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08:22
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Q: If an aircraft were to fly ninety degrees upwards, at what altitude does weightlessness of space begin?

NormLDudeImagining the pilots have fainted in the cockpit and are pointing the aircraft upwards at approximately 90° . Not sure if it would get really cold, or really hot before entering "space". Or if a normal aircraft, like a Boeing777 or Boeing787 could make it past a certain altitude without certain p...

To enter orbit you do need to go fast, not hight
You seem to be labouring under some misconceptions. For a start, the aircraft will not enter space, it will stall. If we could make it to space in jet aircraft, then why would anyone bother with rocket engines? Also, entering space isn't what brings the sensation of weightlessness, but being in the state of constant freefall of orbit.
The question is about altitude. It would be nice to know at what altitude it would stall then, and if it gets hot or cold.
I have not brought up the said "constant freefall of orbit" (of which has never been proven, nor shown, nor seen). That is for another topic.
"constant freefall" is proven, shown and seen. It's what "being in orbit" means. And you need this constant freefall to get weightlessness. If you are not in orbit you don't feel weightlessness at all.
Aside from that, I suspect that this question is not really on topic here, but more appropriate for Space.SE, but I'll wait for the others and see what's the consensus. Otherwise I suspect it can be seen as duplicate of this: aviation.stackexchange.com/q/3425/1467
To answer directly: 1/ Space starts at 100 km, that's a convention, but physics rules don't change at 100 km. 2/ Weightless effect is not a matter of altitude at all, its a matter of trajectory, it can be done on aircraft (see 0G training aircraft in the comment below). 3/ If you fly up, then at some point the aircraft stops climbing and falls like a rock sent upward. You are mixing orbital speed (which creates a "microgravity" environment) which practically happens only at 200+ km (but could theoretically happens at sea level, just it needs a unfeasible heat shield) and altitude.
0G aircraft and dog-dedicated 0G aircraft (I like this one).
08:22
@NormLDude Absolutely wrong. At the altitude of the space station, whose inhabitants clearly experience apparent weightlessness, the strength of gravity is about 90% of that of the surface of the Earth. If it were up to altitude alone, they would drop instantly. And they are, but with their 27,000 km/hr speed, they keep "missing" the Earth. Thus they (the ISS and its people) are in constant freefall, and don't observe the effects of gravity because it's accelerating them both equally. nasa.gov/audience/forstudents/5-8/features/nasa-knows/…
See also : Scale of some orbits - earth's gravity remains significant until around High Earth Orbit (HEO - the red zone in the linked article). See also XKCD : Orbital Speed
@J..., with respect to "earth's gravity remains significant until around High Earth Orbit," I say, tell that to the moon.
Jumping is experiencing weightlessness
@mongo the moon orbits the earth because it is under the influence of earth gravity. Not sure what you are saying. Is it because of the word "significant"? Just trying to understand
08:22
@mongo I mean, of course, that gravitational acceleration remains a significant fraction of that experienced at the surface of the earth. You'll still eventually fall back to earth out there (barring a dance on a lagrange point...)
@Federico - I think this would not qualify for Space.SE. There are just too many issues and misconceptions...
@J... and AytAyt, I was trying to inject some levity. Obviously the moon and earth are inextricably intertwined in their dance as they both orbit the sun and each other. (The quote is from a previous comment by @J...) To retain the moon, the gravitational effect of the earth must be significant, otherwise the moon would depart in a line in an inertial frame from the approximate point where the earth's gravitational effect became insignificant.
@mongo Surely, of course, but the moon orbits once a month while the ISS, for example, orbits 16 times a day. I think there are enough orders of magnitude there to grant a casual throwing about of the word "significant" :)

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