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{composition-setup}{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 in the same direction of motion (e.g. a propeller thrust) or in the opposite direction of motion (e.g. gravity on an ascending ball).   *Note:* Multi-dimensional motion can often be broken into components, as for the case of projectile motion, where there is a constant acceleration along one axis. The constant acceleration model can be used to 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)|gravitation (universal)] is a common example). 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).

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.
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{latex}\begin{large}$v(t) =  v\_{\rm i}+ a (t - t\_{rm i})$\end{large}{latex}\\
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{latex}\begin{large}$x(t) = x\_{\rm i}\+\frac{1}{2}(v\_{\rm f}\+v\_{rm i})(t - t\_{\rm i})$\end{large}{latex}\\
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{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 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

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

h1. Relevant Examples


h4. {toggle-cloak:id=oned}Examples Involving Purely One-Dimensional Motion

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h4. {toggle-cloak:id=freefall}Examples Involving Freefall

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h4. {toggle-cloak:id=catchup}Examples Involving Determining when Two Objects Meet

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h4. {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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