The Catapult

A catapult was a medieval siege weapon used to throw large objects long distances.  We have built a small scale catapult to throw a tennis ball and have determined the physics involved.

Catapults were invented by the ancient Greeks, and were named by them too, as the word "catapult"is derived from the Greek verb "katapeltes" meaning "to toss" or "to hurl."  A catapult is defined by its ability to use torsion (twisting of an object) or tension (pulling or bending of an object) to throw objects. The first greek catapults used a giant crossbow to throw rocks at fortified structures or marching armies.  The crossbow version was then improved on and used by the Romans.  The Romans also created the stereotypical version of the catapult that we know today.  Called an "Onager" it had the rectangular structure, the spoon shaped arm, etc.  With the medieval period came the Mangonel, a larger more powerful  catapult with many variations.  It had continued widespread use until the invention of the trebuchet and finally became obsolete with the invention of gunpowder weapons.  No longer used as an instrument of war, catapults are now built for a variety of reasons.  You have probably heard of the "Pumpkin Chunkin'"contests.  But we have built a small scale catapult to determine the physics involved.

Our Catapult is about 14 inches by 14 inches and about a 12 inches tall and made of solid wooden blocks.  The arm is made of a broken lacrosse stick.  The power of the catapult comes from a 12 inch bungee cord that hooks to the end of the lacrosse stick, goes under the arm, and hooks to the back of one of the blocks.  This creates enough tension so that when we pull the arm back, the bungee stretches, and when we release the arm, the bungee snaps it forward.  There is a wire that is drilled into the sides of the blocks that stops the catapult's arm from going any further, but since the projectile has inertia it will continue to go forward and is thus in flight.

In order to understand the physics involved in the catapult we must first understand all the forces that act on the catapult and the projectile.  

Forces:

Normal Force that exists on every object.

Torsion or Tension: the force that is contained and will act on the catapults arm.

Throwing force: the force that is created when the torsion/tension is released and will act upon the projectile.

Gravity: once the object is thrown, gravity is the only force that will act upon it (besides air resistance, which we will ignore).

Now that we understand the forces involved, we can find out their relations.  The components of this relation are as follows:

1. The angle at which the catapult throws the projectile relative to the horizontal that is the ground. Represented by theta: 


θ

2. The velocity that the object has the instant it leaves the catapults arm. 
VO
 This is a vector that can be divided into vertical and horizontal components.

VYO ,VXO

3. acceleration due to gravity: 9.8 m/s/s.




Time, at any point in the projectile's arc is represented by 


T
The vertical and horizontal displacement of the projectile at any time in its arc is represented by 


ΔY, ΔX

We want to find the horizontal and vertical displacement of the projectile.

This is found by finding horizontal and vertical vectors of the original velocity vector.


These can be found by the following formulas:

Vxo = Cos(θ)*VO

VYo = Sin(θ)*VO

The horizontal component is the easiest, the horizontal velocity will always be the same for a projectile, so we multiply the horizontal velocity by the time the projectile is in the air.  Represented below:

ΔX=Vxo * T

The vertical component is a little harder, since the vertical velocity is always changing due to acceleration due to gravity.

ΔY=Yyo * T + .5(-9.8)*T^2

these formulas can be combined to create two master formulas:

ΔX=Cos(θ)*VO* T

and

ΔY=Sin(θ)*VO * T + .5(-9.8)*T^2

These two formulas will be able to derive any component of projectile motion.

We now understand the history behind a catapult.  We understand the forces that affect it.  We understand the the variables that represent and measure these forces, and the formulas that express the relationship between the forces. The only thing left to do is actually launch the thing!  Will get back to you on how that goes.