Orbital speed of a satellite due to gravity

“Nellie’s Satellite Speed” can be neatly generalized to determine the speed v of a satellite in any circular orbit of radius r about a planet.¹ Suppose the satellite falls vertically by a height h over an arclength s that is small compared to the circumference of the orbit. Then the figure shows that cos ϕ = r / (r + h), which rearranges into h = r (1 − cos ϕ)/cos ϕ ≈ rϕ²/2. Substituting ϕ = s/r results in h = s²/ 2r. (In Hewitt’s case, this relation correctly predicts s ≈ x = 8 km for h = 5 m and r = 6370 km.) Equating it to h = gt²/2, where t is the time it takes the satellite to move along this portion of its orbit and g is the gravitational...