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Suppose I have the following coordinate system:

enter image description here

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} $

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

1 Answer 1

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

  • $\dot x=\frac{d}{dt}(\rho\cos \theta)=\dot \rho \cos \phi$
  • $\dot y=\frac{d}{dt}(\rho\sin \theta)=\dot \rho \sin \phi$
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  • $\begingroup$ Thank you. That's awesome. $\endgroup$ Aug 28, 2021 at 20:47
  • $\begingroup$ @stackoverflowuser2010 You are welcome! Bye $\endgroup$
    – user
    Aug 28, 2021 at 20:49

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