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Composition Setup
 Excerpt
hiddentrue
System: One point particle moving in one dimension either because it's constrained to move that way or because only one Cartesian component is considered. — Interactions: Constant force (in magnitude or in its component along the axis). 
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Introduction to the Model
Description and Assumptions
This model is applicable to a single point particle moving in one dimension either because it is physically constrained to move that way or because only one Cartesian component is considered. The force, or component of force along this direction, must be constant in time. The force can be in the same direction of motion or in the opposite direction of motion. Equivalently, the model applies to objects moving in one-dimension which have a position versus time graph that is parabolic and a velocity versus time graph that is linear. It is a subclass of the One-Dimensional Motion (General) model defined by the constraint da/dt = 0 (i.e. a(t)=constant).
 Info
Multi-dimensional motion can often be broken into components, as in the case of projectile motion. In this manner, the 1-D motion with constant acceleration model can be employed to describe the system's motion in any situation where the net force on the system is constant, even if the motion is multi-dimensional. 
Learning Objectives
Students will be assumed to understand this model who can:
S.I.M. Structure of the Model
Compatible Systems
A single point particle, or a system such as a single rigid body or a grouping of many bodies that is treated as a point particle with position specified by the system's center of mass. 
Relevant Interactions
Some constant net external force must be present to cause motion with a constant acceleration.
Laws of Change
Mathematical Representations
This model has several mathematical realizations that involve different combinations of the variables for position, velocity, and acceleration. 
 Latex
\begin{large}\[v(t) =v_{i}
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h2. Description and Assumptions

{excerpt}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 [gravity] is a common example).  It is a subclass of the [One-Dimensional Motion (General)] model defined by the constraint 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. Prerequisite Knowledge

h4. Prior Models

* [1-D Motion (Constant Velocity)]

h4. Vocabulary

* [position (one-dimensional)]
* [velocity (one-dimensional)]
* [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. Variables and Parameters

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 Change

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

 Latex
\begin{large}
$x
\[x(t) = x_{
\rm 
i}+\frac{1}{2}(v_{
\rm 
f}+v_{
\rm 
i})(t - t_{
\rm 
i})
$
\]\end{large}
{latex}\\
\\
{latex}\

 Latex
\begin{large}
$
\[ x(t) = x_{
\rm 
i}+v_{
\rm 
i}(t-t_{
\rm 
i})+ \frac{1}{2}a(t-t_{
\rm 
i})^{2}
$
\]\end{large}
{latex}\\
\\
{latex}
 Note
In the above expressions, ti is the initial time, the time as which the position and velocity equal xi and vi respectively. Often tiis taken to equal 0, in which case these expressions simplify.
 Latex
\begin{large}
$v^
\[v^{2}
 
(x)= v_{
\rm 
i}^{2}
 
+ 2 a (x - x_{
\rm 
i})
$
\]\end{large}
{latex}

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h2. Diagrammatical Representations

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

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h2. Relevant Examples

{contentbylabel:
 Note
This is an important expression, because time is eliminated.
Diagrammatic Representations
Image Added
Click here for a Mathematica Player application illustrating these representations.
Image Added
Click here to download the (free) Mathematica Player from Wolfram Research
Relevant Examples
 
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Examples Involving Purely One-Dimensional Motion
 
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falsetruetrueAND501d_motion,constant_acceleration,example_problem
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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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Examples Involving Freefall
 
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idfreefall
 falsetruetrueAND50freefall,example_problem
 
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Examples Involving Determining when Two Objects Meet
 
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idcatchup
 falsetruetrueAND50catch-up,constant_acceleration,example_problem
 
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idall
All Examples Using this Model
 
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 falsetruetrueAND50constant_acceleration,example_problem
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Image Added

 Image Added
 Photos courtesy US Navy by:
 Cmdr. Jane Campbell
 Mass Communication Specialist 1st Class Emmitt J. Hawks
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