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The moon does fall toward the earth, but it is in pseudo-equilibrium with its momentum. Think of the moon as a Baseball that's been batted so hard, it actually falls over the horizon -- over the curvature of the earth. It is in freefall around the earth. It circles around the Earth which circles around the sun, but if the Earth and Sun were to somehow suddenly not exist, the Moon would travel in a straight line instead of in a circle.

The Earth actually does the same to the Sun -- the Earth falls over the Sun's horizon.

Earth's gravity is strong, and no human astronaut has actually ever escaped it -- the reason astronauts experience weightlessness in orbit is not because the earth's gravity no longer affects them (if it did not, the spacecraft they are in would fly off into deep space instead of orbitting the earth), it is because they are in freefall; you would obtain the same effect if you were in an airplane plummeting straight down toward the earth.

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