I've learned from school and a lot of articles stating that all objects fall at the same rate under gravity if no other force is taken into consideration ( drag, hydrostatics and stuff ). They often use g=GMm/r2 and F=ma to explain this. We can equate g and F since both show the gravitation force, GMm/r2 = ma, and 'm' cancels out giving g=GM/r2 . When we take an example of Earth and two balls with different masses, we can conclude from this that the gravitational force depends only on the mass of earth. However, when we take an object with a mass x, and lets say Earth's mass is 1000000000000x for convenience. We get from F=ma that the gravitational force is xa. The object should exert the same force on the Earth, from newton's third law, so, xa = 1000000000000x*y, where y is Earth's acceleration towards the ball, which would be really, really close to zero when we take the actual masses. From this, we can conclude that when the mass of the object is lower, the acceleration should be higher because the forces are equal, so if we take an object with 2x mass, then the first object we took should reach Earth faster if both objects are dropped from the same height, lets say 5km height, ( because I believe the difference in acceleration would be really small, so we need a larger height to notice that difference ). Why does this dispute between the formulas happen? or is this interpretation wrong?
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