# Indeterminate velocity components of a particle at the center of a polar coordinate system

Given coordinates of a particle and components of its velocity vector in Cartesian coordinates, I need to convert this data into polar coordinates. I had no problem to deduce the formulas which give $$v_r$$ and $$v_\theta$$ in terms of $$v_x$$ and $$v_y$$ (see, e.g., the answer to this question). The result is: $$v_r = \dot r = \frac{x\dot x+y\dot y}{\sqrt{x^2+y^2}}, \qquad v_\theta = r\dot\theta = \frac{x\dot y - \dot x y}{\sqrt{x^2+y^2}}.$$ But what if a particle happens to be exactly at the center of the coordinate system? In Cartesian coordinates a particle might perfectly have $$x=0$$ and $$y=0$$, and , say $$v_x=1$$, $$v_y=1$$, so that the velocity vector has the magnitude of $$\sqrt{2}$$ and is directed to the center of the first quadrant. Now, when I switch to the polar system, I know that the velocity magnitude should be the same ($$\sqrt{2}$$), but in this case the two components are indeterminate (both are $$0/0$$)! How can I circumvent this "coordinate artifact" and to get sensible values of the velocity components in polar coordinates?

Therefore $$v_r = v_{total}$$ and $$v_{\theta} = 0$$.
Mathematically we have: $$\dot x = \dot r \cos \theta - r\dot\theta \sin\theta = \dot r\cos\theta$$ $$\dot y = \dot r \sin \theta + r\dot\theta \cos\theta = \dot r\sin\theta$$ therefore, $$\dot r = \sqrt{\dot x^2+\dot y^2}$$ which is generally not true but is in this case because $$v_{\theta} = 0$$ by definition.