The Effect of Catapult Arm Length on Distance Ball Is Thrown
Charlotte Hirsch
Takoma Park Middle School 2011-12
Abstract
In this experiment, I am testing whether Leonardo da Vinci’s leaf-spring catapult can be improved, so that it throws a ball farther, by changing the length of the catapult’s arm. My hypothesis was if I make the arm 30.5 centimeters, then the ball will be launched the farthest because the arm will have a bigger range of swing with more force and energy built up than the 17, 21.5, and 26 centimeter-long arms will have, without overwhelming the leaf spring like the 35 centimeter-long arm might. In my experiment, I built a leaf-spring catapult with a kit. I replaced the kit's plastic arm with 17 cm, 21.5 cm, 26 cm, 30.5 and 35 cm long wooden rods (3/8 inch thick) with a measuring spoon at the end to hold the ball. I tested the average distance the ball was thrown with each arm. The results of my experiment showed that as the catapult arm length increased, the distance the ball traveled decreased. The 17 cm arm, the shortest arm, threw the ball the greatest distance: 2.649 meters. The longest arm (35 cm) threw the ball the shortest distance, 1.548 meters. My hypothesis was refuted because the 30.5 centimeter-long arm didn’t throw the ball the farthest distance. These results could be used to improve many object-throwing mechanisms that have arms similar to the one in this catapult. To make these mechanisms throw objects farther, we now know that you have to shorten the throwing arm.
Key terms:
· Catapult arm
· Leaf spring
· Da Vinci
· Potential energy to motion
· Mechanics
· Tension
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Journal Article The Effect of Catapult Arm Length on Distance Ball Is Thrown
Introduction and Review of Literature
I am testing whether or not Leonardo da Vinci’s leaf-spring catapult can be improved with a different length of arm. Also, if it can be improved, what is the length of arm that makes the ball launch farthest? I am interested in this topic because I love testing out simple mechanisms made from materials I find around the house, like rubber bands. Trying to build, improve, and design the arm of Da Vinci’s catapult will definitely be a lot of fun. I am also interested in the historical aspect of this project. Researching the ideas of Da Vinci and other catapult scientists will be very interesting. I have no previous experience with this topic.
The independent variable for this experiment is the length of the catapult’s arm, from the base of the release cup to the end of the dowel rod. The levels are 17 centimeters, 21.5 centimeters, 26 centimeters, 30.5 centimeters, and 35 centimeters. The control level is 26 centimeters. Some scientists that I have researched thought that if the arm length is too long, it will put too much stress on the leaf spring. That is why my control is only 26 cm, the length of arm that came in the kit (I will be remaking that too, since it came in plastic and the rest of the arms will be wood). So, I made the two levels that are longer than the control 30.5 cm (4.5 cm longer than the control) and 35 cm (9 cm longer than the control). To make things symmetrical, I chose 4.5 cm shorter and 9 cm shorter for the other two levels. My dependent variable is the distance from the catapult to where the ball first lands, measured in meters. I will place a roll of paper on the ground in the approximate range that the ball will probably land. I will spot the place where the ball first hits the paper and write down a number for the trial and level on the paper where it hits (for Level 1, numbers one through 7, for Level 2, numbers 8 through fourteen, and so on). I will measure from the starting line (where the catapult is) to the mark using a tape measure with meters and centimeters.
The main scientific principle of this experiment is the mechanics of a catapult arm. The arm of a catapult (the long lever that swings and launches the projectile) is a very important part of the catapult. Catapults vary in the way they power the arm to launch a ball. In the leaf-spring catapult, tension is used to swing the arm. The arm is attached to the drum (the cylinder that the arm pivots on). A string that is wrapped around the drum is attached on either end to two curved pieces that form a slightly open arc. The string is tight, and pulls on the springy arc pieces. When a drum release arm (a piece that holds the drum in place) is released, potential energy (the energy stored in a body of matter as a result of its position) and tension cause the drum to rapidly spin on its axle. The arm attached to the drum rotates and releases the projectile.
