How Siege Weapons Work

Though the definitions for types of catapults vary, each shares this goal: to hurl an object through the air. There are are three primary technologies that fall into the "catapult" category.

1. The catapult: the winched-down bucket that people normally think about when they hear the word "catapult"

2. The mangonel is a type of catapult.

3. A trebuchet is a weighted beam that swings a sling carrying the projectile.

Both catapults and ballistas work by storing tension either in twisted ropes or in a flexed piece of wood (in the same way an archery bow does, but on a larger scale).

A trebuchet tends to be easier to build because it consists simply of a pivoting beam and a counterweight that rotates the beam through an arc.

Catapults can launch things a fair distance -- 500 to 1,000 feet (150 to 300 meters) is common. It is surprising how much energy they can store. The gears are important, because they create a winch. The winch allows a person to put a great deal of energy into the catapult over a period of time. Then all of the energy releases at once, throwing the projectile.

Actually, a catapult and a trebuchet work on completely different physics principles.
A catapult works on the principle of torsion - the catapult arm is inserted into a twisted length of rope. When pulled downward, the increased torsion of the rope stores energy, which is released in the "untwisting" motion of the rope, causing the catapult arm to return to the original position (straight up). The limit of the catapult's effectiveness is defined by the amount of torsion that the rope can hold, which, in general, it defined by how strong the frame of the catapult can be made. Using fiber rope, wood, and human power (to twist the rope itself), ancient catapults were limited to a payload of several dozen pounds, being thrown no more than 200-300 yards.
A trebuchet, on the other hand, is a simple counterbalanced lever. It operates very much like a playground teeter-totter (see-saw). The very long arm has the pivot attached close to one end (generally, about 10% of the length of the arm), with a large counterweight (usually, a box filled with rocks or iron pellets) attached to the short end. The long end has a rope sling attached to it. Pulling the arm don to a horizontal orientation (from it's "natural" vertical one), stores potential energy equal to the weight of the counterbalance. When the arm is released, gravity pulls the counterweight down, and the arm swings up quickly. At the top of the arm's arc, the sling is released, throwing the payload.
Overall, imagine a large person sitting on a see-saw, where the see-saw pivot point has been moved close to the large person. Then, a number of other people push down on the other end, raising the large person into the air. A small person then gets on this end (the one being held down by the group). When the group lets go, the large person crashed back down, and the small person quickly shoots up, often flying off the see-saw. The trebuchet works in an identical manner.