still cant get this one?
so damon i know you wanna help!
or anyone else im open for suggestions
haha
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 3.72E+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?
The distance from the moon to this Lagrange point we can call x
Then the distance from the earth to that point is (3.72*10^8 - x) meters (note conversion to meters)
the gravitational attraction of the earth on our ship is
G * mass earth * mass ship /(3.72*10^8 -x)^2
the gravitational attraction of the moon on our ship is
G *1.200^10^-2 mass earth * mass ship /x^2
Set those attractions equal and you have your point. Notice that the gravitational constant G cancels as does the mass of our spaceship.
Solve for x, convert it to kilometers from meters.
i don't think the conversion will be necessary because it wants the answer to be in scientific notation
To find the spaceship's distance from the center of the Moon, you can use the concept of gravitational forces and Newton's law of universal gravitation.
1. Start by finding the gravitational force between the spaceship and the Earth. The gravitational force between two objects is given by the equation:
F = G * (m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 N·(m/kg)^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
2. Since the spaceship is weightless at that point, the gravitational force between the spaceship and the Moon must also be equal and opposite to the gravitational force between the spaceship and the Earth. So we can set up the following equation:
G * (m_spaceship * m_Moon) / r^2 = G * (m_spaceship * m_Earth) / (3.72E+5 - r)^2,
where m_spaceship is the mass of the spaceship, m_Moon is the mass of the Moon, m_Earth is the mass of the Earth, and the distance between the centers of the Earth and the Moon (3.72E+5 km) is subtracted by r to represent the distance between the spaceship and the Moon's center.
3. We can simplify the equation by canceling out the gravitational constant and the mass of the spaceship. Since we are looking for the distance from the center of the Moon, we can solve for r:
m_Moon / r^2 = m_Earth / (3.72E+5 - r)^2.
4. Now we can substitute the Moon-to-Earth mass ratio (1.200E-2) for the ratio of m_Moon to m_Earth:
(1.200E-2) / r^2 = 1 / (3.72E+5 - r)^2.
5. Cross multiply and solve for r:
(1.200E-2) * (3.72E+5 - r)^2 = r^2.
Expand the right side:
(1.200E-2) * (3.72E+5)^2 - (1.200E-2) * 2 * (3.72E+5) * r + (1.200E-2) * r^2 = r^2.
Rearrange the equation:
(1.200E-2) * (3.72E+5)^2 = (1.200E-2 + 1) * r^2 - (1.200E-2) * 2 * (3.72E+5) * r.
Simplify further:
(1.200E-2) * (3.72E+5)^2 = r^2 * (1 + (1.200E-2) * (2 * (3.72E+5)))),
(1.200E-2) * (3.72E+5)^2 = r^2 * (1 + (1.200E-2) * (7.44E+5)),
r^2 = (1.200E-2) * (3.72E+5)^2 / (1 + (1.200E-2) * (7.44E+5)).
Calculate the right side of the equation to find r^2.
6. Finally, take the square root of r^2 to find r, the spaceship's distance from the center of the Moon.