Where was the first catapult built?

There are several ways in which a catapult works.

A ocounterweight catapult works by droping a weight strafed to a boom, while the thboom acts like a giant lever. The projectile builds up momentum at the other end (usually in a sling or cup). At the apex of the swing, the projectile is released to travel in a ballistic trajectory.

Tortion machines (onager, balista, scorpion) use a twisted bundle of rope(skein) to push the boom foreward. this method generates surprising amounts of power.