To find the distance from Earth at which an astronaut would feel the same gravity pull from the moon as they would on the lunar surface, you can use the concept of gravitational force and Newton's law of universal gravitation.
The gravitational force between two objects is given by the equation:
F = (G * m1 * m2) / r^2
Where: F is the gravitational force between the objects, G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
On the surface of the moon, the gravitational acceleration is approximately 1.622 m/s^2. We can equate this to the gravitational acceleration between the astronaut and the moon when they are a distance 'x' away from the center of the Earth.
G * m_moon * m_astronaut / (x^2) = G * m_earth * m_astronaut / (r_earth^2)
Here, m_moon is the mass of the moon, m_earth is the mass of the Earth, and r_earth is the radius of the Earth.
By rearranging the equation, we can solve for 'x':
x = sqrt((m_earth / m_moon) * (r_earth^2))
Substituting the known values:
- Mass of the moon (m_moon) is approximately 7.342 × 10^22 kg
- Mass of the Earth (m_earth) is approximately 5.972 × 10^24 kg
- Radius of the Earth (r_earth) is approximately 6,371 km or 6,371,000 meters
x = sqrt((5.972 × 10^24 kg / 7.342 × 10^22 kg) * (6,371,000 m)^2)
Calculating the value of 'x' will give you the distance from the Earth at which an astronaut would experience the same gravitational pull from the moon as they would on the lunar surface.