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: $${\displaystyle \mathbf {v} =v{\frac {\mathbf {v} }{v}}=v\mathbf {u} _{\mathrm {t} },}$$ with v equal to the speed of travel along the path, and $${\displaystyle \mathbf {u} _{\mathrm {t} }={\frac {\mathbf {v} }{v}}\,,}$$ 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 and the changing direction of u_t, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation for the product of two functions of time as: