I am trying to convert an equation from Cartesian to polar coordinates. I know for a given $x$ and $y$ Cartesian, we can get polar $r=\sqrt{x^2+y^2}$ and $\theta = \arctan(y/x)$.
However, for a given $v_x$ and $v_y$, I want to get $v_r$ and $v_\theta$. Can we write $v_r = \sqrt{v_x^2+v_y^2}$ and $v_\theta = \arctan{(v_y/v_x)}$?
I know its' probably not that simple, but a simple derivation and explanation will be really helpful.
Best Answer
If $r(t) = \sqrt{x(t)^2+y(t)^2}$ , then $$\dot r = \frac{x\dot x+y\dot y}{\sqrt{x^2+y^2}}$$
If $\theta(t) = \arctan\left[y(t)/x(t)\right]$ , then $$\dot\theta = \frac{x\dot y - \dot x y}{x^2+y^2}$$
Alternatively:
If $x(t) = r(t)\cos\left[\theta(t)\right]$ , then $$\dot x = \dot r \cos \theta - r\dot\theta\sin\theta$$
If $y(t) = r(t)\sin\left[\theta(t)\right]$ , then $$\dot y = \dot r\sin\theta + r\dot\theta \cos\theta$$