Why do objects with different masses fall at the same rate? 
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Confused about the role of mass 

Today we were in our Literature class talking about the Renaissance and the Enlightement and our teacher also said that scientific experiments were being conducted, and she gave as an example the experiment in which they dropped objects with different mass from a tower to see which object would land first. She then said that she herself didn't know the outcome and since I'm known in my class as the #1, they asked me and I said that they landed at the same time (to my shock many classmates even disagreed with me about this fact). The teacher asked me to explain why and we haven't had anything about inertia in our physics class, so I was forced to use my own self-thought knowledge of physics: I said gravity does pull harder on heavier objects, but heavier objects have more resistance to move, in other words, they have more inertia. The entire class except for the teacher disagreed with me even though they usually don't. One fairly annoying kid asked me to explain why a feather falls way slower than a bowling ball, and I explained why and I also made the claim that in a vacuum they would fall at the same rate. Was I right or my class? Of course I know my explanation is lacking, but it has truth to it, right? 
 A: Your teacher was referring to an experiment attributed to Galileo, which most people agree is apocryphal; Galileo actually arrived at the result by performing a thought experiment. Your answer to the feather vs. the bowling ball question is also basically correct.
Two other things to be said here:
In order to answer a question on physics or any other subject, there has to be a minimum knowledge and terminology by the person asking the question and the answerer, otherwise it boils down to a useless back and forth. I suggest watching Feynman's famous answer to see a good example.
The second point is the question why the extra pull of the gravity gets exactly cancelled by the extra "resistance" of the object, as you put it. This leads to the question as to why the $m$ in the $F=GMm/r^2$ is the same as the one in $F=ma$. This is known as the Equivalence Principle.
A: In very simple terms, the force pulling the "heavy" object down is greater BUT it also takes more force to accelerate a heavy object. These two effects cancel out.
In complex terms, why this is true - ie. the reason why gravitational mass and inertial mass are the same - is still a puzzle to physics.
