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{composition-setup}{composition-setup}{excerpt:hidden=true}{*}System:* One [point particle] moving in one dimension either because it's constrained to move in that way or because only one dimension. --- *Interactions:* Constant accelerationCartesian component is considered. --- *NoteInteractions:* Multi-dimensionalConstant motionforce can(in oftenmagnitude beor brokenin intoits 1-D vector components, as forcomponent along the case of projectile motion. axis). {excerpt} 
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h1. One-Dimensional Motion with Constant Acceleration


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

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Technically, thisThis 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 moving in one dimension either because it's 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 positive (e.g. a rocket) or negative (e.g. gravity).   *Note:* Multi-dimensional motion can often be broken into components, as for the case of projectile motion. where there constant acceleration along one axis. The constnt acceleration model 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 (i.e. a(t)=constant).
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h4. {toggle-cloak:id=cues} Problem Cues

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For pure kinematics situations, the The problem will often explicitly state that the acceleration is constant, or else will indicate this buby giving some quantitative information that implies constant 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 in magnitude - in fact if one axis lies along the net force, the perpendicular axes will have no acceleration and hence will exhibit motion with constant velocity.
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h4. {toggle-cloak:id=pri} Prior Models

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* [1-D Motion (Constant Velocity)]

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

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

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h2. Model


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

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A single [point particle|point particle], (or a system such as a rigid body or many bodies that is treated as a point particle with position specified by the center of mass).. (The c of m involves the MOMENTUM MODEL.)

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

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Some constant net external influenceforce must be present which producesto cause motion with a constant acceleration that is directed parallel or anti-parallel to the particle's initial velocity.
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h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

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This model has several mathematical realizations that involve different combinations of the variables for position, velocity, and acceleration.
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{latex}\begin{large}$v(t) =  v_{\rm i} + a (t - t_{\rm i})$\end{large}{latex}\\
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{latex}\begin{large}$x(t) = x_{\rm i}+\frac{1}{2}(v_{\rm f}+v_{\rm i})(t - t_{\rm i})$\end{large}{latex}\\
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{latex}\begin{large}$ x = x_{\rm i}+v_{\rm i}(t-t_{\rm i})+ \frac{1}{2}a(t-t_{\rm i})^{2}$\end{large}{latex}\\
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{latex}\begin{large}$v^{2} = v_{\rm i}^{2} + 2 a (x - x_{\rm i})$\end{large}{latex}
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h4. {toggle-cloak:id=diag} {color:red}Diagrammatic Representations{color}

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* *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. The position at *t = t{*}{*}{~}i{~}* will be *x{*}{*}{~}i{~}* , as shown in the graph below (time at the origin is *t{*}{*}{~}i{~}* ):

| !position v time w constant accel 2.PNG! |

In this case the position is positive. The fact that the plot of position vs. time is increased means that the initial velocity, *v{*}{*}{~}i{~}* , is also positive. 

If the acceleration and the initial position *x{*}{*}{~}i{~}* were the same, but the initial velocity was _negative_ , then the graph of position vs. time would look like this:

| !position v time w constant accel 3.PNG! |



The parabola has a minimum value at the time *t{*}{*}{~}min{~}*

{latex}\begin{large} \[ {\rm t}_{\rm min} = {\rm t}_{\rm i}- \frac{{\rm v}_{\rm 1}}{\rm a} \] \end{large}{latex}

This information is intended to familiarize the reader with the shape of the curve and how it behaves. Obviously, if the object starts out at time *t = t{*}{*}{~}i{~}* its real motion will not be described by the portion of the curve for *t < t{*}{*}{~}i{~}*, and so an object moving with positive initial velocity and positive acceleration will not have such a "minimum" position -- it will move in the same direction, with increasing speed, for all *t > t{*}{*}{~}i{~}* . 

A plot of velocity vs. time for this case looks like this. The position of the intercept with the *time* axis is *t{*}{*}{~}min{~}* :


| !velocity v time w constant accel.PNG! | 


On the other hand, an object with negative initial velocity *v{*}{*}{~}i{~}* and positive acceleration _will_ encounter a "minimum" position, at which it will have zero velocity. After slowing to zero it will reverse direction  and gather speed with increasing time. (Note that the slope of the graph is the same as in the above case, which is what you expect, since it is determined only by the value of the acceleration.) The intercept of zero velocity occurs, again, at *t{*}{*}{~}min{~}* *= t{*}{*}{~}i{~}{*}*\- v{*}{*}{~}i{~}{*}*/a*. Since *v{*}{*}{~}i{~}* *< 0*, the intercept is to the right of *t{*}{*}{~}i{~}*. 

| !velocity v time w constant accel 3.PNG! |


The plots for the case of _negative_ acceleration (*a < 0*) are similar, only the parabola opens _downward_ in that case.

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h2. Relevant Examples


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

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{contentbylabel:1d_motion,constant_acceleration,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h4. {toggle-cloak:id=freefall} Examples Involving Freefall

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{contentbylabel:freefall,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h4. {toggle-cloak:id=catchup} Examples Involving Determining when Two Objects Meet

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{contentbylabel:catch-up,constant_acceleration,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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h4. {toggle-cloak:id=all} All Examples Using this Model

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{contentbylabel:constant_acceleration,example_problem|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50}
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!carrier.jpg!\\
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Photos courtesy [US Navy|http://www.navy.mil] by:
Cmdr. Jane Campbell
Mass Communication Specialist 1st Class Emmitt J. Hawks 
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