So here's my physics conundrum: I am on a ROUND Earth, in a vacuum. If I throw a ball really fast (say 99.9% of the orbital velocity for the ball's altitude), and drop another ball straight down, which takes longer to hit the ground? My basic physics reasoning would say that it takes the same time, since from the thrown ball's reference frame, the distance to the ground is exactly the same whether it is thrown around the earth or dropped in one place. The thrown ball simply spirals around the earth to the ground, but effectively falls the same vertical distance as the dropped ball. However, if you throw the ball fast enough, couldn't you get it into what is essentially a very slowly decaying orbit, thus making the thrown ball hit the ground after the dropped one? How do I reconcile these two seemingly conflicting results, and what is the math behind it? This is not homework or anything, just a thought I had today.