Page History

{composition-setup}{composition-setup}{excerpt:hidden=true}{*}System:* One [point particle] constrained to move in one dimension. --- *Interactions:* Constant acceleration. --- *Note:* Multi-dimensional motion can often be broken into 1-D vector components, as for the case of projectile motion. {excerpt} {table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols}{tr:valign=top}{td:width=365px|bgcolor=#F2F2F2}
{live-template:Left Column}{td}{td}

h1. One-Dimensional Motion with Constant Acceleration


h4. {toggle-cloak:id=desc} Description and Assumptions

{cloak:id=desc}
Technically, this model is applicable to a single [point particle] subject to a constant acceleration that is either parallel to or anti-parallel to the particle's initial velocity. Its real usefulness lies in the fact that it can describe mutli-dimensional motion with constant acceleration if the motion along different orthogonal directions is treated by application of the one-dimensional rules independently along the differect axes. Thus, it can be used 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] is a common example). It is a subclass of the [One-Dimensional Motion (General)] model defined by the constraint da/dt = 0.
{cloak}

h4. {toggle-cloak:id=cues} Problem Cues

{cloak:id=cues}
For pure kinematics situations, the problem will often explicitly state that the acceleration is constant, or else will indicate this bu giving some quantitative information that implies the acceleration is constant (e.g. a linear plot of velocity versus time). This model is always applicable to the vertical direction in a problem that specified gravitational [freefall]. 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.
{cloak}

h4. {toggle-cloak:id=pri} Prior Models

{cloak:id=pri}
* [1-D Motion (Constant Velocity)]

{cloak}

h4. {toggle-cloak:id=vocab} Vocabulary

{cloak:id=vocab}
* [position (one-dimensional)]
* [velocity]
* [acceleration]

{cloak}

h2. Model


h4. {toggle-cloak:id=sys} {color:red}Compatible Systems{color}

{cloak:id=sys}
A single [point particle|point particle] (or a system treated as a point particle with position specified by the center of mass).
{cloak}

h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}

{cloak:id=int}
Some constant external influence must be present which produces a constant acceleration that is directed parallel or anti-parallel to the particle's initial velocity.
{cloak}

h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

{cloak:id=laws}
This model has several mathematical realizations that involve different combinations of the variables.
\\
\\
{latex}\begin{large}$v =  v_{\rm i} + a (t - t_{\rm i})$\end{large}{latex}\\
\\
{latex}\begin{large}$x = x_{\rm i}+\frac{1}{2}(v_{\rm f}+v_{\rm i})(t - t_{\rm i})$\end{large}{latex}\\
\\
{latex}\begin{large}$ x = x_{\rm i}+v_{\rm i}(t-t_{\rm i})+ \frac{1}{2}a(t-t_{\rm i})^{2}$\end{large}{latex}\\
\\
{latex}\begin{large}$v^{2} = v_{\rm i}^{2} + 2 a (x - x_{\rm i})$\end{large}{latex}
{cloak}

h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

{cloak:id=diag}
* *Position as a Function of Time*
From the formulas given in the *Laws of Change*, it is clear that a plot of position vs. time will give a *parabola*. If the acceleration is _positive_ the parabola will open upwards, and have a minimum value at the time

{latex}\begin{large}\[t$t = t_{1)\]$\end{large}{latex}
* Velocity versus time graph.

{cloak}

h2. Relevant Examples


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

{cloak:id=oned}
{contentbylabel:1d_motion,constant_acceleration,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{cloak}

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

{cloak:id=freefall}
{contentbylabel:freefall,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{cloak}

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

{cloak:id=catchup}
{contentbylabel:catch-up,constant_acceleration,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
{cloak}

h4. {toggle-cloak:id=all} All Examples Using this Model

{cloak:id=all}
{contentbylabel: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}
\\
\\
{live-template:RELATE license}
\\