Editing Acceleration (section)

== Tangential and centripetal acceleration ==
{{See also|Centripetal force#Local coordinates|Tangential velocity}}
[[File:Oscillating pendulum.gif|thumb|left|An oscillating pendulum, with velocity and acceleration marked. It experiences both tangential and centripetal acceleration.]]
[[File:Acceleration components.svg|right|thumb|Components of acceleration for a curved motion. The tangential component {{math|'''a'''<sub>t</sub>}} is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) {{math|'''a'''<sub>c</sub>}} is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.]]

The velocity of a particle moving on a curved path as a [[function (mathematics)|function]] of time can be written as:
<math display="block">\mathbf{v}(t) = v(t) \frac{\mathbf{v}(t)}{v(t)} = v(t) \mathbf{u}_\mathrm{t}(t) , </math>
with {{math|''v''(''t'')}} equal to the speed of travel along the path, and
<math display="block">\mathbf{u}_\mathrm{t} = \frac{\mathbf{v}(t)}{v(t)} \, , </math>
a [[Differential geometry of curves#Tangent vector|unit vector tangent]] to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed {{math|''v''(''t'')}} and the changing direction of {{math|'''u'''<sub>''t''</sub>}}, the acceleration of a particle moving on a curved path can be written using the [[chain rule]] of differentiation<ref>{{cite web|last1=Weisstein|first1=Eric W.|title=Chain Rule| url=http://mathworld.wolfram.com/ChainRule.html |website=Wolfram MathWorld| publisher=Wolfram Research| access-date=2 August 2016}}</ref> for the product of two functions of time as:

<math display="block">\begin{alignat}{3}
\mathbf{a} & = \frac{d \mathbf{v}}{dt} \\
           & =  \frac{dv}{dt} \mathbf{u}_\mathrm{t} +v(t)\frac{d \mathbf{u}_\mathrm{t}}{dt} \\
           & = \frac{dv }{dt} \mathbf{u}_\mathrm{t} + \frac{v^2}{r}\mathbf{u}_\mathrm{n}\ ,
\end{alignat}</math>

where {{math|'''u'''<sub>n</sub>}} is the  unit (inward) [[Differential geometry of curves#Normal or curvature vector|normal vector]] to the particle's trajectory (also called ''the principal normal''), and {{math|'''r'''}} is its instantaneous [[Curvature#Curvature of plane curves|radius of curvature]] based upon the [[Osculating circle#Mathematical description|osculating circle]] at time {{mvar|t}}. The components
:<math>\mathbf{a}_\mathrm{t} = \frac{dv }{dt} \mathbf{u}_\mathrm{t} \quad\text{and}\quad \mathbf{a}_\mathrm{c} = \frac{v^2}{r}\mathbf{u}_\mathrm{n}</math>
are called the [[tangential acceleration]] and the normal or radial acceleration (or centripetal acceleration in circular motion, see also [[circular motion]] and [[centripetal force]]), respectively.

Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the [[Frenet–Serret formulas]].<ref name = Andrews>{{cite book |title = Mathematical Techniques for Engineers and Scientists |author1=Larry C. Andrews |author2=Ronald L. Phillips |page = 164 |url = https://books.google.com/books?id=MwrDfvrQyWYC&q=particle+%22planar+motion%22&pg=PA164 |isbn = 978-0-8194-4506-3 |publisher = SPIE Press |year = 2003 }}</ref><ref name = Chand>{{cite book |title = Applied Mathematics |page = 337 |author1=Ch V Ramana Murthy |author2=NC Srinivas |isbn = 978-81-219-2082-7 | url = https://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA337 | publisher = S. Chand & Co. | year = 2001| location=New Delhi }}</ref>