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Composition Setup

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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.

}{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} {table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols} {tr:valign=top} {td} 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}

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. \\ {math}

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{

math

large}

Latex
\

\

begin{

math

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

math} {math}

large}

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

math

large}

{

Note

}


In

the

above

expressions,

t

{~}i~

_i is

the

initial

time,

the

time

as

which

the

position

and

velocity

equal

x

{~}i~

_i and

v

{~}i~

_i respectively.

Often

t

{~}

_i

{~}

is

taken

to

equal

0,

in

which

case

these

expressions

simplify.

Latex
\begin{

note} {math}

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

math

large}

{

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:

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

|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50} {cloak} h6. {

Toggle Cloak

:

id

=

freefall

}

Examples

Involving

Freefall

{

Cloak

:

id

=freefall} {contentbylabel:

	freefall

falsetruetrueAND50

freefall,example_problem

|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50} {cloak} h6. {

Toggle Cloak

:

id

=

catchup

}

Examples

Involving

Determining

when

Two

Objects

Meet

{

Cloak

:

id

=catchup} {contentbylabel:

	catchup

falsetruetrueAND50

catch-up,constant_acceleration,example_problem

|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50} {cloak} h6. {

Toggle Cloak

:

id

=

all

}

All

Examples

Using

this

Model

{

Cloak

:

id

=all} {contentbylabel:

	all

falsetruetrueAND50

constant_acceleration,example_problem

|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50} {cloak} {td} {td:width=235px} !carrier.jpg!\\ \\ !bball.jpg|width=235! Photos courtesy [US Navy|

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Image Added

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

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Child pages

Versions Compared

Old Version 185

New Version 200

Key

Introduction to the Model

Description and Assumptions

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 185

New Version 200

Key

Introduction to the Model

Description and Assumptions

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 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

Toggle Cloak
id oned
Examples Involving Purely One-Dimensional Motion