An Onager of His Own

A couple weeks ago was my nephew Ezra's birthday, and his parents gave him the dream of any future physicist (if I've done my job as an uncle): a catapult! Seeing him play with it got me thinking about the mechanics, and a question popped into my head: Catapults, and later cannons, typically have wheels to make them easier to move and aim, but how does that affect firing? The momentum of throwing something forward pushes the catapult back, and if the wheels are free to turn during firing, that backward motion can be significant.

Wikipedia

Momentum is conserved, so if the catapult and stone start at rest, the product of mass and velocity must be equal and opposite for the two after firing:

where M and v_C are the mass and velocity of the catapult, and m and v_S are those of the stone. Initially, I thought this suggested the wheels would help to give greater range, since a greater backwards velocity for the catapult gives a greater forward velocity to the stone. However, there's also the mass: The wheels allow the catapult to act as a separate entity, but removing (or locking) the wheels connects the system to the Earth. To compare these setups, we need to calculate how the energy stored in the spring (or gunpowder) gets distributed. The total energy is

If we combine this with the previous equation, we can eliminate v_C, and see how the stone's velocity changes with the catapult's mass:

The Wikipedia page for the onager, which is the archetypical catapult design pictured above, gives a stone mass of 26 kg, and a range of 130 meters, which lets us estimate an energy. We can plot this velocity as we vary the mass of the catapult:

This says the bigger the base, the faster we can throw the stone, so locking the wheels is the best option. However, that's assuming we're fixing the catapult to the Earth. In the case of a cannon on a ship, it may be more desirable to let the cannon roll backwards than set your ship rocking side-to-side.

Thanks for the inspiration, Ezra, and happy sieging!