ariels' argument is very old indeed. I believe I have heard it attributed to the venerable

Galileo myself. While attempting to find a citation for that argument (which, I agree, has probably been discovered and re-discovered repeatedly in the last half-millenium) I found a great little write up by

Yaskar Safkan, a physics doctoral candidate at some backwater hick school called

MIT. Safkan's not sure who came up with ariel's argument, either, but he thinks it may have been Galileo, and so I am therefore (by the

principle of repeated assertion) correct.

Safkan then goes on to explain why ariel's argument isn't entirely satisfactory, from a modern physics point of view, and then ties it in to a very accurate and easy to read variation of the material StormHunter presented in the writeup above. I think it's good enough that I am going to run the risk of rampant downvoting for redundancy and node it. The text will be my own, however, and the original was found at physlink.com in the "Ask the Experts" column.

First: the problem with ariel's "elementary proof". It makes no reference to acceleration due to *gravity* at all, only acceleration due to an unknown force. Is that a problem? Well, consider a situation where we are releasing objects which accelerate towards their destination via the electromagnetic force. Each object has a static (unchanging) charge *q* and a mass *m*. The rate at which out objects fall now will be proportional to q/m - an object with no charge won't fall at all, an object with a large charge and little mass will fall very quickly, and an object will a large charge and huge mass will fall quite slowly. And tying two objects together will cause them to fall at a rate proportional to (q_{1} + q_{2}) / (m_{1} + m_{2}), which is an averaged rate for the two objects.

Safkan then suggests an alternate, more accurate conclusion for ariel's argument: "If all objects which have equal weight fall at the same rate, then *all* objects will fall at the same rate, regardless of their weight."

Now we'll jump back to our force equations, staring with gravitational force:

F = GMm_{1}/r^{2}

where G is the constant of gravitation, M is the mass of the attracting body (i.e. Earth), r is the distance between the objects, and m_{1} is the gravitational mass of the object. How does this releate to the acceleration of our objects?

F = m_{2}a

where a is acceleration and m_{2} is the inertial mass of the object.

If we do a bit of algebra we quickly come up with

a = G(m_{1}/m_{2})M/r^{2}

And now we ask ourselves the key question: for any object, is the ratio of inertial mass and the gravitational mass of the object the same? Note that it isn't necessary for the ratio to be one; if inertial mass was always double that of gravitational mass, we could just adjust G by a factor of half and we'd have the same effect: the removal of the mass of our object from the calculation of gravitational acceleration on the object. It so happens, nicely enough, that general relativity requires that inertial mass and gravitational mass be equivalent.

Thus: the answer to the question "Why do objects with different masses fall at the same rate in a vacuum?" is simply "Because the ratio of gravitational and inertial mass is constant."^{*}

^{*} The author of this writeup assumes no responsibility if your 4-year old finds this explanation unsatisfying.