Acceleration in a plane

Acceleration in a curvilinear motion is given by

a=($\frac{d{v}_x}{\mathrm{dt}}$)i + ($\frac{d{v}_y}{\mathrm{dt}}$)j + ($\frac{d{v}_z}{\mathrm{dt}}$)k

The components of the acceleration along X,Y ,Z are

$a_x=\frac{d{v}_x}{\mathrm{dt}}$

$a_y=\frac{d{v}_y}{\mathrm{dt}}$

$a_z=\frac{d{v}_z}{\mathrm{dt}}$

The acceleration vector has the same direction as instantanous change in velocity. Since the direction of the velocity changes in the direction in which the curve bends ,the acceleration is always pointing towards concavity of the curve

The acceleration in curvilinear motion can also be expressed in terms of two components

a) The tangential component a_T which is called as tangential acceleration. It is expressed as change in magnitude of the velocity

b) The normal component a_N which is called as normal or centripetal acceleartion. This is expressed as the change in direction of the velocity

$a_T=\frac{\mathrm{dv}}{\mathrm{dt}}$

$a_N=\frac{v^2}{R}$ Where R is the radius of curvature of the path