Jessy+and+Megan

The Ugly Duckling


 * __ Physics... APPLIED! __**


 * Newton's laws**: These are the basic laws of the universe that are applicable to all objects. All objects stay in their natural state of motion unless acted upon by an outside force (i.e. an object at rest stays at rest or an object in motion stays in motion). Unbalanced forces occur when an outside force (such as gravity) causes an acceleration of the object.... //F=MA.// The larger the object, the greater the force that must be applied. Relating to a trebuchet, the longer the arm or the heavier the system, the greater the counterweight must be.

final velocity = initial velocity + accerlation x time// A faster velocity of the projectile means a greater distance flown. By dropping the counterweight from an increased height, GRAVITY will accelerate the counterweight from 0 to a final velocity. Gravity itself is a constant speed- 9.8 m/s2- what differs is the amount of time that the object is falling for. The more time there is for this acceleration to occur, the faster the resulting final velocity will be. This velocity will be the speed at which the projectile travels. To achieve a further distance, it is beneficial to lengthen the trebuchet's arm and release the counterweight from a higher dropping point.
 * Kinetics:** //distance = velocity x time

Friction occurs when the arm pivots, when the counterweight rubs against something on its way down, etc. Air resistance can slow down the motion of the trebuchet's arm if the arm is too wide.
 * Opposing Forces:** Both friction and air resistance counteract any previous acceleration upon an object, or act in the "opposite direction" as the original motion. To maximize the distance of the projectile, potential friction and air resistance of the trebuchet must be decreased.


 * Potential Energy** is the energy a given object is capable of releasing. We found that dropping the counterweight from a higher release point produced greater energy. In turn, our trebuchet's arm swung at a faster rate and the distance/path of the ball was greatly increased.


 * Terminal Velocity:** An object with a certain mass can only reach a certain velocity before it cannot further accelerate. This is due to the force of air resistance acting on the projectile. Despite the size of the trebuchet or the force exerted on the ball (projectile), the ball can only reach a certain speed.


 * Projectile motion:** The motion of a projectile has two components - horizontal and vertical. The horizontal velocity always remains constant, while the vertical velocity changes according to acceleration/deceleration. To achieve the greatest distance, the release angle of the projectile should ideally be at 45 degrees. This produces an arch that gives the ball time to travel horizontally. A low angle release point is not ideal as the projectile will travel in a high arch, and not achieve great horizontal distance. A high angle release point will produce better results as a lower arch can travel a greater distance. Ultimately, an angle in between the extremes would produce the ideal path of motion - and subsequent greatest distance!

**__ PROCEDURE __** 1. Arrange and cut the styrofoam blocks into two stacks of equal height 2. Use hot glue gun to glue styrofoam blocks together - used one piece of cardboard in addition to styrofoam (see Analysis for explanation) 3. Using scissors, divide the thin wooden board into two equal parts 4. Attach the metal holders in the center of the respective wooden boards using heavy duty scotch tape

5. Use hot glue to glue the newly created holders to the styrofoam base structures 6. Using the rounded wood file, file a round indent into the rectangular wooden dowel (approximately 2/5 of the length from one end) 7. Insert a dab of hot glue into the indent 8. Press the center of the round wooden dowel into the indent 9. Coil wire around the attached point -- this is the pivot point of the trebuchet's arm! 10. Hammer a tack into the end of the trebuchet's arm (rectangular dowel) that is closer to the pivot point 11. Place 24 decorative gems into cloth bag

12. Seal cloth bag by tying cord around it 13. Use duct tape to secure the knotted cord/loose end of the cord to the cloth sack 14. Loop other end of cord around the tack (space between the top of tack and the dowel) and tie the cord 15. Attach measuring cup to other end of rectangular dowel (opposite cloth sack/counterweight) using heavy duty scotch tape 16. Use hot glue to attach bamboo skewer to both styrofoam base structures (measured distance apart) 17. Measure (using ruler) and cut out (using scissors) piece of cardboard equal to height of the can (see Analysis for explanation) 18. Hot glue cardboard piece to bamboo skewer and one of the styrofoam base structures 19. Drill hole in additional styrofoam block (not discussed earlier) using round wood file, equivalent in side to one plier (one handle of pair of pliers)

