Earths rotation

Okay, idle mind, too much time on my hands etc. But I was watching TV tonight,
and they were talking about air planes, 600mph, 12 hour flight to the mid-east
and I got to wondering. What effect, if any, does the earths rotation have on
air flight? My line of thought was this: If an object on earth is moving
approximately 1,000 mph (if my math is right, the earth is APPROX. 24,000 miles
in circumference and makes one revolution in 24 hours...) Why is it that once a
plane lifts off the ground and starts flying east at 600 MPH, that it doesn't
lose 400 MPH because of the speed difference. Or why is it tnat if the plane
lifts off and flies west at 600 MPH, it doesn't cover 1600 miles in an hour,
again, due to the speed of the earths rotation. Does my question even make
sense? Is it like a rocket laucnhed from Canaveral, the extra 1,000 MPH boost
just stays with the plane the whole trip? Just got wondering. Thanks for
humoring me.