Suppose that the technology exists for a spacecraft to launch with low velocity. Horizontal velocity is 0 and after brief acceleration the vertical velocity is constant at 30m/s. After 4 days the craft reaches the high-earth orbit and the propulsion is turned off. What would happen then?
In: 4
It would fall back to earth. There is insufficient horizontal velocity component to maintain an orbit so earth’s gravity would simply pull it back towards the ground. It would only be around 1000km from earth so earth’s gravity would likely dominate.
The simplest to imagine is if this happened at the north or south pole (where there would be essentially 0 horizontal velocity). But even at the equator (max horizontal component at liftoff), that component is only about 1000km/hr and that is not sufficient to maintain orbit.
An object with the trajectory you described would never reach orbit. It would reach high altitude, but what it wouldn’t ever do is start orbiting the Earth. It would just fall back down just like if you shot a gun directly upwards and tracked the bullet.
E: depending on the properties of the object and where it launched from (and what exactly you mean by 30 m/s straight up), when it fell back down it would almost certainly not fall down exactly where it began. It’s hard to say what exactly would happen because the problem is not fully specified.
It would not be in high Earth orbit. It would fall right back down, because orbits happen as something falls towards Earth and misses. If it’s just falling straight back down it won’t miss.
30 m/s * 4 days = 10,400 km. The gravitational acceleration at that altitude is 1.4 m/s^(2) so it would stop moving upwards in 21 seconds, gaining only 0.3 km of additional height before falling down to Earth.
A rocket trying to do that would run out of propellant after ~10-15 minutes, by the way, just to counter gravity. At that time it’s at an altitude of roughly 20-30 km, still far away from space.
It falls back down. Orbit requires horizontal speed: https://what-if.xkcd.com/58/
What you described kind of sounds like a weather balloon: https://en.wikipedia.org/wiki/Weather_balloon minus the propulsion turning off. I’m going to reinterpret your question a bit. “Suppose the technology exists” to mean that you have a magical method of providing a force equal to gravity to have your craft ascend at constant speed. This is equivalent to teleporting it to its final position. It falls. The phrasing of your question with “high earth orbit” as is is impossible. A stable high earth orbit has a sufficient amount of sideways velocity. If you mean it is at the altitude above Earth where high orbit but with insufficient sideways velocity, it will fall.
Khan Academy has this “Satellites 101” with some demos https://www.khanacademy.org/partner-content/nasa/searchingforlife/mars-modern-exploration/a/satellites-101-article You might want to read and watch https://spaceplace.nasa.gov/launching-into-space/en/ for the basics of how we currently get to orbit.
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