Mini Cardboard Catapults

Sam McConnell, Max Dietrich, Patrick Revilla

Chen

Math Applications in Engineering

20 January 2015
Abstract

For this project, the tasked was building a catapult out of cardboard and other household

materials, and using it to hit a target at a set but unknown distance away (another catapult). This

was a direct test of knowledge of ODEs and the ability to understand and manipulate the math

behind creating a catapult such as this one. Though this is a small scale model, the same

principles could also be scaled up to a larger catapult, using real materials like wood, metal, and

high-strength cables instead of cardboard, popsicle sticks, and weak rubber bands. It was

decided to go with a simple yet effective catapult design, using an onager-style catapult. The

flexibility and consistency that this design provided served the needs of the situation well,

although limitations in its maximum range might make it unsuitable for all situations. After

trying to hit a target at range 120in, the projectile came very close, hitting the ground right next

to the target twice. However, the catapult was not quite accurate enough to hit the target directly.

This most likely had to do with the inaccuracy of determining the launch angle. However, the

catapult came very close to its mark and served as a fine example of the principles involved.
Introduction

Catapults were one of the earliest yet most effective war machines, developed by the

Greek and Roman Empires and used in serious combat all the way into WWI, where they were

used to hurl grenades from the trenches to the enemy side of the No Man’s Land. However, they

reached their most popular use before the development of explosive weapons, which were able to

outclass catapults in terms of devastation and accuracy. This occurred in the Middle Ages, when

walled cities were used as a primary defense against invaders. Attackers could use the catapults

to either throw incendiary or biological weapons (decaying carcasses) over a cities walls to harm

the inhabitants, or simply throw heavy objects to bash down the walls. They were used primarily

to break sieges or other longer assaults, as shorter battles often could not benefit as much from

the power of a catapult.

In the past, heavy duty catapults--like the ones previously mentioned--were made out of

wooden beams for strength and iron reinforcements for connecting 2 pieces together. In early

history, the most common type of catapult was the ballista, which functioned like a giant bow.

They also used iron plates and bolts for the structural pieces, and would cut trees near the

attacking point and use the local wood to finish the catapult. They also used ropes as the power

source, and used animal sinew and other materials in order to generate the necessary torsion.

These catapult designs were very prominent in Greek and Roman models, as they could be easily

be deployed just about anywhere. As materials became scarce in the Medieval Ages, these were

replaced by onagers and trebuchets, the former of which the catapult was modeled after.

There are three basic designs of catapults, torsion, ballista, and trebuchet. The simplest is

the trebuchet where you have a short arm on one side of the pivot point with a very heavy

counterweights and a long arm with a comparatively light projectile on the other side. This type
of catapult uses the torque caused by the counterweights to fling the projectile great distances. It

is also very important for this type of catapult that the counterweights are on the short side of the

arm because they can only be accelerated no faster than 9.8 meters per second2 and to cause the

projectile to be accelerated faster than that the projectile side of the arm must be longer. It was

decided not to use this one because it was not know how to make the counterweight or the

structural integrity to support that counterweight.

The most complicated of the threes basic kinds of catapults is the ballistia. it has two

torsion springs each attached to a arm which is inturn attached to a rope that propels the

projectile toward the intended target. Because it has the two torsion springs it is more powerful,

more compact, and much harder to build, also if the two springs are not perfectly calibrated it

will shoot at an angle and not at full force. It was decided against this design because the two

springs would both be hard to use and overly complicated to build.

The chosen catapult was the onager, which relies on torsion power. This catapult uses a single
spring (in this case, the coiled bands) and a single lever to throw the projectile. It is arguably the
weakest of the catapults but that is a very relive term. It was easy to build and calibrate, it was
also strong enough to launch the projectile the distance it needed to. for theses reasons and the
reasons against the other designs it was decided to use the torsion catapult.
Analysis of the Problem

The problem presented is to design, build, and perform calculations for a catapult made

of cardboard, rubber bands, popsicle sticks and a few other household materials. Then problem

is having to use the catapult to hit a target at a previously unknown distance. The objective was

to come up with a design for the catapult, as well as model the equations behind it launching a

projectile (a small ball). This model relies on the exact same principles of a real catapult

launching. It is similar to when medieval engineers had to build catapults to hit a certain city.

They have to be accurate, so as not to miss the first shot.

