Velocity is equal to the derivative of position with respects to time, ${\displaystyle {\vec {v}}={\frac {d}{dt}}\,{\vec {r}}}$

The Tangent Vector is the unit vector tangent to the motion, ${\displaystyle {\vec {T}}={\frac {\vec {v}}{\left\Vert {\vec {v}}\right\|}}}$. The Normal vector, similarly, is the unit vector normal to the motion, ${\displaystyle {\vec {N}}={\frac {\frac {d{\vec {T}}}{dt}}{\left\Vert {\frac {d{\vec {T}}}{dt}}\right\|}}}$

Acceleration is equal to the derivative of velocity with respects to time, ${\displaystyle {\vec {a}}={\frac {d}{dt}}\,{\vec {v}}}$