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The big power requirement of VTOL is not the problem in the overall energy budget because the hovering time is very short compared to the total flight time. Main problem is the additional weight of the more powerful engines.

BUT: If you want to fly supersonic, you need very powerful engines anyways, so if you can re-use them for VTOL it's a win win.

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To fly supersonic you need an engine that is very powerful at supersonic speed.

VTOL requires an engine that is powerful at low speed.

The harrier jump jet, a most famous VTOL aircraft, didn't have supersonic capability when all its contemporaries did and can only take off vertically with reduced payload or it would burn all its fuel just taking off.

While I don't have the numbers I think VTOL supersonic aircraft would be VERY hard indeed and it would be something that has to be seen to be believed.

The F35-B is both supersonic and VSTOL [1]: very hard indeed, but done. Just like the Harrier, the F35-B probably doesn't take off vertically with any significant payload though.

[1] https://en.wikipedia.org/wiki/Lockheed_Martin_F-35_Lightning...

The F-35-B has essentially two different types of engine (the main engines and the lift fan) to make that work.

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