Editing Acceleration (section)

== Definition and properties ==
[[File:Kinematics.svg|thumb|300px|Kinematic quantities of a classical particle: mass {{mvar|m}}, position {{math|'''r'''}}, velocity {{math|'''v'''}}, acceleration {{math|'''a'''}}.]]

===Average acceleration===

[[File:Acceleration as derivative of velocity along trajectory.svg|right|thumb|Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time {{mvar|t}} is found in the limit as [[time interval]] {{math|Δ''t'' → 0}} of {{math|Δ'''v'''/Δ''t''}}.]]

An object's average acceleration over a period of [[time in physics|time]] is its change in [[velocity]], <math>\Delta \mathbf{v}</math>, divided by the duration of the period, <math>\Delta t</math>. Mathematically,
<math display="block">\bar{\mathbf{a}} = \frac{\Delta \mathbf{v}}{\Delta t}.</math>

===Instantaneous acceleration===
[[File:1-D kinematics.svg|thumb|right|'''From bottom to top''': {{bulleted list
 | an acceleration function {{math|''a''(''t'')}};
 | the integral of the acceleration is the velocity function {{math|''v''(''t'')}};
 | and the integral of the velocity is the distance function {{math|''s''(''t'')}}.
}}]]
Instantaneous acceleration, meanwhile, is the [[limit of a function|limit]] of the average acceleration over an [[infinitesimal]] interval of time. In the terms of [[calculus]], instantaneous acceleration is the [[derivative]] of the velocity vector with respect to time:
<math display="block">\mathbf{a} = \lim_{{\Delta t} \to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt}.</math>
As acceleration is defined as the derivative of velocity, {{math|'''v'''}}, with respect to time {{mvar|t}} and velocity is defined as the derivative of position, {{math|'''x'''}}, with respect to time, acceleration can be thought of as the [[second derivative]] of {{math|'''x'''}} with respect to {{mvar|t}}:
<math display="block">\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}.</math>

(Here and elsewhere, if [[Rectilinear motion|motion is in a straight line]], [[Euclidean vector|vector]] quantities can be substituted by [[Scalar (physics)|scalars]] in the equations.)

By the [[fundamental theorem of calculus]], it can be seen that the [[integral]] of the acceleration function {{math|''a''(''t'')}} is the velocity function {{math|''v''(''t'')}}; that is, the area under the curve of an acceleration vs. time ({{mvar|a}} vs. {{mvar|t}}) graph corresponds to the change of velocity.
<math display="block" qid=Q11465>\mathbf{\Delta v} =  \int \mathbf{a} \, dt.</math>

Likewise, the integral of the [[Jerk (physics)|jerk]] function {{math|''j''(''t'')}}, the derivative of the acceleration function, can be used to find  the change of acceleration at a certain time:
<math display="block">\mathbf{\Delta a} =  \int \mathbf{j} \, dt.</math>

===Units===
Acceleration has the [[dimensional analysis|dimensions]] of velocity (L/T) divided by time, i.e. [[length|L]] [[time|T]]<sup>−2</sup>. The [[International System of Units|SI]] unit of acceleration is the [[metre per second squared]] (m s<sup>−2</sup>); or "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.

===Other forms===
An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoing ''centripetal'' (directed towards the center) acceleration.

[[Proper acceleration]], the acceleration of a body relative to a free-fall condition, is measured by an instrument called an [[accelerometer]].

In [[classical mechanics]], for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net [[force]] vector (i.e. sum of all forces) acting on it ([[Newton's laws of motion#Newton's second law|Newton's second law]]):
<math display="block" qid=Q2397319>\mathbf{F} = m\mathbf{a} \quad \implies \quad \mathbf{a} = \frac{\mathbf{F}}{m},</math>
where {{math|'''F'''}} is the net force acting on the body, {{mvar|m}} is  the [[mass]] of the body, and {{math|'''a'''}} is the center-of-mass acceleration. As speeds approach the [[speed of light]], [[Special relativity|relativistic effects]] become increasingly large.