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


h4. {toggle-cloak:id=motconst} One-Dimensional Motion with Constant Acceleration


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

For a time interval during which the acceleration is constant, the instantaneous acceleration at any time will always be equal to the average acceleration.  Thus, by analogy with the definition of [average velocity|velocity], we can write:
{latex}\begin{large} \[ a = \langle a\rangle_{t} = \frac{\Delta v}{\Delta t} = \frac{v - v_{\rm i}}{t - t_{\rm i}} \] \end{large}{latex}

Taking this equation as a starting point and using the relationship between average velocity and position
{latex}\begin{large} \[ \langle v\rangle_{t} = \frac{\Delta x}{\Delta t} = \frac{x - x_{\rm i}}{t- t_{\rm i}} \] \end{large}
{latex}
{warning}We removed the brackets in the acceleration equation above because acceleration is constant. We *cannot* similarly drop the brackets in the average velocity equation, because velocity is *not constant* when acceleration is constant (except for the trivial case of _a_ = 0). {warning}

lets us derive five very important equations.

{info}Three of these equations follow directly from the integrations performed in the section above.{info}

{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. For example, you could be coasting along in a car at a constant 60 mph with zero acceleration when suddenly you see traffic stopped ahead. If you slam on the brakes, your car will still be going 60 mph for an instant, then it will drop to 59, 58, 57, etc. Your acceleration, on the other hand, has almost instantly changed from zero to a substantial acceleration directed opposite your motion. You can feel this abrupt change as a passenger as you are forced against your seatbelt. Similarly, when an airplane begins its takeoff run, you can feel yourself suddenly pressed back in your seat as the plane's acceleration changes almost instantaneously from 0 to a significant forward acceleration. Because of this, it is often reasonable to approximate a complicated motion by separating it into segments of constant acceleration.



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