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?
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
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.