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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. \\ {latex}

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}\\ {latex}

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

{latex}\\ {latex}

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

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

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

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falsetruetrueAND50

1d_motion,constant_acceleration,example_problem

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freefall

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Examples

Involving

Freefall

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	freefall

falsetruetrueAND50

freefall,example_problem

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catchup

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Examples

Involving

Determining

when

Two

Objects

Meet

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	catchup

falsetruetrueAND50

catch-up,constant_acceleration,example_problem

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All

Examples

Using

this

Model

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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|http://www.navy.mil] by: Cmdr. Jane Campbell Mass Communication Specialist 1st Class Emmitt J. Hawks {td} {tr} {table}

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

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