If I understand your idea correctly, you're proposing that one could travel from East to West simply by going up in a helicopter (or a balloon or something) and letting the Earth spin underneath. There are some minor problems with this proposal, but more importantly there's been a major change (due initially to Galileo) in our understanding of how physics operates. The picture you're using is basically that of Aristotle.
Let's say that you jump straight up. Do you then land on a different spot on the floor than where you started? No- because initially you and the floor are moving together. You continue to move together in the absence of some influence changing one of those motions. The same happens if you throw a ball straight up. Even if you're on a 'moving' train, it will fall right back to you. That fact goes by various names: the principle of inertia, Newton's first law, the principle of relativity, etc. There is no such thing as a special state of rest, so there's no tendency for you to "stop moving" while the Earth continues to move.
What are the minor problems? The only important sideways force here is from the wind, and so which way you tend to blow depends on the wind currents in your area. It's true that steady winds in some places allow travel by balloon, without needing any sideways propulsion. That is the closest I can think of anything matching your idea, but the direction of the winds is not always East to West. In my area it usually goes the other way.
Mike W. (with help from Inga and Jon)
(published on 05/16/2013)