# How to convert from radial / polar velocity to cartesian velocities

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}

• Is $\phi$ constant?
– user
Aug 28, 2021 at 20:40
• Yes, $\phi$ is constant. Aug 28, 2021 at 20:40

Since $$\phi$$ is constant the conversion is straightforward, indeed we simply have that
• $$\dot x=\frac{d}{dt}(\rho\cos \theta)=\dot \rho \cos \phi$$
• $$\dot y=\frac{d}{dt}(\rho\sin \theta)=\dot \rho \sin \phi$$