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When we travel in a car upwards on a slope (against gravity), we can keep driving the car at a constant speed without accelerating (without changing its speed). Theoretically, the only thing we need is a power source such as its engine with sufficient fuel. If the slope is long enough to cross the karman line, what is the minimum velocity (remember - velocity, and not acceleration) that we need to have so as to complete the journey? Likewise, if the vehicle has to travel vertically upwards, what will be this value of constant speed?

Niranjan
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    It seems to me that you're suggesting applying an acceleration to cancel gravity, in which case you're left with a constant velocity for the whole ascent (as in your car analogy). In that case, I would think any vertical velocity greater than 0 would eventually get you over the Karman Line, if we ignore fuel requirements. Are you asking something different than this? – Drake P Sep 05 '22 at 15:45
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    Understanding escape speed: you can abandon a celestial body at any speed. You can calculate an escape speed a non-propelled object would need, say, from the surface of Earth, under ideal conditions (no atmosphere). But you don't have to reach this speed. – Quora Feans Sep 06 '22 at 11:31
  • The Spin Launch system gets the projectile to a constant velocity in a near vacuum before launch. I think they have already done some tests that put projectiles just above the Karman Line, IIRC they give an estimate of the RPM of the shell before launch in this BBC Click episode. – Martin Sep 06 '22 at 13:56
  • @Drake: you are almost there. I believe that the car on the upwards slope is also being acted upon by gravitational pull. Since we are maintaining constant upward speed, I also assume we have overcome the G pull. Therefore, as you seem to agree, any speed more than ZERO, should make us cross the karman line. (Lets not be too theoretical. Lets us ignore atmospheric drag etc.) In such a case why do we waste so much amount of fuel in accelerating the spacecraft? we might as well have sufficient time to make it cross atmosphere, and then do what we want.. travel further or orbit? – Niranjan Sep 06 '22 at 14:10
  • @Niranjan your question is unclear and your comment above is a critical clarification and belongs in the body of your original question. It actually radically changes the relevant question being asked. The literal answer to your original question, as asked, is any positive velocity. Your question is ambiguous enough that the The Rocket fan's answer below is actually speculating other (better) questions that are more worth answering. People shouldn't have to guess what you mean. Your question should be clear and concise. This also doesn't have anything to do with the Karman line. – Wyck Sep 06 '22 at 16:52
  • @Wyck: The question is very clear - it says "what is the minimum velocity..." Anyway, the purpose of asking this question is to understand why we tend to impart high velocity to cross the atmosphere (Karman line), and waste much fuel. Instead if we can ascend very smoothly and slowly (without acceleration / bare min. required), we can make better use of fuel for travelling further &/or imparting orbital velocity. Hope things are clear now. – Niranjan Sep 08 '22 at 04:20
  • @Niranjan using a low velocity actually wastes fuel, not saves it. You have to remember that you don't get constant velocity for free while ascending - you're spending fuel to fight gravity. The longer you spend doing that, the more fuel you waste to not falling back to the ground. The faster the better. – Drake P Sep 11 '22 at 16:33

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When you ask about the minimum velocity needed to cross the Karman line there are 2 things you could mean. The first thing is that you could mean the minimum velocity needed on the ground going straight up to cross the Karman line (for example a bullet getting shot upwards) or

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You could mean what is the minimum velocity going at a constant speed upwards. I will go through the 2 options.

The first option is similar to a gun shooting a bullet upwards. According to this site (https://www.quora.com/What-is-the-minimum-velocity-required-for-a-rocket-to-overcome-earths-gravity-and-travel-into-space?share=1)the minimum velocity needed would be 1.4 km/s. That number is ignoring the air resistance. The actual number is higher in real life.

Here (Could any existing gun reach the Karman Line?) it mentioned a gun that shot a bullet at 3.6 km/s and it flew up to 180 km. I haven’t done the math, but I assume you would need around 2-3 km/s to get a bullet pass the Karman line.

