Newton’s Cupcake

Isaac Newton published a famous thought experiment in A Treatise of the System of the World. He imagined what would happen if one were to shoot a cannonball from the top of a very high mountain (so high up that air resistance could be ignored). If the canonball were launched with a relatively small velocity, it would strike the ground at some distance from the mountain. However, if the velocity of the canonball were much larger, it might “miss” the ground entirely and go into orbit around the planet. This thought experiment was the key to linking the force of gravity to orbital motion.

It is easy to determine the speed at which circular orbital motion will occur. Let \(M\) be the mass of the planet, \(m\) be the mass of the cannonball, and \(r\) be the radius of the orbit. We know that the force of gravity provides the centripetal force for a body in uniform circular motion around the planet.

\[\begin{equation} m\frac{v^2}{r} = \frac{GMm}{r^2} \end{equation}\] \[\begin{equation} v^2 = \frac{GM}{r} \end{equation}\] \[\begin{equation} v = \sqrt{\frac{GM}{r}} \end{equation}\]

After learning about the HTML5 canvas tag in a workshop at the AAPT Summer Meeting, I decided to make a little simulation in order to visualize this thought experiment. The planet below has the same radius as the Earth; you can adjust the mass of the planet, the initial velocity of the projectile, and the angle of launch of the projectile. Sadly, we are out of cannonballs, but we do have a brave cupcake volunteer.

Initial Cupcake Speed		7500 m/s
Planet Mass		6.0e24 kg
Angle of Fire		0 degrees