Re: how does a catapult work???

"catapult (n) - ancient device used for hurling cats at the enemy.  
It was later replaced by the rockapult, a much more effective weapon."

But seriously - 

All varieties of catapults rely on some mechanical scheme to store potential
energy, and then convert it suddenly to kinetic energy to throw a projectile.

Types of siege engine are generally identified by their construction.  A catapult
proper generally stores its energy by twisting large bundles of rope, made from very strong
plant fibers, animal sinew, or hair.  Energy is stored by winding the catapult with
a lever & gear arrangement, not unlike a giant version of those rubber-band airplanes.
A projectile is loaded, and the trigger releases the arm.  The arm is designed
in such a way that leverage converts the tremendous stored energy of the coil into
kinetic energy with great speed.

A ballista is arranged like a giant crossbow, and combines the function of throwing
arm and spring in just the same manner.  Again it is ratcheted back with a lever & gear
to load.  A variation on this theme is the springal, which holds dozens of spears
at a time, which are fired with tension from a green plank which is ratcheted back
for tension.

The real monsters of the siege engine world are trebuchets ("trench buckets").
The trebuchet is a long lever, with its fulcrum close to the bottom end.  The
arm is winched into place with pulleys, locked, and the bottom (counterweight)
is loaded with rocks.  At the top end of the long arm, the lighter projectile
is loaded. This projectile can still be quite large (since the counterweight is
tons of rock); common trebuchet loads included boulders, barrels of Greek fire
(a primitive form of napalm), and dead livestock (ever seen "Monty Python & the 
Holy Grail"?  Well, that part was for real.  It was used to break a siege with 
disease, or fear of disease, which is often sufficient.).  When the arm is released,
leverage converts the force of the counterweight's fall into tremendous speed for 
the load.

Some good references about catapults include:"The Way Things Work", by David
Macaulay, "Catapult - Harry & I Build a Siege Weapon", by Jim Paul, and there 
was also an article in Scientific American in 1979.