Tangential acceleration $a_t$ is component of acceleration vector tangent to the trajectory of the particle. $$a_t=\frac{dv}{dt}$$ Where $v$ is the speed of the particle.

Normal acceleration $a_n$ is the acceleration component perpendicular to the trajectory. Therefore, its direction is given by the radius vector of curvature. $$a_n=\frac{v^2}{R}$$ Where $R$ is the radius of curvature of the trajectory.

Tangential and normal acceleration can be calculated for a point with position vector \( \vec{r} \) as,

$$\vec{a}_t=\frac{d\vec{v}}{dt}=\frac{d}{dt}\left(\frac{d\vec{r}}{dt}\right)$$

$$\vec{a}_n=\frac{\vec{v}^2}{R}=\frac{(\frac{d\vec{r}}{dt})^2}{\left| \frac{d\vec{r}}{ds} \right|}$$