Question

1. During the Apollo missions, astronauts traveled from the Earth to the Moon and back. When they were close to Earth, gravity accelerated them toward Earth, but when they were close to the Moon, gravity accelerated them toward the Moon. However, there is a point somewhere between the Earth and the Moon where they experienced no acceleration due to gravity.

   The Earth is 6.0×1024 kg, the Moon is 7.3×1022 kg in mass, and the two are 3.8×108 m apart. How far from Earth were the astronauts when they experienced zero acceleration due to gravity? Hint: remember that zero acceleration requires zero net force.

Answer 1

Let m be the mass of the astronauts. The gravitational force of an object of mass M on the astronauts when they are distance r away from the object is

Let astronauts be distance x from the Earth. The gravitational force of the Earth on the astronauts is

If d be the distance between the Earth and the Moon, the astronauts are distance d-x from the moon. The gravitational force of the moon on the astronauts is

The gravitational force of the Earth and the moon points in opposite directions. At the point of zero net force, the two forces balances each other.

Substituting given values