# Acceleration - Tangential and Centripetal Acceleration

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 ut, 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:

begin{alignat}{3} mathbf{a} & = frac{mathrm{d} mathbf{v}}{mathrm{d}t} \ & = frac{mathrm{d}v }{mathrm{d}t} mathbf{u}_mathrm{t} +v(t)frac{d mathbf{u}_mathrm{t}}{dt} \ & = frac{mathrm{d}v }{mathrm{d}t} mathbf{u}_mathrm{t}+ frac{v^2}{r}mathbf{u}_mathrm{n}, \ end{alignat}

where un 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 Frenet–Serret formulas.