Trebuchets are a type of catapult used in medieval times for besieging castles and walled cities.  They're pretty easy to make, since there is only one moving piece, but pretty complicated to model mathematically.  Here's mine.

For more information/plans/pics on such, goto:  http://www.ripcord.ws/ .  You'll notice I didn't feel like doing all the fancy joinery they go through.  My range is around 140 feet for a tennis ball.  I use a 1.25 ft of railroad track, maybe 50 lbs, for a counterweight.

A trebuchet has the following parts:

When you let the thing go, the counter weight falls, the throwing arm yanks the sling, a pin at the end of the throwing arm releases one end of the sling, the projectile sails and the crowd roars (especially if you hit 'em).

Download a trebuchet simulator 520kb

These are some graphs of data modeled by the simulator:  

• DIRECT VARIATION - Range is directly related to counterweight mass.  That is, as I raise counterweight mass, the projectile goes farther.  There is a  straight line relationship between mass and range.  (nb:  if I had 600 lbs of counterweight, I could throw a one pound rock nearly a half mile.)

• INVERSE VARIATION - Range is inversely related to projectile mass.  That is, as I put bigger rocks in the sling to throw, the projectile range drops off sharply.  

• EXPONENTIAL GROWTH TO A MAXIMUM - By lowering the counterweight, range is increased exponentially to a limit.  At some point lowering the counterweight won't help any more (especially if the counterweight hits the ground).

• GROWTH TO A MAXIMUM - There is a definite "best" value for the length of the sling.  The range falls off if the sling is longer or shorter.