20. Hot glue this styrofoam block (aligned vertically) to one of styrofoam base structures 21. Place one handle of pliers in the hole 22. Use paint to make trebuchet beauteous 23. Remove part of one of styrofoam base blocks using exacto knife (to prevent friction with counterweight - see Analysis for more detailed explanation)

**__MATERIALS__** (does not include those used in failed first model) - 24 decorative gems, ~ 0.02 kg each - cord - cloth bag - duct tape - one tack - one pair of pliers (release of counterweight)
 * //COUNTERWEIGHT//**

- round wooden dowel - rectangular wooden dowel - wire - hot glue - 1/3 plastic measuring cup - duct tape
 * //ARM//**

- styrofoam blocks, various sizes - 2 thin wooden boards - 2 metal holders - scotch tape (heavy duty) - hot glue - 2 pieces of cardboard - bamboo skewer
 * //BASE STRUCTURE//**

- blue paint - red paint - permanent markers - paintbrushes - water
 * //BEAUTY//**

- hot glue gun - markers (for reference points) - small hammer - exacto knife - wood file - scissors - ruler (measure cardboard)
 * //TOOLS//**

-** coffee - chocolate - oranges - loud, terrible radio music - laughing at Megan's dancing to aforementioned loud, terrible radio music
 * //STIMULANTS (Wednesday night, not Sunday night, no worries)//

-** Christmas ball (projectile) - additional 16 decorative gems, ~ 0.02 kg each - 6 rectangular wooden dowels - handsaw - small nail (never was able to use it...) - string - 2 billiard balls - small piece of ribbon (used to tie cloth bag) - can (to control arm's release point) - wooden paddle - additional round wooden dowel - electronic scale - 6 additional bamboo skewers - meter stick - ruler 1. Ensure pliers are straight 2. Place pivot point (point where round dowel attaches to rectangular dowel) equidistant from styrofoam base structures 3. Bend measuring cup back slightly 4. Place projectile in measuring cup 5. Place taped section of cloth bag between pliers, closing handles together 6. Release plier handles (no contact with base structures, only plier handles)
 * __ //Additional materials used in experimentation and various failed models// __
 * __TESTING PROCEDURE__**

Variables we tested (see Analysis): 1. Tightness of cord (attaching counterweight to trebuchet arm) 2. Starting point of measuring cup (finally secured by piece of cardboard and bamboo skewer) 3. Dropping height of counterweight 4. Distance between base structures 5. Placement of ball in measuring cup 6. "Bendiness" of measuring cup 7. Part of cloth bag (counterweight) held between pliers (top of cloth bag vs. taped section) 8. Distance between pliers and base structure (how far out) 9. Weight of counterweight (number of decorative gems used)

**__ MEASUREMENTS and CALCULATIONS __** __**Testing independent variable: weight of counterweight**__ (Note: mass is approximate as we were using Jessy's electronic bathroom scale. For the most part, we calculated the changes in mass by calculating that the weight of 1 decorative gem was ~ 0.02 kg)
 * Number of decorative gems || Approximate mass of trebuchet (kg) || Length of arm (m) || Distance travelled (m) || P value = d/ml ||
 * 40 || 1.14 kg || 0.65 m || 6.7 || 9.04 ||
 * 16 || 0.64 kg || 0.65 m || 3.8 || 9.13 ||
 * 28 || 0.89 kg || 0.65 m || 6.7 || ** 11.58 ** ||
 * 24 || 0.80 kg || 0.65 m || 5.7 || 10.96 ||

