$$
\vec{v}=\frac{d\vec{r}}{dt}=\frac{d(r\vec{e_r})}{dt}=\dot{r} \vec{e_r}+r\frac{d\vec{e_r}}{dt}=\dot{r} \vec{e_r}+r\dot{\theta}\vec{e_{\theta}}
$$
$$
\frac{d\vec{e_r}}{dt}=\dot{\theta}\vec{e_{\theta}}
$$
$$
\vec{a}=\ddot{r}\vec{e_r}+\dot{r}\dot{\theta}\vec{e_{\theta}}+\frac{d(r\dot{\theta})}{dt}\vec{e_{\theta}}+r\dot{\theta}\frac{d\vec{e_{\theta}}}{dt}=(\ddot{r}-r\dot{\theta}^2)\vec{e_r}+(2\dot{r}\dot{\theta}+r\ddot{\theta})\vec{e_\theta}
$$
$$
\tan{\theta}=\frac{dy}{dx},\frac{d\theta}{\cos^2{\theta}}=d(\frac{dy}{dx}),r=\frac{ds}{d\theta}=\frac{\sqrt{(dx)^2+(dy)^@}}{}
$$