Solutions To Retrograde Motion Problems:

 1)Solution:  

We have:   V2/V1 = (a2/a1) (T1/T2)

By convention we assign '1' to the inner planet (Venus) and '2' to the outer (Earth). We have a2 = 1 AU and for Venus (from Kepler's third law):

T1 = (224.69/365.25) yr. = 0.6151 yr.

a1 = {[T1]²}^1/3 = [(0.6151)²]^1/3

a1 = 0.723 AU

Therefore:

V2/V1 = (0.7234)(1/0.6151)

V2/V1 = 1.175

(b) According to a Table of Orbital Velocities in Astrometric & Geodetic Data:

V(Venus) = 35.02 km/s

V(Earth) = 29.78 km/s

Take the ratio of the velocities: 

V(Venus)/V(earth) = (35.02 km/s)/ (29.78 km/s) = 1.175

So, Venus' orbital velocity is 1.175 times Earth's which conforms to the result of part (a).

2. The  component, V_p cos (φ)  doesn't contribute to the observed angular velocity of the planet because the  component vector direction  (along line P'E')  is oblique to the motion vector (itself tangent to the orbit). 

3) If the angular velocity of the planet as observed from Earth is:  - (V_p - V)/ PE and parallel to the orbital motion, then it must also be in a direction opposite to the orbital motion, and hence is retrograde at opposition.  (Since V_p < V )