Motion on a Curve

Velocity and Acceleration

Suppose the position of a particle/body moving along curve C is defined by a vector function \boldsymbol{r}(t) = f(t) \boldsymbol{i} + g(t) \boldsymbol{j} + h(t) \boldsymbol{k}. Then the velocity and acceleration of such particle/body will be defined as

(1)   \begin{equation*}\boldsymbol{v}(t) = \boldsymbol{r}^\prime (t) = f^\prime(t) \boldsymbol{i} + g^\prime(t) \boldsymbol{j} + h^\prime(t) \boldsymbol{k} ,\end{equation*}


(2)   \begin{equation*}\boldsymbol{a}(t) = \boldsymbol{r}^{\prime\prime} (t) = f^{\prime\prime}(t) \boldsymbol{i} + g^{\prime\prime}(t) \boldsymbol{j} + h^{\prime\prime}(t) \boldsymbol{k} ,\end{equation*}


Centripetal acceleration and curvilinear motion in the plane are described in the following video.

8-Minute Lecture on Motion on a Curve

Reference

Dennis G. Zill. Advanced Engineering Mathematics, 6^{th} edition. Jones & Bartlett Learning. 2016.