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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] 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). {excerpt}
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h1. One-Dimensional Motion with Constant Acceleration


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

{cloak:id=desc|visible=true}
This model is applicable to a single [point particle] moving in one dimension either because it's 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 positive (e.g. a rocket) or negative (e.g. gravity).   *Note:* Multi-dimensional motion can often be broken into components, as for the case of projectile motion. where there constant acceleration along one axis. The constnt acceleration model 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 (i.e. a(t)=constant).
{cloak}

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

{cloak:id=cues}
The problem will often explicitly state that the acceleration is constant, or else will indicate this by giving some quantitative information that implies constant acceleration (e.g. a linear plot of velocity versus time).  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 in magnitude - in fact if one axis lies along the net force, the perpendicular axes will have no acceleration and hence will exhibit motion with constant velocity.
{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|visible=true}
A single [point particle|point particle], or a system such as a rigid body or many bodies that is treated as a point particle with position specified by the center of mass. (The c of m involves the MOMENTUM MODEL.)
{cloak}

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

{cloak:id=int|visible=true}
Some constant net external force must be present to cause motion with a constant acceleration.
{cloak}

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

{cloak:id=laws|visible=true}
This model has several mathematical realizations that involve different combinations of the variables for position, velocity, and acceleration.
\\
\\
{latex}\begin{large}$v(t) =  v_{\rm i} 
+ a (t - t_{
\rm 
i})
$
\]\end{large}
{

 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}
In these 
 Note
In the above expressions, 
 
 t
{~}i~ 
i is 
 
 the 
 
 initial 
 
 time, 
 
 the 
 
 time 
 
 as 
 
 which 
 
 the 
 
 position 
 
 and 
 
 velocity 
 
 equal 
 
 x
{~}i~ 
i and 
 
 v
{~}i~ 
i respectively. 
 
 Often 
  
 t
{~}
i
{~}
is 
 
 taken 
 
 to 
 
 equal 
 
 0, 
 
 in 
 
 which 
 
 case 
 
 these 
 
 expressions 
 simplify.


Here's an expression that relates the velocity at initial and final times - it follows algebraically from the two expressions above. 
\\
{latex}
simplify.
 Latex
\begin{large}
$v^
\[v^{2}
 
(x)= v_{
\rm 
i}^{2}
 
+ 2 a (x - x_{
\rm 
i})
$
\]\end{large}
{latex}
This is an important expression, because the velocity can be regarded as a function of initial and final position, hence time is eliminated from the expression.  This realization is the gateway to deriving the relationship between \[work\] and \[kinetic energy\].
{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 increasing speed, for all *t > t{*}{*}{~}i{~}* .

A plot of velocity vs. time for this case looks like this. The position of the intercept with the *time* axis is *t{*}{*}{~}min{~}* :
| !velocity v time w constant accel.PNG! |
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 slowing to zero it will reverse direction  and gather speed with increasing time. (Note that the slope of the graph is the same as in the above case, which is what you expect, since it is determined only by the value of the acceleration.) The intercept of zero velocity occurs, again, at *t{*}{*}{~}min{~}* *= t{*}{*}{~}i{~}{*}*\- v{*}{*}{~}i{~}{*}*/a*. Since *v{*}{*}{~}i{~}* *< 0*, the intercept is to the right of *t{*}{*}{~}i{~}*.
| !velocity v time w constant accel 3.PNG! |
The plots for the case of _negative_ acceleration (*a < 0*) are similar, only the parabola opens _downward_ in that case.
{cloak}

h2. Relevant Examples


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

{cloak:id=oned}
{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
 
 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}
{cloak}

h4. {
 Toggle Cloak
:
id
=
freefall
} 
Examples 
 
 Involving 
 
 Freefall


{
 Cloak
:
id
=freefall}
{contentbylabel:
freefall
falsetruetrueAND50freefall,example_problem
|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{cloak}

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


{
 Cloak
:
id
=catchup}
{contentbylabel:
catchup
falsetruetrueAND50catch-up,constant_acceleration,example_problem
|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{cloak}

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


{
 Cloak
:
id
=all}
{contentbylabel:
all
falsetruetrueAND50constant_acceleration,example_problem
|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{cloak}
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!carrier.jpg!\\
\\  !bball.jpg|width=235!
Photos courtesy [US Navy|
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 Image Added
 Photos courtesy US Navy by:
 Cmdr. Jane Campbell
 Mass Communication Specialist 1st Class Emmitt J. Hawks
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