The catapult used a cardboard base (as implied by the name “cardboard catapult”),

several rubber bands for the torsion within the catapult, a spoon as the catapult arm, and popsicle

sticks to hold it together. Other materials like straws were available, but were not used as they

were not seen useful. The team received a corner piece of cardboard, so it was much easier to

build and reinforce the structure. The rubber bands were stretched across the piece of cardboard,

and used popsicle sticks as cross supports to prevent it from bowing inward under the stress.

Instead of using the rubber bands as a conventional stretching manner, instead they were used as

a form of torsion, and powered the catapult by twisting the rubber bands instead of pulling them.

This simplified the build and the calculations, as well as making finding a linear relation in the

data easier. These calculations were used to find a value for the number of twists of the rubber

bands, and also found a regression in the data to find a value directly from the data.
Turns 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Inches 24 26 39 48 52 60 72 82 84 94 100 108 120 117 132 137 142 146 132 144
Travelled
Conclusion

Based on the experiments with the Torsion based catapult, the mechanism proved to yield

fairly consistent and flexible shot distances, at the cost of durability and maximum range. The

main benefits of the catapult were that shots would travel close to the same distance when the

rubber bands were given the same number of turns and that a wide range of turns are possible

with the design. The fact that the catapult fired fairly consistently was very beneficial to us, as it

would have been to armies in the past. Once a ranging shot has been shown to travel the needed

distance, the user could feel safe in knowing that the next shot would as well. The angle that the

rubber bands on the catapult were twisted could be changed by turning two popsicle sticks on the

machine’s exterior (Figure 3, Appendix A), giving the catapult the ability to fire any distance up

to twelve feet. These popsicle sticks are held in place by the friction between themselves and the

cardboard box. This allows the user to set a certain amount of torsion to fire the catapult with

and fire it over and over again, applying only a phase angle to the lever, without having to reset

the rubber bands. This allows the catapult to fire much faster and with a much higher precision

than many of the other cardboard catapults. These feature in a catapult would be useful to an

army that needed to hit a moving target, or one that wanted to have a mobile catapult that could

be used in varying circumstances.

These benefits did have some tradeoffs however. One of these was the distance that the

projectile could be thrown. The torsion constant k of the rubber bands was calculated to be only

1.364x10-3 Newtons/radian, much smaller than their spring force constant. This meant that even

when 5 rubber bands were twisted over 3000 degrees each, they could not match the distance of

some of the catapults that were not torsion based. Another downside of the catapult that was

associated with this was the durability of the rubber bands. As shown in the graph and table
above, there were drops in the distance the projectile travelled after 7 turns and 9.5 turns. These

decreases were caused by some of the rubber bands being torn apart under the great strain they

were subjected to, as the rubber bands were both stretched past their equilibrium lengths and then

twisted up to 10 times. This was not a major problem when the strains were applied for short

periods of time but if the rubber bands were left in that state for many minutes or hours, then it

was not uncommon for them to break enough so that they needed to be replaced. The large

amounts of torsion also caused the sides of the catapult to bow inwards until rigid gussets were

added in the form of popsicle sticks(figure 3, Appendix A).

During the trials where the catapult had to hit another team’s catapult, some of these

strengths and weaknesses were exhibited. Once the distance to the target was known, the

catapult was able to be quickly set up to fire that distance, 10 feet. Even after the first shot, the

popsicle sticks were able to hold the rubber bands in their pre-firing position, allowing for a

quick second shot. Both shots however fell a few inches short of the target and the second shot

travelled a few inches to the left of the target. These unsuccessful shots might be attributed to

improper set up of the catapult, or weakened rubber bands, nevertheless they did prevent the

catapult from achieving its goal.

Overall, the design gave the catapult greater flexibility, consistency, and rate of fire than

other the catapults of other groups, but caused it to be more prone to breaking, and gave it a

shorter maximum distance. This design fit us well for the needs of the problem, hitting a fixed

target at an unknown distance, but anyone looking for a catapult design to make should consider

whether a large projectile range, or great durability are important for their needs.
 Appendices

Appendix A: Graph/Data

Turns 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Inches 24 26 39 48 52 60 72 82 84 94 100 108 120 117 132 137 142 146 132 144
Travelled
Appendix B: Calculations

The relationship between the torque of the catapult, the catapult’s moment of inertia and the
number of turns given to the rubber bands can be shown by the following differential equation
where τ is the catapults torque, I is the catapult-projectile moment of inertia, k is a rubber band’s
torsion constant, and θ is the number of angles that each rubber band is turned.
Στ=−kθ=Iθ ' '
Στ=Iθ ' ' +kθ=0
This does make several assumptions about the system, however. For one, the term in the net
torque that has to do with the torque caused by gravity in in this lab was ignored. As the spoon is
more horizontal, Gravity acts more on the spoon, slowing it down and providing a component
against the movement of the spoon’s arc.

