Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and 160 km above the surface of the Moon, where the acceleration due to gravity is 1.42 m/s2. The radius of the Moon is 1.70 106 m.

(a) Determine the astronaut's orbital speed.
(b) Determine the period of the orbit.

I know I should use the variation of Newton's second law that says F=ma, or F=m(v^2/r). I just don't really get how to relate these two problems, since I have a radius of the moon, and a radius of orbit.

To solve this problem, we can start by recognizing that the gravitational force experienced by the astronaut orbiting the Moon provides the centripetal force required to keep them in orbit.

(a) To determine the astronaut's orbital speed, we can equate the gravitational force to the centripetal force:

F = m(v^2 / r)

where F is the gravitational force, m is the mass of the astronaut, v is the orbital speed, and r is the radius of the orbit.

The gravitational force is given by:

F = (GMm) / (r + R)^2

where G is the universal gravitational constant, M is the mass of the Moon, and R is the radius of the Moon.

Now, equating the gravitational force to the centripetal force, we have:

(GMm) / (r + R)^2 = m(v^2 / r)

Simplifying,

v^2 = (GM) / (r + R)

Plugging in the known values:

G = 6.674 × 10^-11 m^3/(kg s^2)
M = 7.35 × 10^22 kg
r = 1.70 × 10^6 m
R = 1.70 × 10^6 m

we can calculate v:

v^2 = [(6.674 × 10^-11 m^3/(kg s^2)) * (7.35 × 10^22 kg)] / [(1.60 × 10^5 m) + (1.70 × 10^6 m)]^2

v^2 ≈ 2.426 × 10^6 m^2/s^2

Taking the square root of both sides, we find:

v ≈ 1558 m/s

Therefore, the astronaut's orbital speed is approximately 1558 m/s.

(b) To determine the period of the orbit, we can use the formula:

T = (2πr) / v

Plugging in the known values:

r = 1.60 × 10^5 m
v = 1558 m/s

T = (2π * 1.60 × 10^5 m) / 1558 m/s

T ≈ 6454 s

Therefore, the period of the orbit is approximately 6454 seconds.

To solve these problems, we can use the law of universal gravitation and Newton's second law of motion. Let's break down the steps to find the astronaut's orbital speed and the period of the orbit.

(a) Determining the astronaut's orbital speed:
To find the astronaut's orbital speed, we can equate the gravitational force between the Moon and the orbiting astronaut to the centripetal force acting on the astronaut.

1. Calculate the gravitational force:
The gravitational force between two objects can be calculated using the formula: F = G * (m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant (6.67430 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the objects.

In this case, the mass of the Moon is not given, but we can ignore it because it cancels out in subsequent calculations, leaving us with the acceleration due to gravity of the Moon acting as the centripetal acceleration. Thus, we can calculate the force acting on the astronaut as F = m * a.

Given:
Acceleration due to gravity (a) = 1.42 m/s^2
Mass of the astronaut (m) = not given

2. Determine the centripetal force:
The centripetal force acting on the astronaut is provided by the gravitational force between the Moon and the astronaut, which can be written as F = m * (v^2 / r),
where v is the orbital speed and r is the radius of the circular orbit.

3. Equate the gravitational force to the centripetal force to find the orbital speed:
We can set up the equation F = F, where F is the gravitational force calculated in step 1 and F is the centripetal force calculated in step 2. We obtain:
(G * m1 * m2) / r^2 = m * (v^2 / r)

As mentioned earlier, we can ignore the mass of the Moon (m1) since it cancels out. Also, the mass of the astronaut (m) is not given. Since they didn't provide us with a mass value, we need additional information or an assumption to compute the orbital speed.

(b) Determining the period of the orbit:
To find the period of the orbit, we can use the orbital speed and the circumference of the orbit.

1. Calculate the circumference of the orbit:
The circumference of a circle is given by the formula: C = 2 * π * r,
where C is the circumference and r is the radius of the circular orbit.

2. Use the orbital speed and the circumference to find the period:
The period of an object orbiting a celestial body can be calculated using the formula: T = C / v,
where T is the period and v is the orbital speed.

Given:
Radius of the Moon (r) = 1.70 * 10^6 m

To proceed with solving these problems, we need either the mass of the astronaut or an assumption to compute it.

a=V^2/r

a)find r : 160*10^3 + 1.7*10^6 = ...

V=sqrt(ar)= sqrt(1.42r)= ... That's your answer.

b) T=(2pir)/V