**How Catapults Work**

**THE CATAPULT PROPER**

The name, catapult proper, suggests that this device was the most common form
of siege engine. It employs a large bundle of plant fibers, hair (human or animal),
and leather, which is twisted around a fulcrum in order to store the potential
energy required to throw a projectile.

When the rope has been twisted as tight as it can possibly be, it has stored its
maximum potential energy, which depends on its thickness, length, and tensile
strength. The lever arm of the catapult, which throws the projectile, is attached
to the fulcrum around which the rope is twisted. The arm is pulled back and locked
into position, so as to rotate around this fulcrum.

The arm, attached to the rope at the fulcrum, and having stored a tremendous amount
of potential energy, is quickly released. Because the rope tends to seek out a
state of lower potential energy, it releases the energy it has stored in the only
way it can: by rotating about the fulcrum. Thus, the potential energy is transferred
into rotational kinetic energy. The rotational energy is constant throughout the
arm, so the speed attained will increase further from the fulcrum. The rotational
motion of the arm causes a centripetal force to , based on the speed, to accelerate
the projectile. Depending on where the lever arm is stopped, the projectile will
have a different trajectory. Most catapults have a crossbeam directly above the
fulcrum, which stops the lever arm, giving the projectile a trajectory initially
parallel to the ground.

- By the law of conservation of energy, the stored potential energy (U) is
transferred into rotational kinetic energy (K), with some loss due to friction.
U = K

- The rotational kinetic energy is equal to one half the rotational inertia
(I) times the square of the rotational velocity: (ω) K = (1/2)Iω²

- The rotational inertia of a flat slab of length L, width W, and mass m, and
rotated about its center of mass, which the lever arm is assumed to be is: I =
m(L²+W²)/12

- The distance between the fulcrum and the center of mass of the arm must be
taken into account. This distance is labeled h, and by the parallel-axis theorem,
the rotational inertia of the lever arm is: I = m(L²+W²)/12 + mh²

- The rotational velocity (ω) of a rotating object is equal to its linear
speed (v) divided by the distance from the axis of rotation. For the projectile
this distance is r, the distance between it and the fulcrum. ω = v/r

- Substituting equations (4) and (5) into equation (2), we arrive at: K = (m(L²+W²)/12
+ (1/2)mh²)*(v²/r²)

- In a perfect physics world, where the effects of gravity and friction are
negligible, the linear speed of the projectile will remain constant as it rotates,
and will have the same speed as it leaves the lever arm when the arm stops. Thus,
the equation for the speed of the projectile as it leaves the catapult can be
written as: v = r * sqrt(24U/(m(L²+W²+12h²)))

- Therefore, the velocity of the projectile can be maximized by increasing
the distance between the projectile and the fulcrum, increasing the stored potential
energy (by twisting the rope tighter or using thicker or stronger rope), decreasing
the mass of the arm, decreasing the length and width of the arm, or decreasing
the distance between the center of mass of the arm and the fulcrum.

**BALLISTA**

A ballista, in many respects, like a large crossbow has its projectile placed
on a rack. A rope is attached to two ends of a beam, usually made out of strong
wood or iron, and pulled around behind the projectile. The beam is placed beneath
the rack and above the projectile so it can be hurled without obstruction. The
projectile is pulled back with a crank-thus pulling back the beam as well-and
then locked into place. Depending on the length and strength of the beam, it will
store more or less potential energy. The rope also stores some energy.

When the lock is released, the beam, like the wound rope for the catapult, tends
to expel its stored energy. Unlike the catapult, the beam transfers it directly
through the rope to the projectile in the form of linear kinetic energy. The rack
can be placed at different angles to the ground in order to produce different
trajectories.

- The stored potential energy (U) is transferred to the projectile in kinetic
energy (K). U = K

- The linear kinetic energy is equal to one half the mass of the projectile
(m) times the square of the linear velocity (v). K = (1/2)mv²

- If friction and gravity are negligible, then the initial velocity of the projectile
as it: v = sqrt(2U/m)

- Thus the velocity of the projectile can be maximized by increasing the potential
energy (by strengthening the beam or increasing its length) or by decreasing the
projectile's mass.

**TREBUCHET**

The trebuchet is much like the catapult proper in many respects; however, instead
of storing potential energy in a taut rope, the trebuchet uses a counterweight
to hurl its projectile (thus using gravitational potential energy).

The trebuchet uses a cross beam, much like a seesaw, as its fulcrum. On one side
of the beam is the counterweight, and on the other side is the projectile. Assume
for the sake of example that the projectile is on the left and the counterweight
on the right. The end with the projectile is pulled downwards, which in turn increases
the potential energy of the counterweight because it is raised,. When the lever
arm is released, the weight of the counterweight pulls the lever arm down, producing
clockwise torque on the entire system. This torque then forces the left side of
the beam back up, propelling the projectile nearly straight up at a blazing speed.

- The weight of the counterweight (W) is equal to its mass (M) times the acceleration
due to the gravitational force of the earth, a constant g. W = Mg

- The weight is a downward force that produces a clockwise torque on the entire
system. The torque (T) is equal to the weight times the distance between the counterweight
and the fulcrum (R) times the vertical component of the weight on the beam if
it is at an angle (Θ). T = WRsinΘ

- The torque is constant throughout the system, but produces a force directed
up on the projectile. The force (F) is also different depending on the distance
between the projectile and the fulcrum (r). T = -FRsinΘ

- The force on an object is given by its mass (m) times the acceleration (a)
produced by the force. (newton's second law of motion): F = ma

- Substituting equations (1), (2) and (4) into (3) and solving for a gives us
the equation for the resultant acceleration. If friction is negligible, then the
acceleration can be given by: a = -RMg/(rm)

- Thus the acceleration on the object and the velocity that it leaves the trebuchet
with can be maximized by increasing the distance between the counterweight and
the fulcrum, increasing the counterweights mass, decreasing the distance between
the projectile and the fulcrum, and decreasing the mass of the projectile.