this is ans to malay's question:

          it is not a case of TIR exactly. actually to see the world outside the max angle of incidence is 90 degrees which gives angle of refraction as theta which happens to be the critical angle. so the ans is 36 by root 7 but the above ans have wrong concept.

there are infinite no. of solutions . [ i have assumed k to be a positive integer. as there is no restriction on x.

Consider the particle at a position radius at which makes  with vertical.

     It is being acted by a pseudo force ma[a=g] constantly with respect to sphere.

 

time taken t = root of 2R(1-cos) /g

v_x=at = 2sin into root of gR

v_y=root of 2R(1-cos)/g= 2sin into root of gR[ this velocity is with respect to the sphere as this sphere has no velocity in vertical direction]

 thus the net 'speed' = 2sin into root of 2gR