Newton's cannonball — from falling to orbit

What just happened?

The faster you fire the cannonball horizontally, the farther it travels before it falls. At a certain speed the Earth’s surface “falls away” beneath it exactly as fast as it drops — the cannonball never reaches the ground and circles in orbit. This is Newton’s thought experiment: a cannon on a very high mountain.

Orbital and escape velocity:

For the Earth (R = 6371 km): $v_I \approx 7.9\ \mathrm{km/s}$ (circular orbit) and $v_{II} \approx 11.2\ \mathrm{km/s}$ (escape). Between them the trajectory is an ellipse, and at $v \ge v_{II}$ — an open hyperbola.

An orbit is perpetual falling:

A satellite in orbit is in continuous free fall — it falls “past” the Earth. No magic force balances gravity; it is gravity itself that bends its straight-line motion into a closed trajectory. That is why astronauts are weightless: they fall together with the station.

Why you need this for the exam:

You derive the formula $v_I=\sqrt{GM/R}$ by setting the gravitational force equal to the centripetal force ($\tfrac{GMm}{R^2}=\tfrac{mv^2}{R}$). Problems about satellites, cosmic velocities and weightlessness all reduce to this single equality. Pair it with the exercise “cosmic velocities”.

Remember: an orbit is a fall that never ends — at $v_I \approx 7.9\ \mathrm{km/s}$ the cannonball orbits the Earth, and at $v_{II} \approx 11.2\ \mathrm{km/s}$ it escapes forever.