I was looking through the Mailbag Archive and came across the old-ish question of whether a feather and a hammer would hit the ground at the same time… Not to berate Ian, but I think he missed the point of the questioner. The questioner was saying that by Newton’s third law, the feather exerts the same force on the earth that the earth exerts on the feather, and the same thing with the hammer. Thus, the earth will be accelerating towards the hammer slightly faster than it accelerates towards the feather, and therefore, the hammer will hit the ground first.

In complicated physics language: The acceleration of two bodies towards each other under gravity is given by mu * a = -G*m1*m2/r^2, where a is the acceleration, m1 and m2 are the masses, r is the distance, G is the gravitational constant, and mu is the “reduced mass” of the two bodies. (This can be found in pretty much any first-year undergraduate physics text.) Mu is equal to (m1*m2) / (m1 + m2), which is about (but not quite) equal to m1 or m2 if we decide that one of the masses is negligible with respect to the other. For all practical intents & purposes, this is fine. However, if one decides to be particularly persnickety about this, then we find that the acceleration is equal to -G*(m1 + m2) / r^2.

So, if we assume that r is essentially constant over the distance in question, then… Let’s see, radius of the earth is 6370 km, a feather weighs about 1 gram, a hammer about 1 kg… If we drop these objects 100 m (about 40 stories or so), then the hammer will hit the ground about… 7.5 x 10^-25 seconds ahead of the feather. Talk about your photo finish.