If an aircraft travelled in the opposite direction to the earth so that the earth had turned once would the occupants of the plane be considered as having travelled backwards in time?
The answer to your question: No.
If you flew around the world in the opposite direction to the earth's rotation but flew at a very slow speed (long enough for the earth to have rotated once around) then an entire day would have passed. Thus, you could arrive back where you started earlier than you had left but on the following day. This would just be a terrible case of jet lag. If, you traveled so fast that you went all the way around the earth without it having rotated very much you could arrive back where you started just after you left. Either way, you would not have travelled back in time.
To think of it another way, imagine that you went up in a hot air balloon and floated above the earth. The fact that you are above the Earth does not affect the passage of time in any significant way. You can go any direction you want and yet the people of the Earth still age and their clocks still run. You can go wherever you want, in whatever direction you want, at whatever speed you want and their clocks will still run. No matter what you do, those clocks will run and will thus be later when you get back then they were when you left.
In both of these instances, however, you would be traveling faster than the those persons on the earth. As a result, time in your frame of reference would flow more slowly than it did on Earth. As a result, you would age more slowly than those people on the Earth. This time dilation happens to all travelers but is so minute that it goes unnoticed. However, if you were to travel around the Earth many times at a very high velocity (close to the speed of light), you would age very little while the Earth and the people on it aged at a regular rate. Thus, if you landed after many revolutions around the earth, you would arrive YOUNGER than all the of people would expect you to be. More time would have passed on Earth than would have passed on your ship. Thus, you might be a year older but perhaps 100 years might have passed on Earth. This is about as close to time travel as you can ever get but it works only into the future, not into the past.
David Consiglio, B.S., High School Chemistry and Physics Teacher
'A scientist is happy, not in resting on his attainments but in the steady acquisition of fresh knowledge.'