File talk:Acceleration components.JPG

The basis of the two terms is:

$\mathbf {v} ={\frac {\mathbf {v} }{|\mathbf {v} |}}|\mathbf {v} |\ ,$

${\frac {d}{dt}}\mathbf {v} =|\mathbf {v} |{\frac {d}{dt}}{\frac {\mathbf {v} }{|\mathbf {v} |}}+{\frac {\mathbf {v} }{|\mathbf {v} |}}{\frac {d}{dt}}|\mathbf {v} |,$

or

${\frac {d}{dt}}\mathbf {v} =\mathbf {a_{c}} +\mathbf {a_{t}} \ .$