For a trajectory $\mathbf{r}(t)$, the velocity and acceleration are $\mathbf{v}=d\mathbf{r}/dt$ and $\mathbf{a}=d\mathbf{v}/dt$. In curvilinear coordinates, these derivatives must account for the time dependence of the basis vectors.

In plane polar coordinates, $\mathbf{r}=r\,\hat{\mathbf{e}}_r$. Then $\mathbf{v}=\dot r\,\hat{\mathbf{e}}_r+r\dot\theta\,\hat{\mathbf{e}}_\theta$ and $\mathbf{a}=(\ddot r-r\dot\theta^2)\hat{\mathbf{e}}_r+(r\ddot\theta+2\dot r\dot\theta)\hat{\mathbf{e}}_\theta$. The $2\dot r\dot\theta$ term is the kinematic Coriolis-like coupling arising from rotating basis vectors.

For motion along a space curve, acceleration decomposes as $\mathbf{a}=\dot v\,\hat{\mathbf{t}}+v^2\kappa\,\hat{\mathbf{n}}$, where $v$ is speed and $\kappa$ is curvature. This separation is often more useful than Cartesian components in problems with geometric constraints.