depends on what type of catapult:
lowering the throwing arm raises the weighted end, increasing the gravitational potential of the weighted end. when the catapult is released, gravitational potential energy is turned into kenetic energy thus overcoming the inertia of the projectile. This builds velocity and momentum on the projectile...because the mass of the projectile is free from the catapult and ultimately from the ground, the throwing arm is slowed quicker than the projectile throwing the projectile.
This is the kind of catapult you see in medieval movies and a deviation of this kind is the one made My Leonardo DaVinci as seen on "Doing DaVinci" on Discovery. This time, lowering the throwing arm creates elastic potential energy through a system of ropes (or in DaVinci's case, a bow). When the catapult is fired, the latch releases the arm and the elastic potential energy is converted into, again, kenetic energy the bar on the catapult then stops the throwing arm and the projectile carries its momentum and velocity.
After this the projectiles of the catapults are the same. The projectiles then move in an arc which can be determined using the angle of release, release velocity and a little trig. The projectile flies at a constant velocity in the x direction (negating air resistance) while the velocity in the y direction is countered (or applied depending on the release angle) until the projectile is brought to the ground. NOTE: the projectile is subject to rotational movement while flying as well. After the projectile is brought to the ground, friction works against momentum to bring the kenetic energy back to 0 in which the projectile comes to rest.