On July 19, 1969, Apollo II's orbit around the moon was adjusted to an average orbit of 111km. The radius of the moon is 1785 km and the mass of the moon is 7.3e22 kg. a) How many minutes did it take to orbit once? b) At what velocity did it orbit the moon?

To determine the time it took for Apollo II to orbit once around the moon, we can use the equation for the period of a satellite orbiting a celestial body:

T = 2π√(r^3/GM)

where:
T = period (time taken to complete one orbit)
r = radius of the orbit
G = gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = mass of the celestial body

a) To find the time taken to orbit once around the moon (T), we need to convert the given average orbit radius from km to meters:

r = 111 km = 111,000 meters

Now we have all the necessary values to substitute into the equation:

T = 2π√[(111,000)^3 / (6.67430 × 10^-11 × 7.3 × 10^22)]

Calculating this expression will give us the period in seconds. To convert it to minutes, we will divide by 60:

T (in minutes) = T (in seconds) / 60

b) To determine the velocity of Apollo II as it orbited the moon, we can use the formula for orbital velocity:

v = √(GM/r)

where:
v = orbital velocity

Substituting the known values:

v = √[(6.67430 × 10^-11 × 7.3 × 10^22) / 1785]

Calculating this expression will give us the velocity in meters per second.

Now that we have the steps and formulas, we can proceed to calculate the values.

To find the answers to both parts of the question, we can use the principles of circular motion and the gravitational force. Let's break it down step by step:

a) To calculate the time it takes to complete one orbit, we can use the formula for the period of an object in circular motion:

T = 2πr/v

where T is the period, r is the radius of the orbit, and v is the velocity.

Given that the average orbit of the Apollo II is 111 km (or 111,000 m), we can substitute this value for r:
T = 2π(111,000)/v

To calculate the period, we also need to find the velocity of the orbit. We can use the formula for the centripetal force:

F = (mv^2)/r

where F is the gravitational force between the moon and Apollo II, m is the mass of Apollo II, and r is the radius of the moon.

Set the gravitational force equal to the centripetal force and solve for v:
(G * M1 * M2) / r^2 = (m * v^2) / r

where G is the gravitational constant (approximately 6.67430 × 10^-11 m^3/(kg * s^2)), M1 is the mass of the moon, M2 is the mass of the Apollo II, and v is the velocity.

Rearranging the equation to solve for v gives us:
v = sqrt((G * M1 * M2) / r)

Given the radius of the moon as 1785 km (or 1,785,000 m) and the mass of the moon as 7.3e22 kg, we can substitute these values into the equation to find v.

Substitute the value for v into the equation for T:
T = 2π(111,000) / sqrt((G * M1 * M2) / r)

Finally, calculate T to find the answer in minutes.

b) To find the velocity at which the Apollo II orbits the moon, substitute the values of the mass of the moon (M1) and the Apollo II (M2), the radius of the moon (r), and the gravitational constant (G) into the equation for v:

v = sqrt((G * M1 * M2) / r)

Calculate v to find the answer in units of velocity, such as meters per second (m/s).

Haha w

GM/R^2 = V^2/R

M is the mass of the moon.
R = 111 + 1785 km = 1896*10^3 m
G is universal constant of gravity
b) Solve for V.
a) period * V = 2 pi R