On July 19, 1969, Apollo II's orbit around the moon was adjusted to an average height of 188 kilometers above the moon's surface. The radius of the moon is 1810 kilometers, and the mass of the moon is 7.3 x 1022 kilograms.
a. At what speed did it orbit the moon?
b. How many minutes did it take to orbit once?
m gmoon = m v^2/r
where r is the orbit radius
r = 1.81^10^6 +.188*10^6 meters
g moon = G Mmoon/r^2
so
v^2 = gmoon * r
square root of that is v
then
2 pi r = distance around orbit
v = speed
time = distance / speed
To answer these questions, we can utilize Newton's law of universal gravitation and the equations of centripetal motion. Let's break down the steps to find the answers to each part:
a. To find the speed of the Apollo II spacecraft as it orbits the moon, we can equate the gravitational force between the spacecraft and the moon to the centripetal force.
First, let's find the gravitational force between the spacecraft and the moon using Newton's law of universal gravitation:
Gravitational Force = (G * mass of spacecraft * mass of moon) / distance^2
where:
G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
mass of spacecraft is unknown
mass of moon = 7.3 x 10^22 kg
distance = radius of moon + average height of orbit above the moon
We'll solve for the mass of the spacecraft:
Gravitational Force = (G * mass of spacecraft * 7.3 x 10^22 kg) / (1810 km + 188 km)^2
Solving for the mass of spacecraft will give us an unknown value. Unfortunately, we don't have enough information to calculate the speed of the spacecraft without knowing the mass of the spacecraft.
b. Similarly, we can't determine the time taken to orbit once without the speed. The time taken to complete one orbit can be found using the formula:
Time (T) = (2π * orbit radius) / orbital speed
Since we don't have the speed, we cannot directly calculate the time taken to orbit once.
In summary, without knowing the mass of the Apollo II spacecraft or its orbital speed, we cannot calculate the speed or the time taken to orbit once.