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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. Description and Assumptions

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 (universal)] 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).

h4. Problem 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.

h4. Learning Objectives

Students will be assumed to understand this model who can:

* Explain the difference between how physicists use the term [acceleration] versus the everyday use of the terms "accelerate" and "decelerate".
* Describe the features of a [motion diagram] representing an object moving with constant [acceleration].
* Summarize the givens needed to solve a problem involving motion with constant [acceleration].
* Construct a consistent sign convention for the initial velocity, the final velocity and the acceleration in the case of objects that are speeding up or slowing down.
* Describe the features of a [position versus time graph] representing an object moving with constant [acceleration].
* Given a [position versus time graph], determine whether the object represented is speeding up or slowing down.
* Given a linear [velocity versus time graph], determine the corresponding [acceleration].
* State the equation that relates [position], initial [velocity], [acceleration] and time for motion with constant [acceleration].
* State the equation that relates [position], initial [velocity], final [velocity] and [acceleration] for motion with constant [acceleration].
* Solve a quadratic equation for time.
* Mathematically solve for the meeting time and location of two objects moving with constant [acceleration] by setting up and solving a system of equations.
* Graphically locate the meeting point of two objects moving with constant [acceleration].
* Describe the trajectory of a [projectile].
* Describe the acceleration of a [projectile] throughout its trajectory.
* State the conditions on the [velocity] and [acceleration] that describe the maximum height of a [projectile].


h1. Model


h4. Compatible Systems

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.)

h4. Relevant Interactions

Some constant net external force must be present to cause motion with a constant acceleration.

h4. Laws of Change

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}

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

{latex}\begin{large}$v^{2} = v_{\rm 
 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_{
\rm 
i})
$
\]\end{large}
{latex}
{note}This is an important expression, because time is eliminated.{note}

h4. Diagrammatic Representations

h1. 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
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!carrier.jpg!\\
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Photos courtesy [US Navy|
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 Photos courtesy US Navy by:
 Cmdr. Jane Campbell
 Mass Communication Specialist 1st Class Emmitt J. Hawks
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