There are two ways to think about how gravity affects objects; The Newtonian way and the Einsteinian way:
Newton would say what the other commenters all have already said: More massive objects experience a larger *Force* proportional to their mass, but the *acceleration* itself is force divided by the mass again, so it cancels out in the end. Basically, if you’re heavier, gravity pulls on you more, but also you need to be pulled on more for the same acceleration because you’re harder to move.
Einstein’s approach to gravity is very different: According to him, gravity is not really a force, instead space just accelerates downwards at a constant acceleration, and an object that isn’t accelerated by an actual force moves along with it, downwards. This also explains very well why you’re weightless when you’re falling; you’re not accelerating in space, so there are no net forces acting on you. Standing on the ground, it pushes up against you and accelerates you upwards, so by the principle of action and reaction, you feel a downwards force.
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