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If airplanes' highest speed depends directly on the air it is moving through and not the ground, does it mean we can build a time machine if only the air moves near the speed of light?

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The downvotes are a bit unfair since you ask a good question. You're really asking about the relativistic addition of velocities i.e. if the plane is moving at 0.9$c$ and the air is moving at 0.9$c$ then is the plane moving at 1.8$c$ relative to the ground?

You probably won't be surprised to find that the answer is no (Einstein was right!) because the simple rule for adding velocities $v = v_1 + v_2$ does not work for velocities near the speed of light. You need to use the relativistic formula for adding velocities.

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Well, It may be a good question. But, we've got enough questions on the relativistic addition of velocities. I think you've seen plenty on the topic :-) – Waffle's Crazy Peanut Jan 5 '13 at 18:25
I was actually asking if we can't build a plane that fast, why don't we make air faster? It would still be in total < c. I know we'd have to obey c in small regions of space. And it's still a "time machine" going to the future without having to surpass c or even be very close to it. – j riv Jan 11 '13 at 8:23
@LelaDax: oops, sorry. Making the air travel at near $c$ seems considerably harder than making some form of craft travel at near $c$. How would you do it? A big fan? – John Rennie Jan 11 '13 at 9:27

No. You should add velocities using the relativistic rule, so the velocity of the airplane with respect to ground will be less than the velocity of light.

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