Where Do You Aim to Shoot a Falling Object?

This question should not be confused with the “Rifleman’s rule” or “cosine rule” that approximates the aim point when shooting at a stationary target above or below the elevation of the firing point. In this case, the target is falling under the influence of gravity, dropped at the same time the bullet is fired. The following discussion is a bit of a spoiler, so I’ll put it behind the curtain.

This is similar to the physics puzzle that asks whether a bullet fired horizontally at high velocity strikes the ground before, at the same time, or later than a rock dropped at the moment the gun discharged. (Pedants will note that the usual answer to this does not take into account the curvature of the Earth, but that effect can usually be neglected except in very long range cases such as naval gunnery.)