__**Performance over 9 test trials**__
 * Trial || Mass (kg) || Length of arm (m) || Distance travelled (m) || P value = d/ml ||
 * 1 || 0.9 kg || 0.65 m || 5.15 m || 8.80 ||
 * 2 || 0.9 kg || 0.65 m || 5.27 m || 9.01 ||
 * 3 || 0.9 kg || 0.65 m || 4.78 m || 8.17 ||
 * 4 || 0.9 kg || 0.65 m || 5.15 m || 8.80 ||
 * 5 || 0.9 kg || 0.65 m || 4.62 m || 7.90 ||
 * 6 || 0.9 kg || 0.65 m || 4.20 m || 7.18 ||
 * 7 || 0.9 kg || 0.65 m || 5.05 m || 8.63 ||
 * 8 || 0.9 kg || 0.65 m || 5.17 m || 8.84 ||
 * 9 || 0.9 kg || 0.65 m || 4.89 m || 8.36 ||

**AVERAGE of 9 test trials:** **8.41**

Our **initial design** involved two teepee-like base structures. Balanced on these structures were two (one one each side) metal holders attached to thin wooden platforms. The round dowel would slide through the metal holders. The round openings of the metal holder would allow the arm (attached perpendicularly to the round dowel) to rotate. The counterweight (two billiard balls in a cloth bag) was attached to the end of the arm by a string tied around a nail.
 * __ ANALYSIS - Changes to the Model __**

This design ** failed **. We encountered numerous problems when we tried to execute the plan. First of all, we experienced some difficulty hammering a small nail into the end of the rectangular dowel (labeled 1). After 50 minutes, we were still unsuccessful... the nail had gone in about a millimetre. That was some strong wood. To this date, we still don't understand exactly what happened. Since using nails proved to be too difficult for us, we replaced the nail with a tack (attached to the end of the arm). While hammering in the tack, Jessy chipped off the top of the tack. Silly girl. In the end, this mistake did not matter as the tack was still able to keep the string (and attached counterweight) in place. Our first model required the 6 wooden dowels (3 in each triangular base structure) to be angled on one end in order for the thin wooden platform to sit flat. We tried many times to accomplish this task (by means of exacto-knife, handsaw and scissors..), but in the end the wooden structures were still imbalanced. It was tricky to get the wooden dowels to line up, especially two sets of three!



After creating both of the teepee-like base structures, we realized the unevenness could not support the platforms and the attached arm of the "trebuchet". Furthermore, we realized that on top (pun intended) of the innate imbalance between the structures themselves, we would not be able to nail or attach the thin wooden platforms to the structures. Attaching the platforms to the base structures would be necessary in order for the trebuchet to work at all. Due to our lack of nailing abilities and lack of planning, we deemed this design unfit for success.

When designing our **second model**, we decided wood would not be an option, since both of us fatefully chose Home Economics over Technology in the eighth grade (and also have a general inability to build things that need real tools). Thus, we chose styrofoam. Megan's science fair project last year involved building a house out of styrofoam. She found it to be easy to work with, lightweight and easy to balance. According to the assignment equation, we wanted to build a lightweight machine that could fire the projectile a fair distance, as opposed to building a massive, heavy, complex machine with many nuts and bolts and other materials that would possibly fire the same distance in the end.... cough cough... We thought we would have better luck going with a smaller or a more lightweight model, catering to our strengths! :)

Our second model featured styrofoam blocks instead of wooden dowels assembled into teepees. Instead of using nails, hammers, and handsaws, we used hot glue, glue guns and our bare hands. The upper portion of our design remained the same - the arm (rectangular dowel) would still be joined to the roudn dowel at a 90-degree angle. As in the original design, the round dowel would then attach to the base structures by sliding through the metal holders.

Looking around Jessy's basement, we found several pieces of styrofoam in random shapes. We tried various combinations until we got two stacks that matched up perfectly! Hooray!

