1. History and Physics of Catapults

2. Catapults have been integral to siege warfare since antiquity
Catapults have been integral to siege warfare since antiquity. Ancient Catapults were one of the most effective weapons in siege warfare. Various types of Catapults have been used by the Greeks, Romans, and Chinese. The first catapults were early attempts to increase the range and power of a crossbow. Diodorus Siculus, a Greek historian, was the first to document the use of a mechanical arrow firing catapult (early Ballista) in 399 BC. Catapults as we think of them were introduced to Europe during the Middle Ages. Catapults made their exhibition in England in 1216 during the Siege of Dover, the French crossed the Channel and were the first to use Catapults on English soil. With war prevalent throughout Europe during the Middle Ages the popularity of fortified castle and city wall rose significantly. This made Catapults essential.

3. Catapults were used to launch missiles (many different objects were utilized). These missiles were either launched directly at the wall to cause maximum damage to the fortifications or were launched over the wall to lay siege on the population within the protective walls. Catapults were also used to throw missiles at soldiers.
The first accounted acts of Biological Warfare involved catapults. Bodies of the diseased were hurled over the city walls to infect the residing citizens. Catapults were used extensively throughout Europe (predominantly by the French) until  AD when new defense systems rendered catapults ineffective.

4. There were 3 main types of Catapults:
1.) Ballista
The Ballista is basically a giant crossbow.
The word Ballista comes from the Greek word "Ballistes" meaning throw. Believed to have been invented by the Greeks and later modified by the Romans. The Ballista was created to amply to range and power of the crossbow and was the earliest catapult.
Two wood arms (looks similar to a bow laid on its side, but with a middle section cut out) are attached to a piece of rope. The rope was usually made of human hair or animal sinew.  
The rope was attached to a winch and pulled back, bending the arm back. When released the Ballista would shoot large arrow, or darts toward the enemy with deadly accuracy.
Despite its accuracy it lacked the power of the Mongonel and the Trebuchet. 

5. 2.) Mangonel
The Mangonel is what many people think of when they think of a catapult.
From the Latin word "manganon" meaning engine of war. The Mangonel was invented by the Romans in 400 BC.
The Mangonel consists of a long wood arm with a bucket (early models used a sling) with a rope attached to the end. The arm is then pulled back (from natural 90o angle) then energy was stored in the tension of the rope and the arm. Then the bucket would be loaded.
When released the Mangonel's arm would return to its equilibrium position, when it came in contact with the beam (or block) the arm would stop but the missiles stored in the bucket would continue to launch toward the enemy. 
The Mangonel fired projectiles in an overhead arc, the angle of the path of the projectile could be determined by a block placed on the beam that stopped the Mangonel's arm by using a block that stopped the arm earlier than 90o angle would result in a path angle (above the horizontal) equal to the the angle between the arm and the 90o angle.
The Mangonel was capable of firing 1,300 ft. The Mangonel was easy to construct and wheels were added to the design to increase mobility.
The Onager is a type of Mangonel.

6. 3.) Trebuchet
The Trebuchet was designed for maximum force. The stones it hurled were sent to demolish city/castle walls.
Believed to have been created by the Chinese in 300 BC the Trebuchet was the most powerful of the catapults. The Trebuchet arrived in Europe around 500 AD.
The Trebuchet consisted of a long arm (could be up to 60 ft long) balanced on a fulcrum that was far from center. The short arm was attached to a counterbalance; a heavy lead weight or a pivoting ballist box filled with earth, sand, or stones.  A sling was attached to the end of the long arm. A rope was attached to the long arm and pulled down until the counterbalance was high in the air.
The energy was stored in the potential energy of the counterbalance. The sling was then loaded with projectiles. The rope was released and the counterbalance plummets down. The potential energy is converted into kinetic energy and when the long arm is brought to an abrupt stop the projectile continue with the velocity produced by the kinetic energy.
The Trebuchet was the most feared and hated siege weapon. The men who manned the Trebuchet were called "gynours" and were under constant assault by arrows and missiles. The Trebuchet was the primary target of many reconnaissance missions sent to burn them to the ground. The most powerful and most famous Trebuchet was the WarWolf designed and constructed by Master James of St. George the chief engineer of Edward I.

