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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.

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

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

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_iis taken to equal 0, in which case these expressions simplify.

{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}\[v^{2}(x)= v_{i}^{2}+ 2 a (x - x_{i})\]\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.|

|[!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

{cloak:id=oned}
{contentbylabel:

This is an important expression, because time is eliminated.