:''For the R.E.M. album see [[Accelerate (R.E.M. album)]]''
[[Image:Acceleration.svg|255px|thumb|Acceleration is the time rate of change of speed and/or direction. At any point on a speed-time graph, its magnitude is given by the slope of the tangent to the curve at that point.]]

In [[physics]], '''acceleration''' is defined as the [[Derivative|rate of change]] of [[velocity]], or, equivalently, as the second [[derivative]] of position (with respect to time). It is thus a [[vector (spatial)|vector]] quantity with dimension [[length]]/[[time]]². In [[SI|SI units]], acceleration is measured in [[Metres per second squared|meters/second²]] (m·s^-²). The term "acceleration" generally refers to the change in instantaneous velocity.

In common speech, the term acceleration is only used for an increase in speed. In physics, any increase or decrease in speed is referred to as acceleration and similarly, motion in a circle at constant speed is also an acceleration, since the direction component of the velocity is changing. See also [[Newton's Laws of Motion]].
==Relation to relativity==

After completing his theory of [[special relativity]], [[Albert Einstein]] realized that forces felt by objects undergoing constant [[proper acceleration]] are indistinguishable from those in a gravitational field. This was the basis for his development of [[general relativity]], a relativistic theory of [[gravity]].

This is also the basis for the popular [[Twin paradox]], which asks why one twin ages more rapidly when moving away from his sibling at near light-speed and then returning, since the aging twin can say that it is the other twin that was moving.
[[General relativity]] solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In ''special'' relativity, only [[Inertial frame of reference|inertial frames of reference]] (non-accelerated frames) can be used and are equivalent; ''general'' relativity considers ''all'' frames, even accelerated ones, to be equivalent. (The path from these considerations to the full theory of general relativity is traced in the [[Introduction to general relativity]].)

==Formula==
The '''formula''' for the average acceleration over a time period <math>\Delta t</math> is
:<math>\mathbf{\bar{a}}=\frac{\mathbf{v}(t+\Delta t)-\mathbf{v}(t)}{\Delta t}</math>
where
:<math>\mathbf{v}(t+\Delta t)</math> is the final velocity
:<math>\mathbf{v}(t)</math> is the initial velocity
:<math>t</math> is the initial time and <math>\Delta t</math> is the change in time


The formula for the instantaneous acceleration at time <math>t</math> is
:<math>\mathbf{a}(t)=\lim_{\Delta t \to 0}\frac{\mathbf{v}(t+\Delta t)-\mathbf{v}(t)}{\Delta t}=\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}</math>
Thus acceleration is the first [[derivative]] of velocity. One should note that the expression (Final position - Initial Position) / (Total time taken) is the average velocity, and the [[Limit (mathematics)|limit]] as the time interval approaches zero is the instantaneous velocity. Therefore, velocity is the first derivative of position, making acceleration the second.

One should also note that the average and instantaneous accelerations over a time period <math>\Delta t=t_1-t_0</math> are related through the [[Mean value theorem | Mean Value Theorem for Integrals]]:
:<math>\bar{\mathbf{a}}\int_{t_0}^{t_1}\mathrm{d}t=\int_{t_0}^{t_1}\mathbf{a}(t)\mathrm{d}t</math>


Putting it all together means:
:<math>\mathbf{a} = \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}</math>
where
:<math>\mathbf{a}</math> is acceleration
:<math>\mathbf{v}</math> is velocity
:<math>\mathbf{r}</math> is position
:<math>t</math> is time

==References==
<div class="references-small">
* {{cite book | author=Serway, Raymond A.; Jewett, John W. | title=Physics for Scientists and Engineers | edition=6th ed. | publisher=Brooks/Cole | year=2004 | id=ISBN 0-534-40842-7}}
* {{cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics | edition=5th ed. | publisher=W. H. Freeman | year=2004 | id=ISBN 0-7167-0809-4}}
<references />
</div>

==External links==
* [http://www.lightandmatter.com/html_books/1np/ch03/ch03.html Acceleration and Free Fall] - a chapter from an online textbook
*[http://physnet.org/home/modules/pdf_modules/m72.pdf ''Trajectories and Radius, Velocity, Acceleration''] on [http://www.physnet.org Project PHYSNET]
* [http://www.scienceaid.co.uk/physics/forces/motion.html Science aid: Movement]
* [http://www.physicsclassroom.com/Class/1DKin/U1L1e.html Physics Classroom: Acceleration]
* [http://science.dirbix.com/physics/acceleration Science.dirbix: Acceleration]
* [http://www.ajdesigner.com/constantacceleration/cavelocitya.php Acceleration Calculator]
* [http://www.glenbrook.k12.il.us/gbssci/phys/Class/circles/u6l1b.html Motion Characteristics for Circular Motion]

{{Kinematics}}

[[Category:Physical quantity]]
[[Category:Dynamics]]

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