Physics Extended Essay (Ballistics)

INTRODUCTION

1._Background Information For thousands of years human beings have been fighting in battles, always trying to find new ways of killing each other and winning. Combat was getting more and more sophisticated and human force was not sufficient enough to conquer a castle or destroy a ship. That was the time when ancient engineers introduced siege warfare, when “simple yet complicated hurling machines that rely on the fundamentals of math and physics using levers, force torsion, tension, and traction” were brought to the battlefield. One of the most unique and destructive siege weapons was the ancient Greek ballista, used by Alexander the Great and later improved by the Romans. Ballista is a “torsion-powered machine,” and unlike other ancient weaponry, it casts bolts and hurls stones straight and low, keeping the parabola as horizontal as possible. For that reason, projectiles travel farther and the gradient of the trajectory remains small – not sleep. It is a very accurate missile weapon, but because of its fairly lightweight bolts, the ballista does, however, lack the high momentum of heavy rocks (80 – 140 kg) launched by trebuchets and mangonels. The operating crews were able to load spherical (stones) and shaft (bolts) projectiles according to the situation and the needs. The former were fired in order to destroy masonry and light structures (castle gates, siege towers), while the latter were often used against armored troops. Original models of the ballistae could hit targets 300 or even 400 meters away.

In order to determine which shaft projectiles ancient engineers considered the most effective, I have developed the following research question: “An investigation into the relationship between the wingspans of a projectile stabilizing fin and the force exerted, displacement moved (?) and the velocity of the projectile.” Considering the physical properties described below, I hypothesize that small fins will fly faster and exert more force, while greater wingspan will increase the gliding distance, thus increasing the displacement. Two different fin shapes will be investigated: triangular and quadrangular. Having 4 fins on each bolt would provide much more stability, but we will follow the way that has been used by ancient Greeks, Macedonians and Romans – this ballista was designed for two-finned bolts.

I have chosen this topic because ancient warfare offers a wide spectrum of theories and principles based on physics and mathematics, simultaneously giving me an opportunity to recreate one of the ancient siege machines myself. While ballistae and projectile motion are an interesting topic, it also brings different principles of physics together for a thorough research experiment. Using a representative model of a real ancient ballista I have assembled for this project, I will investigate the measurement (?) of these physical quantities, trying to optimize them in order to find the most effective shaft projectiles for this ancient siege engine.

2._Scope of work:

There are five main physical principles involved behind the science of the ballista:

I. Torque
Ballista’s true power lies in the skeins. A skein is a ‘twisted bundle of rope’, where the actual force is ensured by torsion, while the rope is fully twisted because of the applied torque (rotated about axes). Skeins entangle bowstring arms that help to fire the projectile. Torsion springs, when twisted, store mechanical energy. The more the skeins are twisted, the more energy will be exerted on the projectile. In other words, “the amount of force (actually torque) it exerts is proportional to the amount it is twisted.”[2] Such technique makes it “…possible to shoot lighter weight projectiles with higher velocities over a longer distance.”[3] The objective of the experiment is to keep the amount of torque at maximum all the time by winding up the skein blocks in order to make the experiment as fair as possible and obtain the highest results.

II. Drag
Also known as air resistance, it is the force that opposes an object moving through a fluid (a liquid or gas). This is called fluid friction. Drag forces are dependent on the velocity, shape and size of the body.

III. Lift
In aerodynamics, an airplane wing or a projectile fin exerts a downward force on the fluid (air) while in motion, and the air simultaneously exerts on upward force on the fin. Lift is defined as the component of this surface force exerted by a fluid which is perpendicular to the oncoming flow direction. Sir Isaac Newton published 3 laws of motions in 1687. The principle of lift relies on Newton’s third law of motion, since “to every action there is an equal and opposite reaction.”[4] Lift is always generated by the difference in velocity, so if the projectile is not moving, there is no motion between the object and air, which means no lift.

Diagram 2* (left): Forces on a typical airfoil
Diagram 3* (right): Drag and skin friction dependent on the shape of an object.

