During an Apollo lunar landing mission, the command module continued to orbit the Moon at an altitude of about 165 km. How long did it take to go around the Moon once?
1) Find the radius of the moon. I will call it M.
2) Add the altitude above the moon's surface to the moon's radius. This will give you the radius of the command module's circular orbit (we are assuming the orbit is circular).
So you have r = M + 165km
3) Now find the circumference of the circle, or orbit. Use the equation C = 2(pi)r.
4) Now you know the distance for each orbit. Is that what the question means by "how long"? If so, you are done. If not, you have more to do, such as determining the speed the command module traveled.
To determine how long it took for the command module to orbit the Moon once during an Apollo lunar landing mission, we need to know the Moon's orbital period.
The Moon's average orbital period is approximately 27.3 days (or about 27 days, 7 hours, 43 minutes, and 11.6 seconds). This means that it takes the Moon approximately 27.3 days to complete one orbit around the Earth.
Now, the altitude of the command module during the lunar landing mission is given as 165 km. Since this altitude is relatively low compared to the Moon's mean distance from the Earth (about 384,400 km), we can consider the orbit of the command module around the Moon as circular, neglecting any variations due to the Moon's irregular shape or gravitational anomalies.
To calculate the time it takes for an object to complete an orbit, we can use Kepler's Third Law, also known as the law of periods. It states that the square of the orbital period (T) is directly proportional to the cube of the semi-major axis (a) of the orbit.
In this case, the altitude of the command module above the Moon's surface (165 km), which essentially represents the radius of the orbit, can be used as the semi-major axis (a). We need to convert this altitude to meters by multiplying it by 1000, giving us 165,000 meters.
Using the law of periods, we can set up the following equation:
T^2 = (4π^2 / GM) * a^3
Where T is the orbital period we want to determine, G is the gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2), and M is the mass of the Moon (approximately 7.348 × 10^22 kg).
Plugging in the given values, we can solve for T:
T^2 = (4π^2 / (6.67430 × 10^-11 * 7.348 × 10^22)) * (165,000)^3
Simplifying this equation gives:
T^2 ≈ 1.099 * 10^8
Taking the square root of both sides gives us:
T ≈ √(1.099 * 10^8)
T ≈ 10485 seconds
Therefore, it took approximately 10,485 seconds (or approximately 2 hours, 54 minutes, and 45 seconds) for the command module to orbit the Moon once during the Apollo lunar landing mission.