Momentum. It's all relative (yay! non-lightspeed Relativity talk!). The helicopter's still moving at the speed at which the Earth is rotating, so it doesn't move "back" (wait, it'd be forward, wouldn't it) through the time zones.
You're sitting in a chair? Not moving at all? You're really going a few tens of thousands of miles per hour (across a few different vectors that are all changing constantly, but that's less important). Earth's spinning on its axis (1,000 MPH), going around the sun (90 miles a second), the sun's hurtling around the galaxy (14,000 MPH), and the Milky Way's shooting off into the Universe (can't measure that speed because there's no point of reference; we can only measure the relative speeds of the other galaxies around us).
Speed is just relative to the notion that the Earth isn't moving, because to us, it doesn't look like it is.
Oh, here's an experiment you can do to prove that. So you're in a moving car. Don't do this while you're the driver. You toss a ball up from knee height to eye height. The ball will come right back down into your hand instead of flying backward and hitting you in the face. It has to do with the current moment of all objects involved. Since the ball has the same momentum as the car (because it did when you threw it up), it will continue to have that horizontal momentum even as you add the vertical vector of the throw. If you toss the ball up and have the driver slam on the brakes, the ball will fly forward and miss your hand. Don't try that, though. You'd have to slam fairly hard.
Now all this evolves into neat talk about ships moving at 99.999999995% light speed and what happens if you throw a baseball in the direction the ship is moving at 0.000000006% light speed.
But that gets complicated.