Find angular momentum, energy, and distance of closest approach

[TSny]

I like the initial simplification.

r=$\frac{R}{2sqrt(1-AR_0)}$

I believe that should be $$r = \frac{R_0}{2\sqrt{1-\frac{AR_0}{mw^2}}}$$
With the approximation of the square root, you have $$r = \frac{R_0}{2}(1+\frac{AR_0}{2mw^2})$$

I am comfortable now with the concepts that were touched on and this is how I would/will work out the problem. I must admit, though, that the further simplification for small x is something I would not likely use unless necessary.

Thank you for your advice and guidance!

OK. Good work.