If I had a mass of $100\:\rm{kg}$ accelerating due to gravity, using $F=ma$:
$F = 100\:\rm{kg} \times 9.8\:\rm{m/s^2}$
$F = 980 \:\rm N$...
If I increased the mass to 200kg, the force would be 1960 N:
$F = 200\:\rm{kg} \times 9.8\:\rm{m/s^2}$
$F = 1960 \:\rm{N}$
Now, finally getting to my question: Does this increase in force (which is supposed to be a push/pull) mean that the object would fall faster when it weighs more?
Answer
No, the heavier object does not fall faster. Instead, they heavy and light object fall at the same acceleration (and hence the same speed if they are both simply dropped). This is an example of the equivalence principle.
The more massive object has more gravitational force on it, but it also has more inertia. Specifically, because the object is twice as massive, it has twice the inertial mass.
The force on it is doubled, so the acceleration stays the same.
If we look at
$$F = ma$$
we see that when $F$ and $m$ are both multiplied by 2, $a$ stays the same.
Check these questions for more:
Free falling of object with no air resistance
Why is heavier object more reluctant to get falling down?
Projectile motion without air resistance
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