**Tangential and Centripetal Acceleration**

The velocity of a particle moving on a curved path as a function of time can be written as:

with *v*(*t*) equal to the speed of travel along the path, and

a 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 *v(t)* and the changing direction of **u**_{t}, the acceleration of a particle moving on a curved path on a planar surface can be written using the chain rule of differentiation and the derivative of the product of two functions of time as:

where **u**_{n} is the unit (inward) normal vector to the particle's trajectory, and **r** is its instantaneous radius of curvature based upon the osculating circle at time *t*. These components are called the tangential acceleration and the radial acceleration or centripetal acceleration (see also circular motion and centripetal force).

Extension of this approach to three-dimensional space curves that cannot be contained on a planar surface leads to the Frenetâ€“Serret formulas.

Read more about this topic: Acceleration

### Famous quotes containing the words tangential and/or centripetal:

“New York is full of people ... with a feeling for the *tangential* adventure, the risky adventure, the interlude that’s not likely to end in any double-ring ceremony.”

—Joan Didion (b. 1934)

“There is a relation between the hours of our life and the centuries of time. As the air I breathe is drawn from the great repositories of nature, as the light on my book is yielded by a star a hundred millions of miles distant, as the poise of my body depends on the equilibrium of centrifugal and *centripetal* forces, so the hours should be instructed by the ages and the ages explained by the hours.”

—Ralph Waldo Emerson (1803–1882)