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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
Description
...
. It is a subclass of the One-Dimensional Motion (General) model defined by the constraint da/dt = 0 (i.e. a(t)=constant).
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_{
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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}$v_{\rm f} =  v_{\rm i} + a (t_{\rm f} - t_{\rm i})${latex}\\
{latex} $ x_{\rm f} = x_{\rm 
i}+\frac{1}{2}(v_{
\rm 
f}+v_{
\rm 
i})(t
_{\rm f}
 - t_{
\rm 
i})
${latex}\\
{latex}$ x_{\rm f} 
\]\end{large}

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

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
 
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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
 
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idall
 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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