“Hucbald ap Urp” (pseudonym, real name not available) constructed an experiment called “A Leonardo da Vinci Leaf-spring Catapult” on April 14, 2007. He used a 100-foot tape measure to find the distance the catapult threw a ball (from ground level to ground level). The results ranged from 28 to 33 feet. He found that small rocks tended to stay in the release cup for longer. This study relates to my experiment because it tests a leaf-spring catapult, just like I am. Although he did not change the length of the arm, it is still helpful to know about how far the ball will go if I keep the length of the arm the same.
Alexis Cooley and Kim Jensen constructed an experiment about “The Study of the Catapult” (the date of the experiment is not available). They were finding out the relationship between the length of arm and the velocity of the ball’s flight. The catapult they used was made of a large triangle. For the experiments with the fewest human/weather errors, the data showed that the shorter the throwing arm, the greater distance of flight. This contradicts the r*F= Torque formula (torque is the measure of a turning force on an object/force around a given point, and r is the distance from the pivot point to the point where force is applied), which says that the longer the arm, the farther it should have launched. Their guess of why their data went against this is that the longer arm caused the ball to launch high, but not far horizontally. Or the arm could have been so long that it overwhelmed the small strength of the spring; the longer the arm, the more stress on the weakening spring. This study tested almost exactly what I am testing; but they tested the effect of the length of arm on velocity, while I am testing the effect of the length of arm on just distance. Also, the scientists mentioned that it is possible that if the arm is too long, it will overwhelm the spring. This is very helpful for when I pick my measurements of arms, so I won’t make the arms too long.
Students sponsored by NIST (their names are not available) did a study on statistics, “The Fractional Factorial Example,” and used a catapult to construct their experiment (the date of the study is unknown). The catapult used was powered by a rubber band. They tested and changed the important factors that affect the distance the golf ball is thrown, such as: band height, start angle, number of rubber bands, arm length, and stop angle. They also found out what was needed (what factors/how many or much) for the ball to reach three distances. The results showed that as the arm length increased, the distance (from catapult where ball lands) increased. This relates to my experiment because one of the things these students changed and tested was arm length. Arm length is my Independent Variable too, and the results of this experiment are helpful for writing my hypothesis.
The background research and previous studies have helped me choose the levels of arm length. Research has shown that if the arm is too long, it will be too heavy for the leaf-spring. That led me to choose levels that aren’t extremely long.
If I make the arm 30.5 centimeters, then the ball will be launched the farthest, because the arm will have a bigger range of swing with more force and energy built up than the 17, 21.5, and 26 centimeter-long arms will have, without overwhelming the leaf spring like the 35 centimeter-long arm might. In the NIST experiment, the results showed that a longer arm makes the ball go further. Also, in “The Study of the Catapult,” even though their data showed that the shorter arm launches the ball farther, they admitted that a physics formula proves that the longer the arm, the farther the ball travels. Alexis Cooley and Kim Jensen also said that their longer arms may have been so long that they put stress on the spring. Because of the reasons explained in Cooley and Jenson’s paper, I am not predicting that the longest level of arm will throw the ball the farthest.
Materials and Methods
I built a catapult by following the instructions in the Academy Da Vinci Catapult kit (see appendix for instructions). I pulled off the arch-shaped stopper on the end of the arm included in the kit, and pulled the arm out of the drum. I took a 1/8-teaspoon plastic measuring spoon and attached it with a 3 inch piece of duct tape to the end of the 26 cm dowel rod so that the spoon was in front of the dowel rod. I used the plastic arm from the kit to saw an arm from the dowel the exact same size as the plastic one. I pushed the rod through the hole 5 centimeters into the drum, and placed the small “u” shaped stopper over the end of it so it wouldn’t slide out. I placed the catapult on top of 5 books on the floor. I rolled paper onto the floor directly in front of the catapult (in a straight line).
