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Wiki Markup
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}
}{composition-setup}{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} {table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols}{tr:valign=top}{td:width=365px|bgcolor=#F2F2F2}
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h1. One-Dimensional Motion with Constant Acceleration


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

{cloak:id=desc}
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. Its real usefulness lies in the fact that it can describe mutli-dimensional motion with constant acceleration if the motion along different orthogonal directions is treated by application of the one-dimensional rules independently along the differect axes. 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.
{cloak}

h4. {toggle-cloak:id=cues} Problem Cues

{cloak:id=cues}
For pure kinematics situations, the problem will often explicitly state that the acceleration is constant, or else will indicate this bu giving some quantitative information that implies the acceleration is constant (e.g. a linear plot of velocity versus time). 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.
{cloak}

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

{cloak:id=pri}
* [1-D Motion (Constant Velocity)]

{cloak}

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

{cloak:id=vocab}
* [position (one-dimensional)]
* [velocity]
* [acceleration]

{cloak}

h2. Model


h4. {toggle-cloak:id=sys} {color:red}Compatible Systems{color}

{cloak:id=sys}
A single [point particle|point particle] (or a system treated as a point particle with position specified by the center of mass).
{cloak}

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

{cloak:id=int}
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.
{cloak}

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

{cloak:id=laws}
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}
$\end{large}{latex}\\
\\
{latex}\begin{large}$v^{2} = v_{\rm i}^{2} + 2 a (x - x_{\rm i})$\end{large}{latex}
{cloak}

h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

{cloak:id=diag}
* *Position as a Function of Time*
From the formulas given in the *Laws of Change*, it is clear that a plot of position vs. time will give a *parabola*. 

If the acceleration is _positive_ the parabola will open upwards. The position at *t = t{~}i{~}* will be *x{~}i{~}* , as shown in the graph below (time at the origin is *t{~}i{~}* ):

|!position v time w constant accel 2.PNG!|

In this case the position is positive. The fact that the plot of position vs. time is increased means that the initial velocity, *v{~}i{~}* , is also positive. 

If the acceleration and the initial position *x{~}i{~}* were the same, but the initial velocity was _negative_ , then the graph of position vs. time would look like this:

|!position v time w constant accel 3.PNG!|



The parabola has a minimum value at the time *t{*}{*}{~}min{~}*

{latex}\begin{large} \[ {\rm t}_{\rm min} = {\rm t}_{\rm i}- \frac{{\rm v}_{\rm 1}}{\rm a} \] \end{large}{latex}

This information is intended to familiarize the reader with the shape of the curve and how it behaves. Obviously, if the object starts out at time *t = t{~}i{~}* its real motion will not be described by the portion of the curve for *t < t{~}i{~}*, and so an object moving with positive initial velocity and positive acceleration will not have such a "minimum" position -- it will move in the same direction, with uincreasing speed, for all *t > t{~}i{~}. 

On the other hand, an object with negative initial velocity *v{~}i{~}* and positive acceleration _will_ encounter a "minimum" position at which it will have zero velocity, after which it will reverse direction  and gather speed with increasing time. 
For the case of positive acceleration, with positive non-zero velocity at *t = t{*}{*}{~}1{~}* and non-zero position *x{*}{*}{~}1{~}* this will appear as:

|!position v time w constant accel 2.PNG!|


* *Velocity versus time graph*

A plot of velocity vs. time, with positive velocity at *t{*}{*}{~}1{~}*, is given by:

!OneDMotionConstAccel VelocityPlot.PNG!

{cloak}

h2. Relevant Examples


h4. {toggle-cloak:id=oned} Examples Involving Purely One-Dimensional Motion

{cloak:id=oned}
{contentbylabel:
\]\end{large}
 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}\[v^{2}(x)= v_{i}^{2}+ 2 a (x - x_{i})\]\end{large}
 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
 
 Toggle Cloak
idoned
Examples Involving Purely One-Dimensional Motion
 
 Cloak
idoned
falsetruetrueAND501d_motion,constant_acceleration,example_problem
|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h4. {
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:
id
=
freefall
} 
Examples 
 
 Involving 
 
 Freefall


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=freefall}
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freefall
falsetruetrueAND50freefall,example_problem
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{cloak}

h4. {
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:
id
=
catchup
} 
Examples 
 
 Involving 
 
 Determining 
 
 when 
 
 Two 
 
 Objects 
 
 Meet


{
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:
id
=catchup}
{contentbylabel:
catchup
falsetruetrueAND50catch-up,constant_acceleration,example_problem
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{cloak}

h4. {
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:
id
=
all
} 
All 
 
 Examples 
 
 Using 
 
 this 
 
 Model


{
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=all}
{contentbylabel:
all
falsetruetrueAND50constant_acceleration,example_problem
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!carrier.jpg!\\
\\ !bball.jpg|width=235!
Photos courtesy [US Navy|
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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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