We know that the velocity vector is tangent to the sphere. Given the Cartesian velocity components from equation 3, we would like to compute the latitude vθ and longitude vφ components of velocity. Begin by taking the time derivative of equation 6.
vx = a(− cosφ sinθ vθ − cosθ sinφ vφ )
vy = a(−sinφ sinθ vθ + cosθ cosφ vφ )
vz = a(cosθ
From the last equation in (9), we can solve for the latitude velocity component.
vz a cosθ
Now plug vθ into either the vx or vy equation and solve for vφ.
vy + vz sinφ tanθ a cosθ cosφ
If this equation turns out to be singular, then use the vx equation.
vφ = −
vx + vz cosφ tanθ a cosθ sinφ