The Catapult is similar to the Trebuchet but it is also different. The Catapult has wheels on it so that armies could easily move it with them to battle to have an edge on their enemies. The Catapult launched large rocks or projectiles at the enemies to smash down castle walls or kill mass amounts of enemy solders at one time.

The first catapult was the Mongol, and it was created by the Romans. They based it on the earlier Greek Ballista. The Catapult works by storing tension either through ropes, or bent wood. The crew pulls the arm of the catapult back and locks it in place either by having a rope tied to the arm, or having gears hold it in place. Once that is done, the crew places the projectile in the sling or cup at the end of the arm. Then they cut the rope or release the gears, the arm goes forward with tremendous speed. At a certain point, the projectile flies (from the sling of cup) towards the unsuspecting knights or castle.

 In order to calculate the physics of the projectile of the catapult you must first know these 3 formulas.
V=d/t          Vfinal=Vi+at            d=Vit+(1/2)at2
Now with these 3 formulas, you can describe the physics of the entire system. (This includes the catapult and the projectile.)
 To understand all of this complicated process, see this website.