A satellite orbits Earth at radius r with speed v. What is v in terms of G, M, and r?

• A. v = sqrt(G M r)
• B. v = sqrt(G M / r)
• C. v = G M / r
• D. v = sqrt(r / (G M))

Answer: (B)

For a satellite in a circular orbit, gravity supplies the centripetal force needed to keep it moving in a circle. The gravitational force is Fg = GMm/r^2 toward the center, where M is the mass of Earth, G is the gravitational constant, and r is the distance from Earth's center. The required centripetal force for circular motion is Fc = m v^2 / r. Setting these equal gives GMm/r^2 = m v^2 / r. Cancel m and one r to obtain v^2 = GM/r, so v = sqrt(GM/r).

Note that r is the distance from Earth's center; if you’re using altitude above the surface, you’d replace r with R_E + h. This speed decreases with increasing r, since it goes as 1/sqrt(r).