpecificationSystem

h4. System SchemaConstituents

*Internal Constituents:* The system must beA single [point particle|point particle] (or a system treated as a [point particle] when using this model.

*External Agents:* [External influences|external force] must be absent or else cancel so that no [acceleration] results.

h4. Descriptors

*Object Variables:* None.

*State Variables:* v, x, t

*Interaction Variables:* None. 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 constant external influence must be present which produces a constant acceleration that is directed parallel or anti-parallel to the particle's initial velocity.

h4. Interaction Variables

Acceleration (_a_).

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h2. Model

h4. Laws of InteractionChange

No net interaction is allowed.

h4. Laws of Change
This model has several mathematical realizations that involve different combinations of the variables.
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{latex}\begin{large}$x$v =  xv_{\rm i} + a v(t - t_{\rm i})$\end{large}{latex}\\

h4. Alternative Representations of Laws of Change

The one [Law of Change] for this model can be rearranged to show that the [velocity|velocity] is equivalent to the slope of a [position|position (one-dimensional)]\\
{latex}\begin{large}$x = x_{\rm i}+\frac{1}{2}(v_{\rm f}+v_{\rm i})(t - t_{\rm i})$\end{large}{latex}\\
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{latex}\begin{large}$ x = x_{\rm i}+v_{\rm i}(t-t_{\rm i})+ \frac{1}{2}a(t-t_{\rm i})^{2}$\end{large}{latex}\\
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{latex}\begin{large}$v^{2} = v_{\rm i}^{2} + 2 a (x - x_{\rm i})$\end{large}{latex}

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h2. Diagrammatical Representations

* Velocity versus time graph.  Since this model assumes velocity is constant, the [Law of Change] will result in _linear_ position
* Position versus time graphs (i.e. graphs with constant slope)graph.

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h2. Relevant Examples

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