7. Ammunition
Catapults were known for throwing a multitude of objects at the enemy. These include but were not limited to:
stones (Mangonel/Trebuchet)
sharp wood poles (Ballista)
darts (Ballista) 
pots of greek fire (Mangonel/Trebuchet)  -- ancient Molotov cocktail (composition is still debated) 
quicklime (Mangonel/Trebuchet) -- lit and thrown react with CO2 in air increase heat of the fire when contact water, would continue to burn in water
burning tar (Mangonel/Trebuchet)
burning sand (Mangonel/Trebuchet) -- became trapped inside armor
dung (Mangonel/Trebuchet)
dead animals (Mangonel/Trebuchet)
body parts (Mangonel/Trebuchet)
dead and mutilated bodies (Mangonel/Trebuchet)
diseased bodies (Mangonel/Trebuchet) -- first biological warfare

8. Modern Day Catapults
An aircraft carrier doesn't have the runway space to allow for a plane to accelerate to take off as it would on the ground so a catapult is used to propel the aircraft into the air in a very short distance.
Catapults as siege weapons became ineffective in AD rendered useless by new defense technology, but they still continues to be used in military operations.
The last large scale use of catapults as a weapon delivery device was in World War 1. Catapults were used to throw hand grenades across No Man's Land and into enemy trenches. Unfortunately for catapults they were soon replace with small mortars. Now catapults are used to launch planes into the air.

9. Physics Behind a Catapult - Mangonel
The Most Important Piece of Catapults: Tension
The entire catapulting process depends on the storing of potential energy as tension (of ropes or wooden arm) or gravitational potential energy
5 Easy Steps to Catapulting:
Pull the arm back (rope in case of Ballista)
Place missiles in bucket, sling, or nook
Release potential energy. Work is done on the arm.
Arm collides with base and is brought to an abrupt stop.
Missiles retain the kinetic energy from the work done on the arm. This kinetic energy launches the missiles at the target.

10. The mangonel was best suited for launching projectiles at lower angles to the horizontal, which was useful for destroying walls, as opposed to the trebuchet which was well suited for launching projectiles over walls. However, the mangonel is not as energy efficient as the trebuchet for the main reason that the arm reaches a high speed during the launch. This means that a large percentage of the stored energy goes into accelerating the arm, which is energy wasted. So the only way to waste as little energy as possible is to make the arm and bucket as light as possible, while still being strong enough to resist the forces experienced during launch.

11. The physics behind a mangonel is basically the use of an energy storage mechanism to rotate the arm. Unlike a trebuchet, this mechanism is more direct. It consists of either a tension device or a torsion device which is directly connected to the arm. 
The figure below illustrates a mangonel in which the energy source is a bent cantilever, which is a form of tension device. This can consist of a flexible bow-shaped material, made of wood for example.  The point P in the figure is the pivot axle, attached to the frame, about which the arm rotates. 
The figure below shows the mangonel at the launch point. To launch the payload the restraining rope is released. 

12. The twisted rope is commonly referred to as a torsion bundle
The twisted rope is commonly referred to as a torsion bundle. It consists of several lengths of rope with the arm inserted in between them. The rope is then twisted manually on both sides of the arm using levers. Upon release, the torsion bundle rotates the arm at high speed, launching the payload. The figure below illustrates how a torsion bundle is twisted. 
The other type of energy storage mechanism is a torsion device, which can consist of twisted rope. This allowed for greater throwing power than the tension device, in ancient catapults. The figure below illustrates the torsion device. 
The fact that a mangonel uses an energy storage device that consists of a deforming material, like wood or rope, means that its throwing distance will not be as consistent as a trebuchet. This is because these materials (unlike more modern materials), naturally wear and lose elasticity during their use. This is something that needs to be constantly monitored during a battle, with replacement materials made readily available, if necessary.