IV. Acceleration

Because the velocity of the projectile is changing, it has acceleration (a) as well. When the trigger is pulled, torque that was stored in the skeins exerts force through the bowstring on the bolt, causing it to accelerate from rest (0). Since the mass of the flying projectile remains unchanged, one can calculate acceleration of the object knowing the force which has the same direction. Newton’s Second law of motion is used for this purpose:
F = ma
In other words, “the acceleration of a body is proportional to the force applied and inversely proportional to its mass.”[5]

V. Momentum
Linear momentum (p) can be defined as the mass times velocity of an object, the SI units for which are kg m s-1 or N s. It is used in the following formula:_
Momentum = mass x velocity p = mv_
Momentum, often called ‘mass in motion’, depends on the amount of mass that is moving and velocity. Every moving object has momentum. It helps us to understand what is going on in collisions and explosions. When the ballista is fired, an impulse delivered by the bowstring

2. Experimental variables

Independent variable * Fin size: wingspan (simultaneously affects the lengths of the fin)

Dependent Variables * Force exerted by the projectile * Displacement * Velocity * Acceleration (calculated)

Controlled variables * Dowel mass * Dowel length * Tension of the skeins * Angle * Target surface (“castle gates”): taped book * Fin material: card-stock

3. Experimental Apparatus

* Functioning ancient miniature ballista prototype

* Vernier LabQuest * Vernier Photogate (2) * Vernier Force meter * Wooden dowels * Card stock * Utility knife * Duct tape * Scissors * Meter stick * Lab stand

* Household items (boxes, books, etc.)

4. Experimental Methods

The main objective of the experiment is to find the optimal projectile (bolt) by varying the wingspan of the fins attached to the dowel and changing the shape of the fin itself. These changes will affect displacement, the force exerted by the bolt on the target and the velocity of the flying projectile. This is performed by measuring the three physical quantities first-hand with scientific equipment and a simple meter stick. Using the obtained results will also allow calculating more physical properties like acceleration. There were three experiments which were done independently of each other.

Preparing the projectiles
Each projectile consisted of two parts: A. A wooden dowel 128 mm long with a diameter of 11 mm. For safety reasons, its tip remained blunt (90°). The surface was very smooth after sanding. B. 10 pairs of carton board fins, each with a different wingspan, each pair represented by a number (1-10). 5 of them had a triangular shape, and are always labeled with a letter “T”. The remaining 5 had an additional angle:

11 mm ± 1 mm
128 mm ± 1 mm
Wingspan
of the fins
‘Projectile’ 1 – 5
Quadrangular
‘Projectile’ T6 – T10
Triangular

Experiment 1: Force exerted (F)
In the first experiment I was measuring the pushing forces. Since ballistae were used against lighter structures, I have constructed a very basic representative version of a castle, where a thin book of approximately 150 pages was taped together to serve as shut castle gates. The book was attached to two plastic blocks that were simultaneously affixed to the table, forming a wall line with gates. Right behind the book, the force sensor was positioned on a piece of wood. It was vital to prevent the sensor from moving anywhere, thus it was also attached to the wood which was affixed to the table. I have used the Bumper screw with the sensor; it was directly touching the center of the book – since my goal was to shoot the gates, the bumper screw measured the pushing forces on the book. After the construction was over, I connected the force meter to Vernier LabQuest.
Calibration: The Dual-Range Force Sensor has a switch that lets the user choose between two ranges: ± 10 N and ± 50 N. Because the forces exerted by the shooting bolt often exceeded 10 Newtons, I had to use the ± 50 N range, trading some accuracy in order to be able to measure a greater range of forces. Since the book was slightly touching the bumper, I calibrated the sensor by resetting the measurement to “0” and also reversing it – this way the pushing forces were recorded as positive values, not negative.
The ballista was positioned approximately 3 meters away from the “gates” on the same level above the ground to keep precision relatively high and not to miss the target too often. Data recording was quite simple: cock the ballista, aim for the center of the book, start to record the data on LabQuest, pull the trigger. In order to avoid random errors, I’ve tried to position the ballista in the same spot all the time. Each projectile was launched 10 times.

