If there were no air, and we ignore things like the Coriolis effect (which would make you hit the side of your pit as you fell), you’d fall through the center of the Earth, come up on the other side, and then swing back down. You’d act kind of like a spring or a pendulum (the force ends up proportional to your distance from the center, just as it would in a spring, at least if we treat the Earth as an idealized sphere of constant density).
If you were to start in the middle of the Earth then you would feel no gravity at all, at least if we treat the Earth as an idealized sphere. You’d be floating and could go in whatever direction you wanted for quite a ways before gravity ramped up enough to matter.
Because earth has atmosphere, you will go through air resistance and eventually you will stop in the middle. But if you do it in the moon, given no resistance at all, you will fall, come on the other side, fall again, come on the other side, and then fall again like that back and forth indefinitely.
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