If a stone is dropped from a balloon on a still day, does the stone fall directly below the balloon, or to the west or east of it?
I think the real dilemma of this problem lies with the 'still day'. The balloon is suspended in air, and in order for the day to be still, the ground and the air cannot move with respect to each other, so the air, and the balloon, are also rotating.
Under the assumption that the wind was completely still, the stone would, I think, fall directly under the balloon.
In real life, the air and the ground don't move exactly in unison, and I expect the air 'lags' behind the ground, which would cause the stone to land west.
And depending on the latitude of this experiment, Coriolis may be important too.