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h1. One-Dimensional Motion (Constant Acceleration)

h4. Description and Assumptions

{excerpt:hidden=true}*System:* One [point particle] constrained to move in one dimension. --- *Interactions:* Constant acceleration. --- *Note:* Multi-dimensional motion can often be broken into 1-D vector components, as for the case of projectile motion.{excerpt}


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h1. One-Dimensional Motion (Constant Acceleration)

h4. {toggle-cloak:id=desc} Description and Assumptions

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Technically, this model is applicable to a single [point particle] 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 [Gravitation] is a common example).  It is a subclass of the [One-Dimensional Motion (General)] model defined by the constraint da/dt = 0. 
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h2.h4. {toggle-cloak:id=cues} Problem Cues

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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. Prerequisite Knowledge

h4. Prior Models{cloak}

h4. {toggle-cloak:id=pri} Prior Models

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* [1-D Motion (Constant Velocity)]

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h4. {toggle-cloak:id=vocab} Vocabulary

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


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h2. Model

h4. System

 {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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h2. Interactions
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h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}

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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.

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h2. Model

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h4. {toggle-cloak:id=laws} Laws of Change

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This model has several mathematical realizations that involve different combinations of the variables.
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{latex}\begin{large}$v =  v_{\rm i} + a (t - t_{\rm i})$\end{large}{latex}\\
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{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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h4. {toggle-cloak:id=diag} Diagrammatic Representations

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* Velocity versus time graph.
* Position versus time graph.


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h2. Relevant Examples

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