Heavier objects fall faster myth or fact?

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sigh hens
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Heavier objects fall faster...myth or fact?

(I'm not too good with history so correct me if I'm wrong.)

pre-gallileo this is what people generally assumed. Gallileo actually did some experiments to show that all objects fall at the same acceleration. Then Newton provided the math and it was realized that wind resistance was that factor that made everyone think heavier objects fall faster. And now it seems everyone says that if you neglect wind resistance that every object regardless of mass, will fall at the same acceleration. But is this technically false?

The Earth pulls everything at the exact same acceleration (assuming the distance from the Earth's core is the same). But the object being dropped is also pulling on the earth. So wouldn't a heavier object cause the Earth to accelerate towards it more than a lighter object? Which would mean that a heavier object would converge with the Earth sooner? The difference in time would be extremely extremely small but that doesn't make it less true.
 
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