How far do these astronauts drift apart?

Question

Two astronauts are each in a circular orbit around an unknown planet. Astronaut "A" is displaced 100 meters "vertically" from Astronaut "B" (that is, the two astronauts are 100 meters apart, on a line that intersects the planet's center).

How far apart will the astronauts be from one another, after they have completed one orbit?

(Surprisingly, the answer does not depend on the planet's mass or on the radius of the orbit. You may assume, however, that the radius of the orbit is much, much greater than 100 meters (and use any resulting mathematical approximations).)

Answer 1

KE = -1/2 PE
v²/2 = 1/2 mgR²/r
v² = mgR²/r
v = √(mgR²) / √r
T = 2πr/v = 2π r√r /√(mgR²)

dT = 3/2 2π√r / √(mgR²) dr
ΔT = 3π√r / √(mgR²) Δr
Δx = ΔT v = 3π√r / √(mgR²) Δr x √(mgR²) / √r

Answer:
Δx = 3πΔr = 942.5 m