Experiment 2: Displacement (s)
The following experiment, just like the other two, was performed indoors in order to avoid the main external factor, wind. The ballista was positioned on a box, 1 meter above the ground. We have prepared 1-meter marks on the floor in order to make the measuring process easier. Since my goal was to measure the displacement, the distance moved in a particular direction (?), my colleague was responsible for recording the results using a meter stick while I was firing the ballista. This experiment lacked accuracy, because each time the projectile landed on the ground, it either slipped or bounced, thus observation was approximate, done by eye. There was a high systematic error.
In aerodynamics this concept is known as the gliding distance. Larger wings should result in longer gliding distance. Space Shuttles, for example, have short wings and minimal and a very ‘steep’ glide, while sailplanes have long wings and extremely long gliding distance.

Experiment 3: Velocity (v)
The goal of the final experiment was to measure the velocity of a flying projectile – the rate of change of displacement. This was done using two Vernier photogates (light gates), both positioned at a known separation. During the first attempt, photogate #1 was located 20 cm away from the trigger beam, from which the projectile is fired, while photogate #2 was positioned also 20 cm away from #1. Because it was extremely hard to hit both photogates with a relatively large bolt, I had to reposition the second device, having it 10 cm away from the first, the latter also now being closer to the ‘siege weapon’. Both photogates were mounted on a pile of books, making it possible for the projectile to pass though them. Very high precision was required, because the flying bolt had to simultaneously break 2 narrow infrared beams located inside the photogates. Internal Gate Mode was used, providing very accurate signals for timing. After having spent a lot of time trying to find the best trajectory to cut both beams with the same projectile, I could finally be able to determine the velocity (between the breaking of the first to the breaking of the second). Once the data recording started, it was necessary to keep the ballista stable, carefully spanning and firing. Each projectile was cast 5 times.

Once the experiments have been carried out, the following data was recorded, arranged into a table, graphed and analyzed. The analysis is broken down into individual experiments first, and then joined together for an overall conclusion and evaluation.

Experiment 1: Measuring force Projectile # | 1 | 2 | 3 | 4 | 5 | 6T | 7T | 8T | 9T | 10T | Wingspan (mm) ± 0.5 mm | 19 | 23 | 27 | 30 | 34 | 17 | 22 | 28 | 33 | 38 | Mass (g) ± 0.01 g | 10.77 | 11.00 | 11.32 | 11.65 | 11.89 | 11.02 | 11.45 | 11.79 | 12.28 | 12.92 | Force exerted (N) ± 0.01 N | Launch 1 | 17.37 | 10.03 | 8.99 | 5.35 | 8.19 | 16.71 | 8.10 | 15.41 | 9.15 | 6.15 | | Launch 2 | 15.18 | 8.47 | 12.67 | 12.17 | 9.26 | 9.36 | 11.59 | 10.15 | 9.96 | 4.02 | | Launch 3 | 17.84 | 7.19 | 16.22 | 7.52 | 10.06 | 10.50 | 16.75 | 12.55 | 15.38 | 7.63 | | Launch 4 | 10.61 | 8.98 | 7.17 | 10.43 | 7.01 | 8.23 | 19.94 | 13.29 | 13.70 | 5.29 | | Launch 5 | 12.03 | 14.27 | 9.26 | 9.24 | 5.09 | 11.97 | 20.97 | 9.92 | 11.23 | 9.52 | | Launch 6 | 16.03 | 11.17 | 15.00 | 19.49 | 3.27 | 15.68 | 14.23 | 11.63 | 12.66 | 11.08 | | Launch 7 | 15.21 | 16.01 | 6.21 | 4.17 | 7.96 | 7.56 | 10.41 | 19.65 | 14.94 | 9.96 | | Launch 8 | 26.71 | 18.13 | 15.83 | 10.08 | 6.47 | 11.32 | 15.60 | 10.52 | 12.10 | 10.39 | | Launch 9 | 12.52 | 21.50 | 13.23 | 13.06 | 8.69 | 14.39 | 12.36 | 16.37 | 8.83 | 9.77 | | Launch 10 | 15.65 | 16.05 | 11.05 | 11.03 | 6.02 | 13.37 | 11.08 | 13.14 | 11.12 | 8.24 | | Average | 15.92 | 13.18 | 11.56 | 10.25 | 7.20 | 11.91 | 14.10 | 13.26 | 11.91 | 8.21 |
Table 1

