Fire a cannonball horizontally from a mountaintop and it lands some distance away. Fire it harder and it lands farther. Now keep going. Isaac Newton asked what happens if the cannon is so powerful that the ball travels far enough for the Earth's surface to curve away beneath it. The ball is still falling the whole time. It just never gets any closer to the ground.
That is all an orbit is: falling, and missing. Newton imagined it in the 1680s, and it still explains why a spacecraft needs the speed it does.
Near the ground, anything you throw sideways falls about 4.9 metres in its first second, no matter how fast it is moving sideways. The Earth is a sphere with a radius of 6,371 kilometres, and its surface curves down by that same 4.9 metres over a horizontal distance of about 7.9 kilometres. So a ball moving sideways at 7.9 kilometres per second, which is roughly 28,500 km/h, falls 4.9 metres in the first second and the ground has dropped away by 4.9 metres too. It is the same height as before. Repeat that every second and the ball circles the planet.
The cannon sits 200 kilometres above the surface, high enough that air would barely matter, and this simulation has no air at all. Pick a muzzle speed and press Fire. The dashed line is where the ball would go if there were no gravity. Everything else is Newton's law of gravity, worked out step by step. Time runs about 300 times faster than real life.
Earth to scale. Blue dot: the ball. Dashed line: the path with gravity switched off.
Try 4 km/s first: the ball arcs down and hits the ground well around the curve. At 6.5 it goes much farther. Then press the 7.79 button. The ball stays at the same height all the way round, and it keeps going. Push beyond it and the path stretches into an ellipse, with the ball swinging out to a far point and falling back toward the cannon. At the top end it does not come back at all.
For a circular path, gravity has to supply exactly the inward pull that keeps something moving in a circle. That pull needed is the speed squared divided by the radius. Gravity at 200 km up is a little weaker than at the surface (about 9.2 m/s² instead of 9.8), and the orbit radius is 6,571 km. Set the two equal and the speed that comes out is 7.79 km/s. At ground level the same sum gives 7.91 km/s, but the air makes that useless in practice.
Higher orbits are slower, not faster. The International Space Station, at about 400 km up, moves at roughly 7.7 km/s and takes about 92 minutes to go round. A satellite in a geostationary orbit, 35,786 km up, crawls along at about 3.07 km/s.
At the height of the Space Station, gravity is still about 88% as strong as on the ground. Astronauts float because they, and the station, are falling together around the Earth. Nothing is pushing up on their feet, so there is nothing to feel. The cannonball has no weight on the way round for the same reason. Weightlessness is free fall that never ends.
Try 9 km/s. The ball rises to a much higher point, slows down there, and comes back. The faster you fire, the farther out the far point gets. At a certain speed it never turns back. That is the escape speed, and it is exactly the circular speed multiplied by the square root of two. From 200 km up it is 11.0 km/s (11.2 at the ground). Fire at 11.1 and watch the path open up and leave.
Getting 100 kilometres straight up needs a speed of only about 1.4 km/s, ignoring air. Getting into orbit needs 7.8 sideways. That gap is why a rocket does not climb vertically for long: after clearing the thickest air it tilts over and spends most of its effort going sideways. Reaching space is the easy part. Staying there is the speed.
What the simulation leaves out: air drag, the Earth's spin (which gives a launch from the equator a free 0.46 km/s eastward), and the pull of the Moon and Sun. Newton's cannon is the clean version, and it gets the main answer right.