"Catapult" is the name given to a variety of related siege weapons used to hurl projectiles without the use of explosives. Crucial during Medieval times and even earlier, these simple machines make use of stored energy to release a projectile, or "payload." This process can be described in the language of everyday physics, chiefly in terms of tension, torsion and gravity.
The most efficient and accurate type of catapult, a trebuchet makes use of a counterweight that is much heavier than the payload to provide the energy for launch. The apparatus is set up like a see-saw, with the pivot point much closer to the counterweight end at the front than to the payload and sling at the rear. This arrangement is an example of the principle of mechanical advantage: The linear velocity of the sling -- in this case the speed with which the payload traces out its arc just before it is launched -- is far greater than that of the counterweight, simply because of the vastly greater mass of the latter.
Probably the most familiar type of catapult, the mangonel fires projectiles at lower angles than does a trebuchet, making it better suited for destroying walls rather than firing objects over them. In physical terms, tension is created in opposing directions by ropes, one in the direction of launch and the other toward the ground beneath the payload. When soldiers cut the rope lashed to the ground, the arm of the mangonel accelerates quickly toward the target and the payload is launched. The potential energy itself originates in the elastic properties of device -- for example, flexible wood -- joining the pivot arm to the rest of the apparatus.
You may never have seen pictures of this type of catapult, but if you've used a crossbow or seen one fired, you can envision how a ballista works. Like a mangonel, its uses twisted ropes to create torsion, or tension, about an axis of rotation. Instead of having one arm that rotates through a vertical plane, it uses twin arms that move through a plane elevated above the horizontal to an angle of its crews' choosing. This device can fire stones like other catapults, but is especially suited to firing enormous darts and spears.
You can built your-own table-top, mangonel-style catapult -- also called a "siege engine" in the old days -- using only a few easily found parts and tools. You'll need a dozen or more pieces of wood six to 12 inches long, a supply of small nails or screws, wood glue, a few tough elastic bands, a pair of eye-hook screws, a six-inch-long metal bar, and a piece of cardboard about 4" square. For a special challenge, you can try assembling these without instructions, but you can find assembly directions in the second Reference.