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The Moon, in the absence of the Earth, would move in a straight line. Earth's pull at a given instant of time pulls the moon closer to Earth, the new direction of Moon now being the resultant of it's old motion and Earth's inward pull. The moon would then like to move in a straight line, but along this new direction, on account of its inertia as enunciated by Newton's first law, but the next instant, the Earth pulls again the moon a bit more. If you keep doing this exercise, you will realize that the Moon describes an ellipse. So, the answer to your question is yes and no, yes because Earth makes the Moon move around it in a particular way due to gravity, but the cause of the initial motion of the Moon till Earth got hold of it is probably is buried in the formation of the Solar System!

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Q: Does Moon move due to Earth's gravity?
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