Question

An exoplanet is in an elliptical orbit around a distant star. At its closest approach, the exoplanet is 0.530 AU from the star and has a speed of 54.0 km/s. When the exoplanet is at its farthest distance from the star of 33.0 AU, what is its speed (in km/s)? (1 AU is the average distance from the Earth to the Sun and is equal to 1.496 ✕ 10¹¹ m. You may assume that other planets and smaller objects in the star system exert negligible forces on the exoplanet.)

km/s

(b)

What If? A comet is in a highly elliptical orbit around the same star. The comet's greatest distance from the star is 20,900 times larger than its closest distance to the star. The comet's speed at its greatest distance is 2.20 ✕ 10⁻² km/s. What is the speed (in km/s) of the comet at its closest approach?

km/s

Answer 1

a)

r_c = distance at closest approach = 0.530 AU

v_c = linear speed at closest approach = 54 km/s

r_f = distance at farthest point = 33 AU

v_f = linear speed at farthest point = ?

using conservation of angular momentum

m r_c v_c = m r_f v_f

(0.530) (54) = (33) v_f

v_f = 0.87 km/s

b)

for comet:

r_c = distance at closest approach

v_c = linear speed at closest approach = ?

r_f = distance at farthest point = 20900 r_c

v_f = linear speed at farthest point = 2.20 x 10^-2 km/s

using conservation of angular momentum

m r_c v_c = m r_f v_f

r_c v_c = (20900) r_c (2.20 x 10^-2)

v_c = 459.8 km/s