Kids come up with the greatest questions. (I guess I'm assuming your daughter is still a kid. For all I know, she could be a 45-year-old award-winning nuclear physicist.)
Most people's instinctive answer to this question would be "No, it wouldn't go around the world, it would stay right where it started." If you picture a hovering helicopter, it stays above one certain spot on the ground (a burning building, a busy intersection, Catherine Zeta-Jones' backyard); that's why we say it's "hovering" as opposed to "flying." And since we know that that spot (like all other spots) is revolving around the Earth's axis, we should deduce that the helicopter revolves with it.
What's not so obvious is why this is happening. After all, if the helicopter is in the air, isn't it freed from the Earth's rotation? Won't the Earth just spin underneath it, like a merry-go-round that you're not riding? That's where the question can start to get confusing.
Actually, when the Earth spins, it's not just the solid rock beneath our feet that is revolving. The atmosphere surrounding that rock is also revolving, moving in the same direction and with the same speed as the Earth's surface. We tend to think of air as being "nothing," but it is much more than that. Even though you can't see it (unless you live in Los Angeles), it has mass, it moves, and it has a definite physical presence. If the air stayed in one place while the Earth spun, we would perceive it as wind moving against us. (You feel the sensation of wind when you're riding on a roller-coaster or in a convertible, even when the air is perfectly still.) And this wouldn't be a light breeze — these winds would be nonstop, at speeds of more than 1000 km/hr.
At this point, we should go back to an example previously seen in the Answer Man column: What happens when you throw a ball up in the air on a bus travelling at 80 km/hr? Does the ball go flying backward and splat on the rear window? No, of course not. It drops right back into your hand. (Click the link above for a more detailed explanation.)
Your helicopter works much like this ball. Although it appears to be going straight up and staying there, hovering in the sky, it's actually in motion, just like everything on the Earth's surface. The helicopter was already moving around the Earth's axis even before it took off, just like your ball was already moving forward even when you were holding it "motionless" in your hand. When it elevates into the air, it still retains its rotational motion (around the Earth's axis) and is pushed/carried along by the air that is also rotating with the Earth.
Your hovering helicopter will actually make a complete rotation around the Earth's axis in 24 hours (most of us do this every day), but it will see the exact same terrain beneath it for the duration of the "trip." Relative to the Earth's surface, it's not moving at all. Relative to an outside observer (maybe someone on Mars), it's moving in a big circle.
Realize that the Earth is spinning at the equivalent of 1,500 km/hr at the equator (it slows down as you approach the poles). That's more than 400 metres per second. If losing contact with the Earth's surface meant you could escape this motion, then you could toss a ball straight up in the air and it would land a kilometre away. You could jump and land on the next block. But we know this doesn't happen. That's because everything associated with the Earth, including the ground, the air, the birds, the balloons, and the helicopters, is in a constant state of motion. But when everything is moving in the same direction at the same speed, we don't detect any motion at all.
Sources: Around the World in 24 Hours, PhysLink.com