The other option would be the minimum velocity needed at a constant speed. So If you wanted to do that there probably would be 3 options: a space elevator, vacuum ballon or a big rocket using a lot of fuel but moving slowly. There would be no minimum speed needed with any of these options expect for the rocket because you can only fly it up as long as it has fuel. The other 2 options can literally fly at around 1 km/h upwards until it passes the Karman line.

Regular weather ballons fly for around 2 - 3 hours before reaching their maximum height.

Joseph Sible-Reinstate Monica
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The Rocket fan
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  • 2nd option about balloons. Do they accelerate > 9.8 M/S2? They do not have any external propulsion, which pushes them upwards except that they rise up being lighter than air. (I am assuming they do not accelerate and still cross or at least reach close to karman line. If I am correct, then would it not be simpler and cost effective to use them (or some similar low cost thrusters) as (although slower) first stage for rockets? Once above the perceivable atmosphere (negligible drag) we can use other stages to go up further and orbit capsules? Its cheaper, if time is available. – Niranjan Sep 05 '22 at 10:01
  • @Niranjan Didn´ t you forget that the higher you go, the thinner the air becomes ? So less air weight is replaced by the balloon and the uplift force on the balloon is decreasing..The balloons speed will decrease to zero and that will be long before reaching the Karman line. – Cornelis Sep 05 '22 at 12:25
  • @Cornelis: Oh OK. that means the Balloons will never reach Karman line. Doesn't matter. May be we can use some other type of thruster - may even be a LF rocket. My point was to see if we can avoid wasting fuel in "accelerating" the spacecraft, or giving it high speed so that we reach above the atmosphere (or close to that, where drag is substantially lower) ASAP, and then use the more powerful thrusters to go further up &/or provide the spacecraft its orbiting velocity. Is this possible? Off course assuming we don't really bother for time spent. What is the min. acceleration / speed needed? – Niranjan Sep 05 '22 at 15:00
  • @Niranjan a normal balloon could not pass the Karman line, but since the atmosphere goes beyond the Karman line a vacuum balloon could pass it. – The Rocket fan Sep 05 '22 at 15:49
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    @Niranjan I think what you're "talking about" is a Rockoon ? https://en.wikipedia.org/wiki/Rockoon I don't know what the initial speed would have to be. Maybe you can find it in questions about rockoons on Space SE or ask a new question. – Cornelis Sep 05 '22 at 17:22
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    @Niranjan At 71 km above sea level the air density is 0.000064 kg/m³. That means a 100 m. diameter vacuum balloon could carry a weight of no more than 60 kg, depending on its shape. Much of that weight would need to be for the balloons hull. – Cornelis Sep 05 '22 at 18:52
  • @Niranjan: Terminal velocity for a balloon is always very low because it has to be large and low density. So its acceleration is not relevant for long. Pretty much just how high you can get it before reaching neutral buoyancy, as the air thins out. Perhaps high enough for a medium-sized model rocket to launch from it to the Karman line, so the speed at any point in the flight is fairly low. – Peter Cordes Sep 05 '22 at 22:08
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    @Niranjan If a rocket hovers,say, 40 km above the surface, it has to produce a force F=mg to withstand gravity. Any temporary, additional force will give it a temporary acceleration with which you can determine a certain speed, so for that, no min. acceleration or speed is needed. So it's possible, see the Rockoon. Also, an initially hovering rocket applying a constant force will soon go up faster and faster because it gets lighter by spending fuel. – Cornelis Sep 06 '22 at 09:28
  • Would your starting elevation make any significant difference? e.g. If you fired a high-powered rifle straight up from the peak of Mt. Everest, would the thinner atmosphere at that height give you an advantage? (Obviously nobody's firing a bullet into orbit from any height on the ground, but if the goal is just to reach a certain height, it might make a difference...) – Darrel Hoffman Sep 06 '22 at 14:04
  • @Darrel Hoffman: YES, the atmospheric drag experienced by the bullet, when fired from mount Everest, will be lower than that if the bullet was fired from sea level. As a result, the bullet is expected to go faster and higher. However, as a point of interest, while the oxygen levels at that altitude (Everest) are much lower, atmospheric drag does not reduce so significantly. – Niranjan Sep 06 '22 at 14:18