Graph 1

On graph 1, red points represent quadrangular fins, while blue are triangular. If we calculate average results for the entire ‘shape group’, one can say that the triangular on average (results summed up and divided by 5) exert 0.24 N more than the quadrangular. There is a noticeable outlier – the 19 mm quadrangular fin. Its shape and size has bounded the best with the dowel, giving the best results of all. In general, triangular fins, because of the smaller angle in the front part, tend to encounter less form drag, which allows them to strike harder. On the other hand, the smallest triangular fin with a wingspan of 17 mm exerted the least force, most likely because its size did not give the projectile enough inertia to hit hard.
[59.3 / 5 = 11.86 blue]
[58.11 /5 = 11.62 red]
Using the force results obtained, we can refer to Newton’s second law of motion and calculate acceleration. Having constant force and constant mass (which will be changed from grams to kilograms), we will divide the former by the latter: a = F / m Force exerted (N) | 15.92 | 13.18 | 11.56 | 10.25 | 7.20 | 11.91 | 14.10 | 13.26 | 11.91 | 8.21 | Mass in Kg-1 | 0.0108 | 0.0110 | 0.0113 | 0.0117 | 0.0119 | 0.0110 | 0.0115 | 0.0118 | 0.0123 | 0.0129 | Acceleration (ms-2) | 1478 | 1198 | 1021 | 880 | 606 | 1081 | 1232 | 1125 | 970 | 635 |
Table 2

Practically proven and theoretically correct, the torque in the skeins fires a lighter projectile with greater acceleration and higher velocity.

Graph 2

Experiment 2: Displacement
The concept of gliding distance, introduced earlier, was studied here. Projectile # | 1 | 2 | 3 | 4 | 5 | 6T | 7T | 8T | 9T | 10T | Wingspan (mm) ± 1 mm | 19 | 23 | 27 | 30 | 34 | 17 | 22 | 28 | 33 | 38 | Mass (g) ± 0.01 g | 10.77 | 11.00 | 11.32 | 11.65 | 11.89 | 11.02 | 11.45 | 11.79 | 12.28 | 12.92 | Displacement (m) ± 0.1 m | Launch 1 | 9.3 | 13.6 | 11.6 | 10.3 | 6.0 | 7.90 | 13.20 | 11.00 | 8.20 | 5.10 | | Launch 2 | 9.2 | 11.8 | 12.8 | 10.7 | 6.0 | 8.30 | 12.80 | 10.60 | 8.90 | 4.90 | | Launch 3 | 10.7 | 13.7 | 12.1 | 11.4 | 5.3 | 8.00 | 13.50 | 10.10 | 9.00 | 4.60 | | Launch 4 | 9.0 | 12.2 | 12.2 | 11.0 | 5.5 | 7.80 | 12.20 | 9.80 | 8.70 | 6.00 | | Launch 5 | 9.4 | 12.9 | 12.0 | 10.3 | 5.7 | 8.60 | 12.70 | 10.90 | 8.00 | 5.70 | | Average | 9.5 | 12.8 | 12.1 | 10.7 | 5.7 | 8.12 | 12.88 | 10.48 | 8.56 | 5.26 |
Table 3

Graph 3

In airplanes, the shape of the plane’s wings allows it to fly. Longer wings glide longer, while short wings have very little lift and their gliding is minimal. In my experiment, it proved to be true to a certain extent. Medium-sized fins with wingspans of 22 and 23 mm had the largest displacement. Projectiles 4, 5, and T10 were simply too big and heavy to glide property. It is vital to remember that the bolt does not just glide, but keeps rotating around its axis while in motion. Like in a space shuttle, using relatively small fins worked out the best – just enough lift, not too much drag allowed to ‘thrust’ the bolt far away.

