A particle P of mass m is projected horizontally with speed √7/2ga from the lowest point of the inside of a fixed hollow smooth sphere of internal radius a and centre O. The angle between OP and the downward vertical at O is denoted by θ. Show that, as long as P remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is 3/2 mg(1 + 2cos θ).

Find the speed of P

i. When it loses contact with the sphere,

ii. When, in the subsequent motion, it passes through the horizontal plane containing O. You may assume that this happens before P comes into contact with the sphere again.