I pushed the arm as far back as it could go. I placed a plastic ball with a mass of less than 2 grams and a 1.5 cm diameter inside the release cup (the plastic measuring spoon). I pulled the release arm so that the drum spun on its axle, and the arm launched the ball. The ball hit the paper and I marked the spot where it hit with its level and trial number. I repeated this process six more times (for a total of 7 trials). I took the arc shaped stopper off the end of the arm sticking out of the drum. I slid this arm out and un-taped the plastic measuring spoon.
I repeated that process with the 17 cm rod: taping on the spoon, securing the rod into the drum, launching the ball by pushing the release arm, and marking the spot. I did this for 7 trials total. I took the stopper off, slid the arm out, and un-taped the measuring spoon to prepare for the next length of arm. For the 21.5 cm, 30.5 cm and 35 cm arms, I followed the same entire procedure. I measured the distance from the end of the base of the catapult closest to where the ball lands to the mark on the paper. I took the lowest and highest results off to make only five trials count.
Throughout my entire experiment, my catapult stayed 10 cm off the ground. I kept the projectile the same so that it wouldn't affect the flight of the ball. The material of the arm (wood), release cup (plastic), and base of the catapult (plastic) were kept as controlled variables. I kept the length of the piece of duct-tape attaching the spoon to the arm controlled. For all 5 levels, the catapult (aside from the arm) remained exactly the same. The design (but not length) of the arm also stayed the same.
Results
The catapult with a 17 cm arm threw the ball an average distance of 2.649 meters, having thrown the ball the greatest distance. The catapult with a 21.5 cm arm had an average distance of 2.473 meters. The control level arm (26 cm) threw the ball an average distance of 2.342 meters. The 30.5 cm arm threw the ball an average of 1.896 meters, followed by the 35 cm arm, which had an
Distance ball is thrown (meters)Arm length (cm) / Trial
1 / Trial
2 / Trial 3 / Trial
4 / Trial 5 / Trial 6 / Trial
7 / Mean
17 / 2.890H / 2.656 / 2.714 / 2.614 / 2.670 / 2.592 / 2.563L / 2.649
21.5 / 2.539H / 2.458 / 2.461 / 2.462 / 2.512 / 2.473 / 2.452L / 2.473
26 / 2.409H / 2.361 / 2.402 / 2.365 / 2.294 / 2.290 / 2.273L / 2.342
30.5 / 2.003H / 1.586L / 1.967 / 1.947 / 1.992 / 1.638 / 1.938 / 1.896
35 / 1.614 / 1.703H / 1.361 / 1.225L / 1.561 / 1.614 / 1.591 / 1.548
average of 1.548 meters. Overall, as the catapult arm length increased, the distance the ball traveled decreased. The arm length had a negative correlation with the distance the ball was thrown.
Discussion and Analysis
One of the possible reasons for the results of this experiment has to do with the height of the path of the ball. A shorter arm put most of its power into the distance the ball traveled, as opposed to the height, therefore sending the ball at a lower height and a longer distance. But as the length of the arm increased, more power was put into increasing the height of the ball instead of the distance. This would cause the distance that the ball travels to decrease as the length of the arm increases.
My hypothesis is refuted. I predicted that as the length of the arm increased, the distance the ball was thrown would increase as well. However, the results proved that as the length of the arm increased, the average distance the ball was thrown decreased. The 17 cm arm, the shortest arm, threw the ball the greatest distance: 2.649 meters. The longest arm (35 cm) threw the ball the shortest distance, 1.548 cm.
My testable question was answered. Da Vinci's catapult can be made to throw the ball a greater distance – but by shortening, not lengthening, the length of the arm. The length of arm that threw the ball the greatest distance was not the length of arm that the kit came with (which would be “Da Vinci’s arm length”).
The results of my experiment corresponded with some of my background research. "The Study of the Catapult" by Alexis Cooley and Kim Jensen had the same results; the shortest arm threw the ball the farthest. Their best guess for why this happened had to do with the main scientific principle of this experiment. The mechanics of the catapult arm caused the ball to travel higher (and not longer) as the length of the arm increased.