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That is the difference between ramjets and scramjets, not necessarily the definition of hypersonic. And some "hypersonic" missiles don't breath air but rather conventional rocket motors to muscle their way to higher speeds.

The oldest definition of hypersonic I have heard is those speeds where compression heating of the atmosphere ahead of the craft generates plasma. A space capsule on reentry hits such speeds, causing the classic radio blackout. So hypersonic aircraft are those that must deal with plasma-related issues that are irrelevant to slower supersonic aircraft.



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