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h1. Acceleration
{excerpt}The time rate of change of [velocity] of an object, or alternately the net [force] on the object divided by the object's [mass].{excerpt}

h3. Mathematical Representation 

{latex}\begin{large}\[ \vec{a} = \frac{d\vec{v}}{dt} \qquad \mbox{or} \qquad \vec{a}=\frac{\sum \vec{F}}{m} \]\end{large}{latex}

h3. One-Dimensional Acceleration


h4. Utility of the One-Dimensional Case


As with all [vector] equations, the equations of kinematics are usually approached by separation into components.  In this fashion, the equations become three simultaneous one-dimensional equations.  Thus, the consideration of motion in one dimension with acceleration can be generalized to the three-dimensional case.


h4. Useful Digrammatic Representations 

Several diagrammatic representations are commonly used to represent accelerated motion.

* [*Position vs. Time Graph*|position versus time graph]: {excerpt-include:position versus time graph|nopanel=true}

* [*Velocity vs. Time Graph*|velocity versus time graph]: {excerpt-include:velocity versus time graph|nopanel=true}

* [*Motion Diagram*|motion diagam]: {excerpt-include:motion diagram|nopanel=true}

h4. Deceleration

In physics, the term _acceleration_ denotes a vector, as does [velocity].  When the acceleration of an object points in the same direction as its [velocity], the object speeds up.  When the acceleration of an object points in the direction opposite the object's [velocity], the object slows down.  In everyday speech, we would call the first case "acceleration" and the second case "deceleration".  In physics, both cases represent acceleration, but with a different relationship to the [velocity].  

h4. Constant Acceleration 

h5. {color:maroon} Integration with Respect to Time {color}

If acceleration is constant, the definition of acceleration can be integrated:

{latex}\begin{large}\[ \int_{v_{\rm i}}^{v} dv = \int_{t_{\rm i}}^{t} a\: dt \] \end{large}{latex}

For the special case of constant acceleration, the integral yields:

{latex}\begin{large} \[ v - v_{\rm i} = a(t-t_{\rm i}) \] \end{large}{latex}

which is equivalent to:

{latex}\begin{large} \[ v = v_{\rm i} + a (t-t_{\rm i}) \] \end{large}{latex}

We can now substitute into this equation the definition of velocity,

{latex}\begin{large}\[ v = \frac{dx}{dt}\]\end{large}{latex}

which gives:

{latex} \begin{large} \[ \frac{dx}{dt} = v_{\rm i} + a t - a t_{\rm i} \] \end{large}{latex}

We can now integrate again:

{latex} \begin{large} \[ \int_{x_{\rm i}}^{x} dx = \int_{t_{\rm i}}^{t} \left( v_{\rm i} - at_{\rm i} + a t\right)\:dt \]   \end{large}{latex}

to find:

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

We finish up with some algebra:

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

which is equivalent to:

{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}


h5. {color:maroon} Integration with Respect to Position {color}

The definition of acceleration can also be integrated with respect to position, if we use a calculus trick that relies on the chain rule.  Returning to the definition of acceleration:

{latex}\begin{large}\[ \frac{dv}{dt} = a \] \end{large}{latex}

we would like to find an expression for _v_ as a function of _x_ instead of _t_.  One way to achieve this is to use the chain rule to write:

{latex}\begin{large} \[ \frac{dv}{dx}\frac{dx}{dt} = a \] \end{large}{latex}

We can now elminate _t_ from this expression by using the defnition of velocity to recognize that _dx_/_dt_ = _v_.  Thus:

{latex}\begin{large} \[ \frac{dv}{dx}v = a \] \end{large}{latex}

which is easily integrated for the case of constant acceleration:

{latex}\begin{large} \[ \int_{v_{\rm i}}^{v} v \:dv = \int_{x_{\rm i}}^{x} a \:dx \] \end{large}{latex}

to give:

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


h5. {color:maroon} Four or Five Useful Equations {color}

The integrations performed above can be combined with the relationship between average velocity and position:
{latex}\begin{large} \[ \bar{v} = \frac{\Delta x}{\Delta t} = \frac{x - x_{\rm i}}{t- t_{\rm i}} \] \end{large}
{latex}

to give five very important equations.

{panel:title=Five (or Four) Equations for Kinematics with Constant Acceleration}
{latex}\begin{large} \[ x = x_{\rm i} + \bar{v}(t-t_{\rm i}) \] \[ \bar{v} = \frac{1}{2}(v+v_{\rm i}) \]
\[ v = v_{\rm i} + a(t-t_{\rm i}) \]\[ x = x_{\rm i} + v_{\rm i}(t-t_{\rm i}) + \frac{1}{2} a (t-t_{\rm i})^{2} \]
\[ v^{2} = v_{\rm i}^{2} + 2 a (x-x_{\rm i}) \]\end{large}{latex}
{panel}

{note}Because the first equation is not specific to the case of constant acceleration (it is simply a definition of average velocity) it is combined with the second equation in the summary on the model specification page for [one-dimensional motion with constant acceleration|1-D Motion (Constant Acceleration)]. {note}

h5. {color:maroon}The Utility of Constant Acceleration{color}

Stringing together a series of constant [velocity] segments is not usually a realistic description of motion, because real objects cannot change their velocity in a discontinuous manner. This drawback does _not_ apply to constant acceleration, however. Objects can have their acceleration changed almost instantaneously.  Because of this, it is often reasonable to approximate a complicated motion by separating it into segments of constant acceleration.



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