At one point, the glue gun overheated and the hot glue melted away the styrofoam, not allowing two pieces to attach. (In our completed project, we tried to make it look artsy... but failed). We found a piece of cardboard to be the same thickness as that particular piece of styrofoam, and proceeded to glue the cardboard beside the styrofoam piece, attaching the next styrofoam block to the cardboard.

In our design, we planned to attach the two dowels (rectangular dowel for the arm and otating round dowel) by making a round indent into the square dowel using a wood file, and gluing the two together. We quickly realized this would not be sufficient in attaching the two dowels, so we wrapped the pivot point with wire, securing the structure. After assembling the machine, we **began testing every variable to find the ideal conditions** for launching the christmas ornament!

Jessy's mom would not let us use billiard balls for the counterweight, but she found us decorative gems (similar to marbles) to use instead. This proved to be a beneficial change, as we were easily able to alter the counterweight by simply removing or adding a few gems (whereas a single billiard ball accounts for a large weight). We initially tested the trebuchet using 24 gems as our counterweight. The string we used to attach the counterweight (cloth bag filled with decorative gems) to the arm was not strong enough; we changed the string to a thicker cord-like string (referred to earlier as a piece of cord).

The first independent variable we tested was the **length of the cord**. The ball is released wherever the rope is fully extended/pulled taut. If the string is short, the release point will be at a lower angle to the pivot point, projecting the ball in a higher arch (and subsequent shorter distance).

In this project, we wanted to achieve maximum distance, as height plays no part in the equation. A longer string, on the other hand, would create a higher-angle pivot point, projecting the ball in a shorter arch and further distance. Realizing the physics principles, we lengthened the cord. At the right cord-length (we experimented to find this variable), the cord is not pulled taut when the counterweight hits the ground. This allows the arm to follow through in its path of motion, as opposed to jerking back with the impact of the string being extended to its maximum length. This means the ball is released at the ideal flight angle. The next variable we tested was the **starting point of the measuring cup** (opposite end of arm as the counterweight). We initially used a can to determine how far back the arm would start at. The arm only rotates so far - it stops once it loses momentum/the counterweight hits the ground. When the can was close to the pivot point (directly underneath), the ball was released at a lower angle/sooner in its path of motion. When the can was placed further away and the arm was pulled back, the release point was at a higher angle. This caused the same benefits as the earlier paragraph. However, the can was not attached to the trebuchet itself. We thought Mr. Kapphahn would consider this cheating. We used a piece of cardboard (same height as can) and a bamboo skewer to hold it in place. Not only did this make the release point a constant variable (can is easily displaced), it actually meant our machine had a lighter weight. This is of course beneficial when you consider the equation of the trebuchet to be d/ml.

We then tested the **dropping height of the counterweight**. We realized that dropping the counterweight from a higher distance than its resting point (when the other end is attached to the machine like in a standard trebuchet model) resulted in a greater force. Since the counterweight has further to fall, it is able to experience a greater acceleration. In turn, the projectile will be released at a higher speed. We know that distance = velocity x time -- therefore, a higher velocity would mean a greater distance.

Again, we realized that dropping the counterweight by hand from an increased height might not be considered within the constraints of this project. We tried many options to develop a method of release that would be considered "part of the machine".

