Solve Elliptical Motion Homework: Find r', r'', θ', θ

[camorin]

Homework Statement

A particle of mass m is subjected to an isotropic, two dimensional, harmonic central force, F=-kr.
(r=(x,y)). At t=0 the particle is at r=A

and has velocity V y^hat.

Find r' r'' θ' θ'' in 2D spherical polar coordinates.

Homework Equations

F_x=-kx
F_y=-ky

The Attempt at a Solution

I have found the equations of motion, as well as the particular solutions using the initial conditions.
So far I have:
x(t)=-Acos(wt-π)
y(t)=V/wcos(wt-π/2)

From here I found r' to be Awsin(wt-π)-Vsin(wt-π/2)
and r'': Aw²cos(wt-π)-Vwcos(wt-π/2)
Im fairly certain everything up to this point is correct, but I have no idea what to do about theta.

I know in spherical polar coordinates θ=arctan(y/x) but I don't think I can just take the derivative in this form. I have tried setting x(t)=-Acos(wt) and y(t)=V/wsin(wt) by using the relation between sines and cosines offset by pi/2.This has brought me to
θ=arctan((V/wa)tan(wt)). Again, I hit a wall with finding the derivatives.

 

[Simon Bridge]

Is the primed notation a space or a time derivative?

I have no idea what to do about theta

Write down an expression for theta and take the appropriate derivative.

I know in spherical polar coordinates θ=arctan(y/x) but I don't think I can just take the derivative in this form.

Why not? Did you try?

I have tried setting x(t)=-Acos(wt) and y(t)=V/wsin(wt) by using the relation between sines and cosines offset by pi/2.This has brought me to
θ=arctan((V/wa)tan(wt)).

... what happens when you take the appropriate derivative of that then?

 

[camorin]

Im not sure of how to take the derivative of that arctan expression. Is there a way to simplify it? I've tried finding methods but I'm having trouble.

The primed notation is a time derivative, sorry.

 

[Simon Bridge]

Look it up.
... that's what everyone else does.