1. The problem statement, all variables and given/known data I'm currently working on this question: A particle moves on the smooth inside surface of the hemisphere z = -(a^2 - r^2)^(1/2), r <= a, where (r, theta, 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 theta-direction. 1. Show that the particle moves between two heights in the subsequent motion, and find them. 2. Show, too, that if the parameter b = (V^2)/4ga is very large then the difference between the two heights is approximately a/(2b). 2. Relevant equations Conservation of energy 1/2 m(r'^2 + r^2 theta'^2 + z'^2) + mgz = constant rtheta' = V r^2 theta = aV theta ' = aV/r^2 ( ' = dot, differentiation) 3. The attempt at a solution I'm really stuck on question 1, I got to an answer using conservation of energy that ended with solving a quadratic, this is the particle moves between z=0 and z= (V^2 + sqrt(V^4 + 16g^2a^2))/4g but when I sub in the parameter in (2) I get to the wrong answer, ie 2ab. So I guess I've done 1 wrong. Any ideas guys?