In order to make the equations simple enough to solve, it was assumed that k is the same for all
rubber bands, that all rubber bands are twisted the same amount, and that the torque created from
the weight of the lever, and projectile are negligible.

It was also assumed that the moment of inertia for the spoon and lever can be found by adding
the moments of a point mass with the mass of the projectile and of a rod with uniformly
distributed weight.
2
MR 2
I= +mr
3

Where M is the mass of the lever (spoon), R is the length of the spoon, m is the mass of the
projectile, and r is the distance from the projectile to the axis of rotation. For the catapult,
M=1.772 g, R=5.75 in, m=7.8 g, and r=4.25 in.

This gives us a moment of inertia of 1.035 x 1 0−4 k g m2= 19.670 g ⋅in 2= 4.99*10.

k was calculated by attaching the catapult up to a spring scale, which measured the force that the
rotated catapult arm produced on a spring. At 8 rotations, the arm produced .6N of force. Given
that this occurred at distance R = 4.25in, k the torque constant can be calculated:

FR = kθ
(.6N)(4.25in) = k(8π)
k = .0812 N in

Knowing these values it is possible to,estimate the distance that a projectile will travel given a
certain amount number of turns on the rubber bands using the kinematic equations, and the
conversions between angular and lateral motion.

Given the values for k and I, can be calculated the angle θ in terms of a time, t. Given the ODE
is of the form Iα + kθ = 0, and that α = d 2 θ/d t 2, the solution to this differential equation is
θ = a cos(
√
k
I √ k
t) + b sin( t), for unknown constants a, b.
I
The initial angle is θ0 at t = 0, so a = θ0 . By differentiating, a similar equation was calculated for
the velocity. However, at t = 0, the angular velocity is 0, so b = 0. Hence the equation is θ = θ0

√k
cos( t)
I

To find the velocity, it is necessary to differentiate, to obtain

ɷ = θ0
√ √
k
I
k
sin( t).
I
However, it is not wanted to have velocity in terms of time, but in terms of theta. Therefore, use
the identity cos(x )2 +sin(x )2 = 1 was used to derive that

ɷ=
√
k
I
√ θ02−θ 2
Multiplying by the radius gives us v = rɷ to get

v=r
√ k
I
√ 2
θ0 −θ
2

Here it was assumed that the ball is being released when the spoon is at an angle of ψ from
vertical. This means that the velocity at launch is given by
v=r
√ k
I
√ θ02−ψ 2

This is now a problem in projectile motion. For an object released at angle ψ from horizontal (is
the spoon is offset from vertical, the ball’s trajectory is offset from horizontal), the time it is in
the air is 2v sin(ψ )/g (by solving for t in the equation -v sin(ψ ) = v sin(ψ ) -gt). Based on this, the
distance it travels is given by d = v cos(ψ ) t = 2cos(ψ )sin(ψ )/g * v 2. This means that the final
equation for the distance is given by

d = sin(2ψ ) k r 2 / I g (θ02−ψ 2) for angles θ0 and ψ , with d in inches.

However, this can also be factored as

❑ ❑ ❑ ❑
d = sin(2ψ ) k r 2 / I g (θ0 −ψ )(θ0 +ψ )
❑ ❑
= sin(2ψ ) k r 2 / I g (θ1 −ψ 0 )(θ1 +ψ 0 +4 πn ) with θ1being the displacement from 0 vertical of
the initial twist ( π /2), ψ 0being the final release angle from 0 vertical (measured to be around .436
from high frame rate footage), and n being the number of twists of the rubber bands. All of
these values are constant except for n, which means that the graph of distance v. twists should
have a nice linear regression. When the shots were taken, d = 120 was plugged in, and used the
values of k, I, g, ψ 0, and θ1to find the number of twists, n.

Finally, an equation was found to relate the number of turns given to the rubber band and
projectiles displacement in inches experimentally obtained from the testing (graph as appendix
A). The equation is:
Expected displacement =0.0379 (degrees turned )+21.305 .