Even after incorporating the pliers into our trebuchet design, we still had to adjust other things to keep its release consistent. For example, we had to adjust the **position of its styrofoam piece** to make sure the tip of the pliers would be centered between the trebuchet's two major bases. When it wasn't, the counterweight was likely to hit the edge of one of the bases on its path downwards, resulting in the disruption of the projectile's motion. Apart from centering the tip of the pliers, we realized we could prevent this possibility of collision by both **spreading apart the two base structures**. However, the length of the rotating circular dowel prevented us from moving them too far apart, because this risked the ends of the dowel slipping out. Instead what we did to create room was cut away at some of the styrofoam on the bases' interior sides. On the one hand, the more we cut away the less stable the base became. On the other hand, this (although minimally) decreased the weight of our trebuchet!
 * 1) We found a //wooden paddle// in Jessy's basement. We thought the counterweight (cloth sack of decorative gems) could rest on the paddle part. By turning the handle, the weight would be dropped. There were a few problems with this idea. First of all, the handle was a rectangular dowel (not round) - meaning it couldn't rotate easily when attached to the base structure. Also, the release of the counterweight was awkward, it slid off irregularly. The subsequent delayed rotation of the arm resulted in a slow release of the projectile.
 * 2) We then tried sticking a round dowel //through// the counterweight (we constructed two holes in the cloth bag). The release was even worse, as it was difficult to pull out the dowel in a smooth and quick fashion. Furthermore, it would have been tricky to attach this (failed) part to the base structure. [[image:Physics_trebuchet_028.jpg width="363" height="290" align="center" caption="Attempt #2 - dowel through counterweight"]]
 * 3) Next, Jessy had a creative yet unachievable vision of constructing some sort of //trap-door release// of the counterweight. Although in theory this would have been the most consistent to operate, the actual process of designing this was beyond the capabilities of our weary minds.
 * 4) Finally, A stroke of genius hit us. **PLIERS**. Pliers are the best imitation of hand-held release, which is simplest and most effective. This release ensures the counterweight is able to drop directly down (without sliding awkwardly and creating friction with another surface). Pliers would also be easy to incorporate into the base section. To do so, we merely cut a hole in an additional piece of styrofoam, inserting one handle of the pliers through the hole. Since this handle was steadily secured, we could use the other handle to open or close the pliers at will. In order to ensure that the pliers would be releasing the counterweight from the optimum height, we attached the styrofoam piece vertically on top of one of the styrofoam bases (see picture).

Another thing we took into account while releasing the counterweight was to **line up the bag with its point of attatchment**- the pivot point where the two sticks are wired together. In the situation where the counterweight is dropped slightly off to the side, some of the force of the bag would be used up on the natural horizontal shifting of the round dowel to compensate.

To make for an easier release of the cloth bag, we **taped around the string which tied the bag shut**. Doing so meant the pliers could not accidentally get caught on either the string or the bag and disrupt the trebuchet's motion. We also like to think that tape decreased possibility of air resistance and resulted in a faster drop (therefore faster projectile flight), although realistically at best this would have a very minimal effect on the overall performance of our trebuchet.

We also did tests to determine whether or not a specific **"placement" of the projectile ball** in our measuring cup affected its flight. However, there was no conclusive evidence to support this and therefore we didn't take this variable into account during the performance of our trebuchet.

Another adjustment we made to the flight path angle of the projectile was to **bend back the measuring cup** prior to launch. We generally found that the more we bent it back the farther the projectile would fly, although we are unsure as to why this is. We suspect that it has to do with the angle of release.

The last and most prominent variable we tested was the actual **weight of the counterweight**. We intentionally did this once all other variables were set to their maximum potential and kept constant. We deduced that it would make sense to start with the heaviest and lightest combinations and narrow it down from there, as we suspected that the maximum range would be achieved by a weight somewhere in the middle. The lowest number of decorative gems we tested was 5. Logically, the projectile barely traveled any distance. The maximum number of gems we tested was 40, which resulted in a distance of 6.7 meters. This originally excited us so much that we didn't want to reduce the weight at all, but thankfully common sense took over and we tested 16 gems, 24 gems, and 28 gems just to make sure. As it turned out, the average distance achieved with the 28 gem weight gave us the best point score in the formula (see section on Measurements and Calculations for details).

**__CONCLUSION__** By taking into consideration the physics principles behind the project itself, our unique trebuchet was able to perform well, even when compared with many "standard" models. Using unconventional materials and innovative concepts (not to mention a beautiful paint job), our trebuchet surpassed expectations - the Ugly Duckling is more than meets the eye!