Answer 1:
According to Newton's Law of Universal Gravitation, the combined pull exerted by the Earth and by the Moon on a third body cancels out if M/R^{2} = m/r^{2} where M is the mass of the Earth, m that of the Moon, R is the distance from the 3rd body to the center of the Earth and r the distance from the body to the centre of the Moon. When solving that equation, you have to take into account that R + r = D, the distance between the center of the Earth and the center of the Moon (which changes with time, but not that much, relatively speaking). You may want to try your algebraic skills in solving the two simultaneous equations for R and r. Having said that, according to Einstein's principle of equivalence, whenever you are freely coasting in the gravitational field of one or more bodies, you will experience ZERO gravity. This means that as soon as you turn off the engines of your spacecraft, you will start to experience ZERO gravity regardless of your position relative to the Earth and the Moon. In other words, the pull exerted by the Earth and by the Moon does not have to cancel out in order for you to experience ZERO gravity. Astronauts on board the shuttle experience ZERO gravity even though they are very close to the Earth.
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