Consider a spaceship located on the Earth-Moon center line (i.e. a line that intersects the centers of both bodies) such that, at that point, the tugs on the spaceship from each celestial body exactly cancel, leaving the craft literally weightless. Take the distance between the centers of the Earth and Moon to be 3.92E+5 km and the Moon-to-Earth mass ratio to be 1.200E-2. What is the spaceship's distance from the center of the Moon?

you don't need the q

Let M1 = moon mass; M2 = Earth mass,
r = distance from spaceraft to moon, and
D = Earth-moon distance

When the two gravity pulls are equal,

M1/r^2 = M2/(D-r)^2
[(D-r)/r]^2 = M2/M1 = 1/(0.012)
(D-r)/r = 9.13 = R/r -1
D/r = 10.13
Look up D and compute r.

To find the spaceship's distance from the center of the Moon, we can start by calculating the gravitational forces acting on the spaceship from both the Earth and the Moon. Since the tugs on the spaceship from each celestial body cancel out, the net gravitational force on the spaceship is zero.

The gravitational force between two objects can be calculated using Newton's Law of Universal Gravitation:

F = (G * m1 * m2) / r^2

Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the two objects
r is the distance between their centers

The gravitational force from the Earth on the spaceship is given by:
F_earth = (G * M_earth * m_spaceship) / r_earth^2

The gravitational force from the Moon on the spaceship is given by:
F_moon = (G * M_moon * m_spaceship) / r_moon^2

Since the tugs on the spaceship from each celestial body cancel out, we have:
F_earth = -F_moon

Substituting the given values:
(G * M_earth * m_spaceship) / r_earth^2 = - (G * M_moon * m_spaceship) / r_moon^2

Simplifying:
M_earth / r_earth^2 = - M_moon / r_moon^2

The mass of the Earth (M_earth) divided by the square of the distance between the Earth and spaceship (r_earth^2) is equal to the negative of the mass of the Moon (M_moon) divided by the square of the distance between the Moon and spaceship (r_moon^2).

Now, we can solve this equation to find the spaceship's distance from the center of the Moon (r_moon).

Using the given values:
M_moon / M_earth ≈ 1.200E-2
r_earth = 3.92E+5 km

To find r_moon, we need to know the mass of the Earth (M_earth).

Unfortunately, the given information does not include M_earth. Without this value, we cannot solve for r_moon.