Newton asked what would happen if you fired a cannon horizontally from a mountain high enough to clear the air, harder and harder. The ball falls the whole way, always. What changes is how far the ground has curved away beneath it by the time it gets there — until the ball falls around the world instead of into it. Fire a few shots and leave them on the screen.
What to try. Start slow and walk the speed up, firing as you go and leaving the old shots on screen. The early paths look like ordinary projectile arcs, because over a few hundred kilometers the Earth is nearly flat. They are not arcs though — each one is a piece of a very long ellipse whose far focus is the center of the Earth, and you are only ever seeing the last sliver before it runs into the ground.
Creep up on circular speed and watch the impact point race away from you. Then find the moment it stops coming back at all. That moment arrives before circular speed: all that is needed is for the perigee to clear the surface, so the ball skims the far side of the world and returns to the mountain from behind. Watch the perigee readout cross zero. Only at circular speed exactly does the path become a circle, and only then does the ball stay the same height the whole way round.
Push past that and the mountain becomes the near point of the orbit rather than the far one — the ball now climbs away and comes back, and the period stretches out fast. Keep going to escape speed and the ellipse never closes at all. Notice that escape speed is only about 1.41 times circular speed, however high the mountain: the same √2 every time.
What this leaves out. No air, which is the whole reason Newton specified a mountain above the atmosphere; a real shot at these speeds would burn up. The Earth is a point mass with no bulge, it does not rotate, and it is alone in the universe — no Moon, no Sun. The cannonball's own mass never enters, which is why it is missing from every formula here. The mountain is drawn to scale but is far taller than any real one.