During an Apollo lunar landing mission, the command module continued to orbit the Moon at an altitude of about 113 km. How long did it take to go around the Moon once?
To determine how long it took for the command module to orbit the Moon once during an Apollo lunar landing mission, we need to consider the radius of the orbit and the orbital speed of the command module.
The radius of the orbit is given as 113 km, which means the total circumference of the orbit is 2*pi multiplied by the radius. Let's calculate it:
Circumference = 2 * pi * radius
Circumference = 2 * 3.14159 * 113 km
Circumference ≈ 710.841 km
Now, we need to find the orbital speed of the command module. The orbital speed can be calculated using the formula:
Orbital speed = 2 * pi * radius / orbital period
However, we are given the orbital altitude instead of the orbital period. To calculate the orbital period, we will need to know the gravitational parameter of the Moon, which is the product of the gravitational constant and the mass of the Moon.
Gravitational parameter of the Moon (μ) = 4.9048695 x 10^12 m^3/s^2
Using the concept of vis-viva equation, the orbital period (T) can be calculated using the formula:
T = 2 * pi * √(radius^3 / μ)
Converting the radius to meters:
radius = 113 km * 1000 m/km
radius = 113,000 m
Now, let's calculate the orbital period of the command module:
T = 2 * 3.14159 * √((113,000 m)^3 / 4.9048695 x 10^12 m^3/s^2)
T ≈ 107.215 minutes
Therefore, it took approximately 107.215 minutes, or about 1 hour and 47 minutes, for the command module to go around the Moon once during the Apollo lunar landing mission.