A lot of what I said in describing how we might fall into the Moon, if you and I were in the same room and suddenly the rest of the world stopped existing, was incorrect. That isn’t to say it was wrong or even bad to consider; it just means that the equations that I produced and the numbers that came out from them aren’t exactly what would happen if the sudden-failure-of-planet-Earth case were to happen. I knew the wouldn’t be exactly right going in, which leaves us the question of what I thought I was doing and why I bothered doing it.
The first reason, and the reason why it wasn’t a waste of time to consider these simple approximations of how strongly the Moon is attracting us — how fast we are falling into it, and how fast we would be falling if the Earth weren’t falling into the Moon along with us — is thanks to something which Isaac Asimov perfectly described. In an essay called “The Relativity Of Wrong”, he wrote about — well, the title says it. Ideas are not just right or wrong; they can be wrong by differing amounts, and can be wrong by such a tiny amount that it isn’t worth the complications to get it exactly right. Probably the most familiar example is the flatness of the Earth. To model the globe, or a large nation, the idea that the Earth is nearly flat is sufficiently wrong as to produce measurable, important errors where plots of land are justifiably claimed by multiple owners, maybe from multiple governments, or aren’t claimed at all and form the basis for nowhere towns in which mild fantasy or comic stories can be set. But if one wants to draw a map of the town, or of one’s own property, the curvature of the Earth is not worth considering. We can pretend the Earth is flat and get our work done a lot sooner. Other sources of error will mess up the precise result before that does.