Versions Compared

Old Version 118

changes.mady.by.user Andrew E Pawl

Saved on

compared with

New Version Current

changes.mady.by.user Andrew E Pawl

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin
{
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). 
 HTML Table
border1
cellpadding8
cellspacing0
rulescols
framevoid
 Table Row (tr)
valigntop
 Table Cell (td)
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}
{live-template:Left Column}
{td}
{td}


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, 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. 
{cloak}

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

{cloak:id=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.

{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}
 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}
{cloak}

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

{cloak:id=diag}

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

{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}

{td}
{td:width=235px}
!carrier.jpg!\\
\\
!bball.jpg|width=235!
Photos courtesy [US Navy|
 Table Cell (td)
width235px
Image Added

 Image Added
 Photos courtesy US Navy by:
 Cmdr. Jane Campbell
 Mass Communication Specialist 1st Class Emmitt J. Hawks
 Wiki Markup
{html}
<script type="text/javascript">
var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www.
...
");
document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E"));
</script>
<script type="text/javascript">
try {
var pageTracker = _gat._getTracker("UA-11762009-2");
pageTracker._trackPageview();
} catch(err) {}</script>
{html}