Why do all objects fall with the same acceleration regardless of mass?

[themadquark]

I am well aware that objects of varying masses, shapes, and surface areas will fall at different speeds and accelerations in an environment with a gas in the way such as air due to air resistance. Why is it though, that gravity causes all objects to fall with the same acceleration in a vacuum? Objects that fall further and have more energy and less time to decelerate have much more impact force, so why is it that this happens?

 

[Dale]

This happens due to the equivalence of inertial mass and gravitational mass. For inertial mass we have $\Sigma F = m_i a$. For gravitational mass we have $F_g=G M m_g/r^2$. If the object is in free fall then $\Sigma F = F_g$ so we have $m_i a = G M m_g/r^2$. Then, because inertial mass and gravitational mass are the same we can set $m=m_i = m_g$ and get $a = G M/r^2$, which is independent of $m$.

 

[Nugatory]

This is an FAQ over in the General Physics section: https://www.physicsforums.com/showthread.php?t=511172

 

[ZapperZ]

I am well aware that objects of varying masses, shapes, and surface areas will fall at different speeds and accelerations in an environment with a gas in the way such as air due to air resistance. Why is it though, that gravity causes all objects to fall with the same acceleration in a vacuum? Objects that fall further and have more energy and less time to decelerate have much more impact force, so why is it that this happens?

Please start by reading this FAQ entry:

https://www.physicsforums.com/showthread.php?t=511172

Zz.

 

[Cosmic Al]

Newton's Laws of motion & gravitation give F = ma =GMm/r^2 where r is distance from center of Earth (roughly constant for dropping light & heavy objects). The mass m of the object cancels out, so its acceleration doesn't depend on its mass. Assume that air resistance isn't a factor.
Galileo showed a non-mathematical proof: Aristotle says that heavy objects fall faster than light objects. So what if we tie together a heavy object with a light object. By Aristotle's reasoning, the light object would then slow down the heavy object and at the same time, the heavy object would speed up the light object. The composite light-heavy mass would fall somewhere between the speed of the two alone, say an average. But the mass of this composite is greater than the mass of either part of the composite, so it should fall faster than either the light or heavy object. Thus, we have a problem in which we have proved that the composite both falls slower than one of its components and also falls faster than either component.