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?

It depends whether the question means:
a) what will be the relative positions of the stone and the vertical of the balloon once the stone lands.
or
b) what will be the relative positions of the stone once it lands the vertical of the balloon when the stone was dropped.

If it means b) then coriolis makes the stone fall slightly westward, because the earth moves eastward. Even if there's absolutely no wind, both balloon and stone have inertia, so the earth will have moved eastward the initial vertical of the balloon when the stone lands. So Sophie would be right.

If it means a) then the stone will fall slightly eastward. Nor only because, as Kenny rightly says, the stone has a greater tangential velocity than the earth, but also because as it falls, its tangential velocity increases further to make up for the loss of momentum involved in the diminishing radium of its circular motion respect the earth.
A beautiful problem.

Posted by carlos on 2005-11-27 14:50:49