I'm also really struggling with this question:
A particle moves on the smooth inside surface of the hemisphere z = −sqrt(a^2 − r^2), r <= a,
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.
a)Show that the particle moves between two heights in the subsequent motion, and find
them.
b) Show, too, that if the parameter β = (V^2)/4ga is very large then the difference between
the two heights is approximately a/2β.
a) Using conservation of energy I found that the particle moves between z=0 and z=[(V^2)+sqrt{(V^4) +16(g^2)(a^2)}]/4g but I am not at all confident in this answer.
b) Substituting in the parameter I get 2βa not the correct answer. This makes me believe I have gone wrong in part a.
Any advice on how to attempt this question/the correct answer would be great. I can supply more details on what I have tried if necessary.
Thank you.