Rowe! Your back! I thought you might have gone to bed. Not sure what time zone you are in. Anyway! do I end up with:
or am I completely off track again?
Almost, if my calculations are correct (this is quite a tedious one!) you should have . You want to try and isolate , start by getting rid of that fraction by multiplying both sides by . Once that's "on the flat", simplify / expand where necessary so you can subtract the term on its own side of the equation and work from there.
Looks like this is the final/accepted version
Let k = SQRT[(1 - r^2)^2 + (2zr)^2] ; then:
x = MRr^2 / (mk)
k = MRr^2 / (mx) ; square both sides:
k^2 = M^2 R^2 r^4 / (m^2 x^2) ; substitute back in:
(1 - r^2)^2 + (2zr)^2 = M^2 R^2 r^4 / (m^2 x^2) ; rearrange:
4 z^2 r^2 = M^2 R^2 r^4 / (m^2 x^2) - (1 - r^2)^2
z^2 = [M^2 R^2 r^4 / (m^2 x^2) - (1 - r^2)^2] / (4r^2) ; sooooooooooo:
z = SQRT{[M^2 R^2 r^4 / (m^2 x^2) - (1 - r^2)^2] / (4r^2)}
Can be slightly simplified to:
z = SQRT[M^2 R^2 r^4 - (m^2 x^2)(1 - r^2)^2] / (2rmx)