It is often stated that masses fall at the same rate towards the Earth at the same rate (barring things like friction), and this seems to be quoted as the Equivalence Principle. Is this statement correct? I was thinking that the force of $M$ on $m$ causes $m$ to accelerate towards $M$ with $a_m = GM/r^2$, and similarly the force of $m$ on $M$ to accelerate towards $m$ with acceleration $a_M = Gm/r^2$. Isn't the relative acceleration $a_m + a_M = G(M+m)/r^2$, and if you're stationary on $M$ it will appear that $m$ is accelerating towards you with an acceleration that depends slightly on $m$? If correct, I assume the correct statement should be something more like "objects of different masses fall in a constant gravitational field at the same rate"? (It may even be that it is?)
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