Newton’s thought experiment. A cannonball fired sideways from a mountain falls in a curve to the ground. Fire it faster and it lands further away. The Earth is round, so its surface curves away below the ball. If the ball is fast enough, the ground curves away as fast as the ball falls, and the ball never reaches the ground. It is in orbit: it is falling all the time, but it keeps missing the Earth.

The speeds. For a circular orbit the gravity of the Earth supplies the centripetal force:

mv²/r = GMm/r², so v = √(GM/r), and the period is T = 2π√(r³/GM)

where r is the distance from the centre of the Earth (the radius of the Earth plus the height). Close to the Earth this is about 7.9 km/s (28,000 km/h), and the orbit takes about 90 minutes. The speed needed to escape from the Earth’s gravity is √2 times as big (1.41 times):

v_esc = √(2GM/r)

Between the two speeds the path is an ellipse, with the launch point at its closest point to the Earth. Below the circular speed it is also part of an ellipse, but one that goes through the Earth, so the ball lands. At the escape speed the path is a parabola and above it a hyperbola: the ball does not come back.

What the model leaves out. There is no air. The Earth is treated as a perfect sphere, with all its mass at the centre (so no oblateness, no Moon and no Sun). The paths are computed with a fourth-order Runge-Kutta method with a step of 1 second. At the surface the model gives a circular speed of 7.91 km/s, and a period of 84.5 minutes.

Things to try. (1) Find the speed that gives a circular orbit at 300 km. (2) Fire the cannon horizontally at 8.5 km/s. Where is the ball furthest from the Earth? Is it moving faster or slower there? (3) What speed do you need to escape?