Galileo's apocryphal Leaning Tower of Pisa experiment, as well as his actual experiment rolling objects down ramps, are incredibly profound, demonstrating that an object's gravitational mass and inertial mass are equal, canceling out to result in acceleration constant across different objects. (Incidentally, because the interior of the Leaning Tower is hollow, it would be better to run the experiment on its inside where wind, especially updraft, is less likely to interfere.)
Currently the best explanation for the effect is General Relativity, with objects in free fall (including at rest) always traveling along geodesics in spacetime, independent of mass. Though that still requires explaining why the more massive falling ball, inducing itself a greater deformation of spacetime, doesn't behave differently from the less massive ball.
It's fun to imagine Galileo, thinking deeply about the experiment, deriving general relativity himself. Previously, imagining Newton.
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