Experiment 3: Velocity

Velocity (ms-1)± 0.05 ms-1 | Launch 1 | 21.3 | 19.6 | 18.0 | 16.2 | 16.2 | 20.1 | 18.9 | 16.8 | 16.2 | 15.8 | | Launch 2 | 20.3 | 20.7 | 18.2 | 16.8 | 15.0 | 19.6 | 18.1 | 17.8 | 15.9 | 14.3 | | Launch 3 | 22.0 | 20.7 | 17.9 | 17.1 | 16.9 | 20.9 | 18.5 | 16.3 | 15.5 | 14.7 | | Launch 4 | 21.9 | 19.9 | 18.8 | 18.3 | 15.8 | 21.2 | 17.6 | 18.0 | 16.1 | 15.1 | | Launch 5 | 20.4 | 21.0 | 19.1 | 17.5 | 16.8 | 20.3 | 18.2 | 17.2 | 15.0 | 14.4 | | Average | 21.2 | 20.4 | 18.4 | 17.2 | 16.1 | 20.4 | 18.3 | 17.2 | 15.7 | 14.9 |
Table 4

Graph 4

In this experiment, quadrangular fins proved to be more efficient by an average of 1.36 ms-1.
18.66 Red
17.30 Blue

Projectile # | 1 | 2 | 3 | 4 | 5 | 6T | 7T | 8T | 9T | 10T | Momentum (kg ms-1) | 0.228 | 0.224 | 0.208 | 0.200 | 0.192 | 0.225 | 0.209 | 0.203 | 0.193 | 0.192 |
Table 5

Investigate drag force of the form: F=-kv^2

Comparing actual and hypothesized results:
Before the experiment was carried out, I was not able to support my predictions about the best fin shape with valid evidence or theory. Referring back to the introduction, we were expecting projectiles with smaller wingspan to exert more force and have a higher velocity, leading to a greater acceleration value and higher momentum. Larger wingspans, theoretically, would glide further and thus the displacement will be greater.
Looking at the results, we can say that smaller fins actually do cause the projectile to exert more force and travel faster. Acceleration and momentum, derived from experimental results, back up this idea. Because bolts 4,5 and T10 were slightly too large for the ballista, they were not fired well; otherwise larger (or, actually mid-sized) wingspans glided the furthest because of lift - “to every action there is an equal and opposite reaction,” hence the size let the bolt glide further.

Bibliography

1. Gurstelle, William. The Art of the Catapult. Chicago: Chicago Review Press,
Inc., 2004. Blurb.

2. http://www.rlt.com/10501; http://www.therthdimension.org/AncientRome/Roman_Army/Artillery/artillery.htm
Date accessed: 22/10/2010

3.

[2] http://www.absoluteastronomy.com/topics/Torsion_spring
[3] http://www.absoluteastronomy.com/topics/Ballista
[4] Feynman, RichardP.; Leighton, Robert B.; Sands, Matthew (1963), The Feynman Lectures on Physics, Reading, Mass.: Addison-Wesley, ISBN 0-201-02116-1 , Vol. 1, §10–1 and §10–2. (Wikipedia, Lift article).
[5] HL book
Diagram 2: http://en.wikipedia.org/wiki/File:Aeroforces.svg
Diagram 3: http://en.wikipedia.org/wiki/Drag_%28physics%29

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[ 1 ]. Gurstelle, William. The Art of the Catapult. Chicago: Chicago Review Press, Inc., 2004. Print.
[ 2 ]. http://www.rlt.com/10501; http://www.therthdimension.org/AncientRome/Roman_Army/Artillery/artillery.htm

Bibliography: 1. Gurstelle, William. The Art of the Catapult. Chicago: Chicago Review Press, Inc., 2004 [3] http://www.absoluteastronomy.com/topics/Ballista [4] Feynman, RichardP.; Leighton, Robert B.; Sands, Matthew (1963), The Feynman Lectures on Physics, Reading, Mass.: Addison-Wesley, ISBN 0-201-02116-1 , Vol [ 1 ]. Gurstelle, William. The Art of the Catapult. Chicago: Chicago Review Press,      Inc., 2004