Versions Compared

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin
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

Description

...

. It is a subclass of the One-Dimensional Motion (General) model defined by the constraint da/dt = 0 (i.e. a(t)=constant).

Page Contents

Table of Contents
stylenone

Prerequisite Vocabulary

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}\[v(t) =v_{i}+ a (t - t_{i})\]\end{large}


Latex
\begin{large}\[x(t) = x_{

...

Panel
borderColor#000000
bgColor#FFFFFF
titleBGColor#e8e8ff
titleModel Specification
borderStylesolid
Wiki Markup
h4. System Schema Internal Constituents:  None.  Object must be treated as a point particle. \\ External Agents:  Some constant external influence must be present which produces the acceleration. h4. Descriptors Object Variables:  None. \\ State Variables:  Time (_t_), position (_x_) , and velocity (_v_) are possible state variables.  Note that in some cases only two of the three possible state variables will be needed. \\ Interaction Variables:  Acceleration (_a_). h4. Laws of Interaction Acceleration must be a constant. h4. Laws of Change {latex}\begin{large}$v_{\rm f} =  v_{\rm i} + a (t_{\rm f} - t_{\rm i})$\end{large}{latex}\\ {latex}\begin{large}$x_{\rm f} = x_{\rm
i}+\frac{1}{2}(v_{
\rm
f}+v_{
\rm
i})(t
_{\rm
 
f}
- t_{
\rm
i})
$
\]\end{large}
{latex}\\ {latex}


Latex
\begin{large}
$
\[ x
_{\rm f}
(t) = x_{
\rm
i}+v_{
\rm
i}(t
_{\rm f}
-t_{
\rm
i})+ \frac{1}{2}a(t
_{\rm f}
-t_{
\rm
i})^{2}
$
\]\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_{\rm f}^
\[v^{2}
(x)= v_{
\rm
i}^{2}
+ 2 a (x
_{\rm
 
f}
- x_{
\rm
i})
$
\]\end{large}
{latex}
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
Toggle Cloak
idfreefall
Examples Involving Freefall
Cloak
idfreefall
falsetruetrueAND50freefall,example_problem
Toggle Cloak
idcatchup
Examples Involving Determining when Two Objects Meet
Cloak
idcatchup
falsetruetrueAND50catch-up,constant_acceleration,example_problem
Toggle Cloak
idall
All Examples Using this Model
Cloak
idall
falsetruetrueAND50constant_acceleration,example_problem
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}