How to convert from radial / polar velocity to cartesian velocities

Question

Suppose I have the following coordinate system:

My input is:

• Radial length $\rho$
• Radial velocity $\dot{\rho}$ (constant velocity)
• Angle $\phi$, where $\tan(\phi) = \frac{y}{x}$

Desired output:

• x-velocity $\dot{x}$
• y-velocity $\dot{y}$

How do I convert the radial velocity $\dot{\rho}$ to Cartesian velocities $\dot{x}$ and $\dot{y}$?

I've already computed $x$ and $y$, but I'm not sure if they're helpful:

$ \begin{align} \sin \phi &= \frac{opp}{hyp} = \frac{y}{\rho}\\ y &= \rho \sin \phi \\ \end{align} $

and

$ \begin{align} \cos \phi &= \frac{adj}{hyp} = \frac{x}{\rho}\\ x &= \rho \cos \phi \end{align} $

Answer 1

Since $\phi$ is constant the conversion is straightforward, indeed we simply have that

• $\dot x=\frac{d}{dt}(\rho\cos \theta)=\dot \rho \cos \phi$
• $\dot y=\frac{d}{dt}(\rho\sin \theta)=\dot \rho \sin \phi$