Coordinate transform given $r$,$\theta$, $\dot\theta$, $\ddot\theta$,

Question

given $r$,$\theta$, $\dot\theta$, $\ddot\theta$ I need to find $x$,$y$,$v_x$,$v_y$,$a_x$,$a_y$, what are the equations for those variables?

$\dot r = 0$

Answer 1

I am not very sure about the meaning of your symbols. But if my understanding is correct. Then I have

$$x=r\cos\theta$$

$$y=r\sin\theta$$

$$v_x=\dot r\cos\theta+r(-\sin\theta)\dot\theta=-r\dot\theta\sin\theta$$

$$v_y=\dot r\sin\theta+r(\cos\theta)\dot\theta=r\dot\theta\cos\theta$$

$$a_x=-\dot r\dot\theta\sin\theta-r\ddot\theta\sin\theta-r\dot\theta(\cos\theta)\dot\theta=-r\ddot\theta\sin\theta-r(\dot\theta)^2\cos\theta$$

$$a_y=\dot r\dot\theta\cos\theta+r\ddot\theta\cos\theta+r\dot\theta(-\sin\theta)\dot\theta=r\ddot\theta\cos\theta-r(\dot\theta)^2\sin\theta$$