Drop a cannonball from the Tower of Pisa. Where will it land?

An apocryphal story has Galileo dropping two cannonballs, one large and one small, from the top of the Tower of Pisa in 1589, to show people that small objects fall as fast as large ones.  Whether Galileo in fact performed the experiment, it's been done by many others.  Where did the cannonballs land?  Ignoring the effects of wind, the answer seems obvious; they fell straight down, and landed directly below, just as surely as two plus two equals four.  Indeed that’s a very accurate approximation.  Believe it or not, it’s only an approximation.

I find this weirdly comforting, like when you discovered that you could create many more colors than the six provided at the kindergarten fingerpaint easel. It's an affirmation that the world is full of endless, fascinating details, in every direction and at every scale.  No matter which direction you point your mental lens, no matter what magnification you set it to, the more you look the more patterns, intricate and beautiful, you will see.  Okay, so I like physics.  Sue me.

For the sake of discussion, let's keep it to one cannonball.  Assume there’s no wind resistance, the earth is spherical and of uniform density, the cannonball is as well, and there are no other gravitational forces.  Even in this simplified, imaginary world, calculating the exact landing spot, as opposed to a very accurate approximation, is hard.  (The fact that the Tower of Pisa has a noticeable lean has no bearing on the problem.)

The first complication is that the earth is rotating on its axis, and the tower is rotating along with it.  That means the top of the tower (i.e. the place where you drop the cannonball) is moving laterally faster than the plaza below.  Atop a 100 meter tower standing on the equator, you will travel roughly 628 meters farther every 24 hours (23 hours, 56 minutes, 4.0916 seconds, but who's counting).  This difference in lateral speed means the trajectory of a dropped cannonball will follow a slight curve.

The Tower of Pisa is north of the equator, which means its speed differential relative to the plaza below is smaller.  But it also leads to the second complication: the cannonball's trajectory will curve slightly south, due to Coriolis, an effect which is also responsible for the counter-clockwise rotation of hurricanes in the northern hemisphere.  So the cannonball will land slightly to the east and south of the spot below where it was dropped.

The cannonball's exact trajectory will depend on the forces it is subjected to.  The third complication is the centrifugal force produced by the earth's rotation, which counteracts to some degree the earth's gravity.  When weighing yourself, you will get a smaller number at the equator than at either of the poles (if the earth spun about 16 times faster, those at the equator would be weightless).  When you drop the cannonball, centrifugal force counteracts, somewhat, the downward acceleration due to gravity.

The fourth complication also lowers the downward acceleration at the moment the cannonball is dropped.  Over four centuries ago, in one of the most famous scientific advances, Isaac Newton showed that the force of gravity varies in inverse proportion to the square of the distance from the earth's center.  The top of the tower is farther from the center of the earth, which means the gravitational force is lower.  As the cannonball gets closer to the earth, the gravitational force increases.

The fifth complication has to do with the gravitational force caused by the cannonball itself.  All mass, no matter how small or how distant, exerts a gravitational force. While the cannonball is falling toward the earth, the earth is falling toward the cannonball.

The earth moves!  Two masses - the earth and the cannonball, in this case - accelerate toward each other.  Since the cannonball has less mass, it moves more than the earth does.  But the earth still moves. This universal law of gravitation was discovered by Isaac Newton.

Drop a cannonball from a 100 meter tower, and it will land after traveling slightly less than 100 meters, because the earth has been pulled ever so slightly toward the cannonball.

Einstein's theory of relativity introduces yet more complications, which I'll skip because I don't know enough about them.  And there may be more complications I haven't thought of.

These five complications have small effects, even when aggregated.  In almost any physics class, you can safely ignore them all, and pretend a dropped cannonball will fall straight down.  But rest assured that no matter where you look, if you choose to look further, there is more to find.