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{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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h4. Introduction to the Model

h5. 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)|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|system] motion in _any_ situation where the [net force] on the [system] is constant, even if the motion is multi-dimensional. {info}

h5. 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].


h4. S.I.M. Structure of the Model


h5. Compatible Systems

A single [point particle|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]. 

h5. Relevant Interactions

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

h4. Laws of Change

h5. Mathematical Representations

This model has several mathematical realizations that involve different combinations of the variables for position, velocity, and acceleration.
\\
{latex}\begin{large}\[v(t) =v_{\rm i}+ a (t - t_{\rm i})\]\end{large}{latex}\\
{latex}\begin{large}\[x(t) = x_{\rm i}+\frac{1}{2}(v_{\rm f}+v_{\rm i})(t - t_{\rm i})\]\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}\[v^{2}(x)= v_{\rm i}^{2}+ 2 a (x - x_{\rm i})\]\end{large}{latex}
{note}This is an important expression, because time is eliminated.{note}

h5. Diagrammatic Representations

* [motion diagram]
* [position versus time graph]
* [velocity versus time graph]

|[!images^MathematicaPlayer.png!|^ConstAccel.nbp]|[Click here|^ConstAccel.nbp] for a _Mathematica Player_ application illustrating these representations using _phase_.|

|[!images^download_now.gif!|http://www.wolfram.com/products/player/download.cgi]|[Click here|http://www.woldfram.com/products/player/download.cgi] to download the (free) _Mathematica Player_ from [Wolfram Research|http://www.wolfram.com]|

h4. Relevant Examples


h6. {toggle-cloak:id=oned}Examples Involving Purely One-Dimensional Motion

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h6. {toggle-cloak:id=freefall}Examples Involving Freefall

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h6. {toggle-cloak:id=catchup}Examples Involving Determining when Two Objects Meet

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h6. {toggle-cloak:id=all}All Examples Using this Model

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!carrier.jpg!\\
\\ !bball.jpg|width=235!
Photos courtesy [US Navy|http://www.navy.mil] by:
Cmdr. Jane Campbell
Mass Communication Specialist 1st Class Emmitt J. Hawks
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