This is known as Ground Effect. Not to be confused with flaring, which is a technique used by pilots to gain lift by increasing the angle of attack as airspeed decreases.
Technicality, you can flare an aircraft at any altitude. The higher the altitude, the faster the airspeed of which you can flare an aircraft before stalling due to air thinning as altitude increases.
The speed of an aircraft during the flight is ideally kept constant relatively to the air surrounding the aircraft. However, in the moderate zone - between 30 and 60 degrees of latitude on both hemispheres - the dominant winds are called westerlies
http://en.wikipedia.org/wiki/Westerlies
for a good reason. They blow from the West to the East. So it's faster to get from Perth to Melbourne because it's in the direction of the wind; but it takes a longer time to fly from Melbourne to Perth because it's against the wind and, again, the natural speed of the aircraft is measured relatively to the air (i.e. wind).
Why do the westerlies have this direction?
It's because of the Coriolis force. The Coriolis force - a "fictitious" force in a rotating frame that adds to the centrifugal one - steers winds (and rivers) in the clockwise direction on the Northern Hemisphere and counter-clockwise direction on the Southern hefmisphere.
http://en.wikipedia.org/wiki/Coriolis_effect
Why does the Coriolis force create westerlies - which are particularly strong on the Souther Hemisphere e.g. in Australia? It's because the normal air that heats up near the equator expands and it would travel to one of the poles. However, this air going towards the poles gets steered by the Coriolis force, so before it reaches e.g. the Southern pole, it's already moving in the East direction.
The westerlies are weaker on the Northern Hemisphere as well as on the summer-having hemisphere because when it's so, the wind from the equator is strong enough (amplified by the land on the Northern Hemisphere, or by the summer) so that it usually manages to reach the pole (without drifting to the East), anyway.
It's very, very far but my uncle lives in Melbourne while a generous IT expert who lives in Perth sent me an iPod Touch for a relatively modest and doable help with a project half a year ago. ;-)
Best Answer
During the flight, you need to get up to use the restroom. There's one 10 rows in front of you, and another 10 rows behind you. Does it take longer to walk to the one that's moving away from you at 600 mph than the one that's moving towards you at 600 mph?
No, because you're moving at 600 mph right along with it -- in the ground-based frame of reference. In the frame of reference of the airplane, everything is stationary.
Similarly, the airplane is already moving along with the surface of the Earth before it takes off. The rotation of the Earth has no direct significant effect on flight times in either direction.
That's to a first order approximation. As others have already said, since the Earth's surface is (very nearly) spherical and is rotating rather than moving linearly, Coriolis effects can be significant. But prevailing winds (which themselves are caused by Coriolis and other effects) are more significant that any direct Coriolis effect on the airplane.