Answer by Tim Hibbetts, F/A-18C pilot:
The first time I went supersonic was by accident and only knew it because the Mach meter said "1.0."
Modern jets that routinely punch the number do so very smoothly, and unless you're flying with very few pylons and external stores or perhaps the F-22, you're not going to really notice a great sense of speed at the really high Mach numbers because you're likely at altitude. Now, if you're making a Mach run close to the deck, it's incredibly invigorating. You're going to feel the bumps associated with thicker air more like a staccato, and you're going to be putting on quite a show. Here's a good example of what that looks like and a good explanation.
Taking off from a carrier in the daylight is one of the single greatest thrills I've experienced. Going from 0 to roughly 150 kts in about 2.5 seconds is even more exhilarating than it sounds. During the stroke, there's a buildup of sound, and you have this amazing visual rush as the deck and personnel around you whiz by at ever-increasing speeds. (Your acceleration actually increases, too, not just the speed.) Then, in the space of a microsecond, the sound decreases dramatically, almost like silence in comparison, the deck goes away, and you're left with this feeling of hanging over the sea 80 feet below you. At night, those lights zooming past you go away, and you're hanging in nothing. Inky blackness envelopes you. You scan your engine instruments because it got so quiet so quickly. You make sure you're still pointing up, because there is no outside reference. And you restart your heart, because your brain's going to need more oxygen soon.
It's a rush.
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