physics planetary motion

Question

On the Apollo missions to the Moon, the command module orbited constantly at an altitude of 110 km above the lunar surface. How much time did it take for the command module to complete one orbit?

Answer 1

For this we need the know the moon's mass and radius.

Once we know that then we can calculate the orbiter's speed,

and from that the period.

Let

m (unknown) be the mass of the orbiter,

v (unknown) be the speed of the orbiter,

M = 7.342×10²² kg be the moon's mass,

r = 1737 km be the moon's radius,

h = 110 km be the altitude of the orbiter

R = r+h = 1847 km = 1,847,000 m be the radius of the orbiter's orbit,

G = 6.67430×10^-11 m³kg^-1s^-2 be the gravitational constant.

The moon's gravity provides the centripetal force for the orbiter;

therefore those two must equal.

GMm/R² = mv²/R

v = √(GM/R)

The period T is then the time it takes to travel the distance 2πR with speed v.

T = 2πR/v = 2πR/√(GM/R)

Substituting actual numbers

T = 2×π×1847000/√(6.67430×10^-11×7.342×10²²/1847000) = 7124 s

which is a little less than 2 hours.