8

# Velocity Azimuth

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 v_{θ and }longitude v_{φ components of velocity. Begin by taking the time derivative of equation 6. }

v_{x = a(− }cosφ sinθ v_{θ − cosθ sinφ }v_{φ ) }

(9)

v_{y = a(−sinφ }sinθ v_{θ + }cosθ cosφ v_{φ ) }

v_{z = a(cosθ }

v_{θ ) }

From the last equation in (9), we can solve for the latitude velocity component.

v_{θ = }

v_{z }a cosθ

(10)

Now plug v_{θ into either the vx or vy equation and solve for vφ. }

v_{φ = }

v_{y + vz sinφ tanθ }a cosθ cosφ

(11)

If this equation turns out to be singular, then use the v_{x }equation.

v_{φ = − }

v_{x + vz cosφ tanθ }a cosθ sinφ

(12)