Page History

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).

HTML Table

border	1
cellpadding	8
cellspacing	0
rules	cols
frame	void

Table Row (tr)

valign	top

Table Cell (td)

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.

Definition

Excerpt
This model applies for a point particle subject to a constant acceleration that is either parallel to or antiparallel to the particle's initial velocity.

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

*position (one-dimensional)
*velocity (one-dimensional)
*acceleration (one-dimensional)

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:

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.

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_{

Panel

borderColor	#000000
bgColor	#FFFFFF
title	h3. Model Specification
borderStyle	solid

Wiki Markup

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, 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
\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
id oned
Examples Involving Purely One-Dimensional Motion

Cloak

id	oned

falsetruetrueAND50

1d_motion,constant_acceleration,example_problem

Toggle Cloak
id freefall
Examples Involving Freefall

Cloak

id	freefall

falsetruetrueAND50

freefall,example_problem

Toggle Cloak
id catchup
Examples Involving Determining when Two Objects Meet

Cloak

id	catchup

falsetruetrueAND50

catch-up,constant_acceleration,example_problem

Toggle Cloak
id all
All Examples Using this Model

Cloak

id	all

falsetruetrueAND50

constant_acceleration,example_problem

Table Cell (td)

width	235px

Image Added

Image Added
Photos courtesy US Navy by:
Cmdr. Jane Campbell
Mass Communication Specialist 1st Class Emmitt J. Hawks

Wiki Markup

{html}
<script type="text/javascript">
var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www.");
document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E"));
</script>
<script type="text/javascript">
try {
var pageTracker = _gat._getTracker("UA-11762009-2");
pageTracker._trackPageview();
} catch(err) {}</script>
{html}

Child pages

Versions Compared

Old Version 21

New Version 201

Key

Introduction to the Model

Description and Assumptions

Definition

Prerequisite Vocabulary

Learning Objectives

S.I.M. Structure of the Model

Compatible Systems

Relevant Interactions

Laws of Change

Mathematical Representations

Diagrammatic Representations

Relevant Examples

Toggle Cloak
id oned
Examples Involving Purely One-Dimensional Motion

Toggle Cloak
id freefall
Examples Involving Freefall

Toggle Cloak
id catchup
Examples Involving Determining when Two Objects Meet

Toggle Cloak
id all
All Examples Using this Model

Child pages

Page History

Versions Compared

Old Version 21

New Version 201

Key

Introduction to the Model

Description and Assumptions

Definition

Prerequisite Vocabulary

Learning Objectives

S.I.M. Structure of the Model

Compatible Systems

Relevant Interactions

Laws of Change

Mathematical Representations

Diagrammatic Representations

Relevant Examples

Toggle CloakidonedExamples Involving Purely One-Dimensional Motion

Toggle CloakidfreefallExamples Involving Freefall

Toggle CloakidcatchupExamples Involving Determining when Two Objects Meet

Toggle CloakidallAll Examples Using this Model

Toggle Cloak
id oned
Examples Involving Purely One-Dimensional Motion

Toggle Cloak
id freefall
Examples Involving Freefall

Toggle Cloak
id catchup
Examples Involving Determining when Two Objects Meet

Toggle Cloak
id all
All Examples Using this Model