A comet of negligible mass travels around the Sun in an elliptical orbit with perihelion (closest-approach) distance r_p and aphelion distance r_a. Let v_p and v_a be its speeds at perihelion and aphelion, respectively.
(a) Derive an expression for the ratio vavp in terms of r_p and r_a only.
(b) Show algebraically that the product r_p v_p equals r_a v_a and interpret the physical significance of this equality.
(c) Express the eccentricity e of the orbit in terms of r_p and r_a, and then rewrite your result from part (a) solely in terms of e.
(d) For Halley’s Comet, r_p = 8.9×10^{10} m and r_a = 5.3×10^{12} m. Compute v_p, v_a, and their ratio, taking the Sun’s mass to be 2.0×10^{30} kg. Comment on how the calculated ratio compares with your general expression from part (c).