The common understanding is that setting air resistance aside, all objects dropped to Earthfall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can’t then fall more slowly just by being sliced in two.

However, I believe the answer is that when two objects fall together, attached or not; they do “fall” faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward themselves, but the objects also accelerate the Earth toward themselves. Considering the formula:

\(F_{g} =\frac{Gm_{1} m_{2} }{d^{2} }\)

Given F = ma thus a = F/m, we note that the mass of the small object doesn’t seem to matter as when calculating acceleration, the force is divided by the mm term, it’s mass. However, this overlooks that the force is applied to both objects, not just to the smaller one. The acceleration on the second, larger object, is found by dividing F, in turn, by the larger object’s mass. The two objects’ acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:

\(a_{{\rm closing}} =\frac{F}{m_{1} } +\frac{F}{m_{2} }\)

Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is 5.972×10²⁴ kg, a falling object of 5.972kg(just over 13 pounds) would accelerate the Earth about 1/10²⁴ as much, which is one part in a trillion.

Of course, in everyday situations, we can, for all practical purposes, treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn’t even detect. But I’m hoping not for a discussion of practicality or what’s measurable or observable, but what we think is happening.

What clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn’t one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus, “falling” consists of multiple objects mutually attracting in space.