conservation of energy/cylindrical polar coordinates

A particle moves on the smooth inside surface of the hemisphere

z = −sqrt(a^2− r^2)

where (r, θ, z) denote cylindrical polar coordinates, with the z-axis vertically upward.

Initially the particle is at z = 0, and it is projected with speed V in the θ-direction.

Show that the particle moves between two heights in the subsequent motion, and find

them.

Show, too, that if the parameter β = V^2/4ga is very large then the difference between

the two heights is approximately a/2β.

Ok

I've spent days trying to do this and I cant seem to get it!

I end up with **z**^2** = **z^2.v^2 - 2.g.z.a^2 - 2gz^3

where bold denotes 'z dot' (derivative of z wrt time)

factorising this I get z((z - ((v^2 + sqrt(v^2 - 4ga))/4g)(z - ((v^2 - sqrt(v^2 - 4ga))/2z^2 + a^2

getting the particle lying between((v^2 + sqrt(v^2 - 4ga)) and ((v^2 - sqrt(v^2 - 4ga))

is this correct?

how should I go about the next bit?

many thanks