I learned that mass is a property of objects that specifies how much resistance an object exhibits to accelerate.
So, pushing an object with a greater mass would cause a smaller acceleration than pushing an object with a smaller mass, when the force is constant.
But, why this does not hold for free-fall acceleration? Why free-fall acceleration does not vary with varying mass?
Because, in Newtonian gravity, the gravitational force on an object is proportional to its mass.
So twice as much mass means twice as much force as well as twice as much resistance to acceleration, leading to exactly the same acceleration!
For example, for two point masses $M$ and $m$ separated by distance $r$, the magnitude of their attractive Newtonian gravitational force is
$$F=\frac{GMm}{r^2}$$
where $G$ is Newton’s gravitational constant.
If we use $F=m a$, the $m$ appears on the left side of the equation as well as the right side, so it cancels out and we get
$$a=\frac{GM}{r^2}$$
for the acceleration of $m$. Its acceleration depends not on its mass, but on the mass of the other particle!
The fact that gravity works this way seemed like a miraculous coincidence to Einstein. He invented General Relativity as a better explanation of why objects with different masses fall with the same acceleration in a vacuum.
