Our assignment was to create the most ideal trebuchet or catapult that could launch a projectile the farthest distance possible. The project could not be more than a meter in any dimension, yet be able to throw a small ball of clay several meters. We then identified problematic variables after creating a successful first machine and adjusted to create a machine that eliminated most of the major challenging aspects.
Our Design
The design of our machine was simple, yet rather effective after modifications were made. As a class, we investigated certain variables that could be changed on the machine that might have some effect on the results.
CLEAR PARAGRAPH
For our experiment, we calculated the ideal load to ratio effort of the fulcrum on our trebuchet was to shoot the mass of the clay ball as far as possible. When the load to effort ratio of the trebuchet arm was one to one, as close to half as our group could possibly get, the projectile was launched the farthest. We tested each of the nine notches on the arm through three trials of distance. We changed the hole the peg was in after determining exactly how far the clay ball went on average out of the three measurements, starting with only five notches at the beginning of our experiment. We calculated the load to effort ratio by dividing the distance from the hole to the end of the arm by the distance from the start of the arm to the hole. With this information, the highest mechanical advantage for the trebuchet was at the halfway point, the 1:1 ratio, at hole five when the load was 24 centimeters over 25 centimeters, almost exactly half of the arm. The fulcrum at the fifth hole resulted in the clay ball reaching about 29.6 meters on average. Because the fifth hole was so successful and the distance of the ball increased with the number of holes, we drilled in another four holes to see if the results would continue to be so successful. However, these next holes’ distances significantly dropped to the point where we did not even test the eighth and ninth hole. The other holes actually showed to be rather linear in velocity on our graph until the fifth notch, before the rate of distance began to plummet at the sixth hole with a ratio of 0.7:1. For example, the first hole had a 9:1 ratio that shot a consistent 9 meters, the second hole shot 17 meters with a ratio of 4:1, and the third hole had a ratio of 2.4:1 to get the clay ball to reach an average of 22.3 meters. We believe this was due to the balance created by the 1:1 ratio, allowing the majority of potential energy to be converted into kinetic energy. With this experiment, we know exactly where the fulcrum should be to reach the optimal fire of our trebuchet.
CALCULATIONS
We calculated with this formula because it found how high the ball went while gravity’s effect on the mass (acceleration) and how fast the projectile moved (time) were also found. Our trebuchet kept the ball rather close to the ground due to the arm’s slight angle of release and the projectile’s rather small mass, furthermore affected by gravity’s strong pull in particular.
With an average horizontal distance of 26.6 meters, we needed to find the rate of distance covered in an increment of time. Using the time in air of 1.423 seconds, we found the ball travels at a velocity of about 41.8 miles per hour.
Because our group had already found the time rising/falling by dividing the total time in air by two to get 0.7115 seconds and knew that the acceleration due to gravity was 9.8 meters/second squared, we used the acceleration formula to isolate the velocity variable. By multiplying 9.8 meters/second squared and 0.7115 seconds, we found that the projectile moves at about 13.8 miles per hour vertically, showing how little our ball rises.
After finding the vertical and horizontal velocity vectors, we created a triangle with the vertical velocity as the height and the horizontal as the base to calculate the hypotenuse using the Pythagorean Theorem to find that the ball moves at a total velocity of 44 miles per hour.
The spring constant is a measurement of how long and strong the rubber bands we used to fire the arm of the trebuchet are. We found that the three rubber bands’ force is 28 Newtons, or 6.29 pounds, by stretching them as far as they could go on a spring scale to measure the magnitude of the force through weight. We then pulled the rubber bands to their highest point next to a meter stick to find that the bands can stretch to 0.4 meters. Dividing the force of 28 Newtons by 0.4 meters, we measured that the rubber bands have a spring constant of 70 Newton/meters, or about 51.63 foot-pounds.
To find how much energy the trebuchet has due to its positioning height wise in the gravitational field, we substituted the spring constant of 70 Newton/meters and found that the spring (rubber bands) compressed and expanded 0.535 meters, resulting in a potential energy of 10.02 Newton(meters), or Joules.
With a 7 gram ball and a total velocity of 19.67 meters/second, we calculated that the ball has 1.35 Joules of energy due to motion.
Having calculated the potential energy of 10.02 Joules and the kinetic energy of 1.35 Joules, our group knew that energy was lost throughout the process of launching the ball due to mainly air resistance and friction. However, our percentage of energy transferred from potential to kinetic energy was much greater than anticipated and rather larger than some of the other groups.
ADVANTAGES OF OUR TREBUCHET
Our Design
- Flat plywood base (58x31x1.5 cm)
- Two legs (44.5x5x2 cm) with four holes at 27, 32, 37 and 44.5 cm
- 31 cm wooden cylinder for axel
- Two nails with small heads
- Arm/ axle (48.5x1.5x2 cm) with nine holes five cm apart
- Three rubber bands
The design of our machine was simple, yet rather effective after modifications were made. As a class, we investigated certain variables that could be changed on the machine that might have some effect on the results.
- HIGHER AXLE: We moved the axle of the arm up to 0.445 meters after trying 27, 32, and 37 centimeters. The arm’s 44.5 centimeter height allows the axle to completely rotate, circular motion fully able to act on the arm. Friction is minimized when the arm has a clear path without scraping along the ground, avoiding chances of being slowed down.
- NAIL ANGLE: The nail we used has a very small head, so one group found that having the nail at a 10° angle forward rather than straight relative to the arm. This allows the projectile to be launched at a consistent 30°, the optimal angle for the mass to be released at to decrease the effect of air resistance on the ball.
- NO STOPPER: A stopper has no effect on the machine because the ball is released much sooner before coming anywhere near the stopper.
- PROJECTILE: We started with a ball of about 5 grams, but made the mass 2 grams heavier to reduce air resistance. The heavier the ball is, the more the mass can travel and not be as affected by air resistance. However, the amount of inertia on an extremely heavy mass drags the projectile down to a short distance. 7 grams is the happy medium between the two, air resistance and inertia both not as major of problems on the mass.
- STRING LENGTH: The length of the string we began with was fifteen centimeters long before we made the length much longer to 40 centimeters. If too long, the string will drag the ball along the ground, creating friction and preventing the mass from gaining the momentum it needs to launch. If too short, the string cannot fully rotate around the axle.
CLEAR PARAGRAPH
For our experiment, we calculated the ideal load to ratio effort of the fulcrum on our trebuchet was to shoot the mass of the clay ball as far as possible. When the load to effort ratio of the trebuchet arm was one to one, as close to half as our group could possibly get, the projectile was launched the farthest. We tested each of the nine notches on the arm through three trials of distance. We changed the hole the peg was in after determining exactly how far the clay ball went on average out of the three measurements, starting with only five notches at the beginning of our experiment. We calculated the load to effort ratio by dividing the distance from the hole to the end of the arm by the distance from the start of the arm to the hole. With this information, the highest mechanical advantage for the trebuchet was at the halfway point, the 1:1 ratio, at hole five when the load was 24 centimeters over 25 centimeters, almost exactly half of the arm. The fulcrum at the fifth hole resulted in the clay ball reaching about 29.6 meters on average. Because the fifth hole was so successful and the distance of the ball increased with the number of holes, we drilled in another four holes to see if the results would continue to be so successful. However, these next holes’ distances significantly dropped to the point where we did not even test the eighth and ninth hole. The other holes actually showed to be rather linear in velocity on our graph until the fifth notch, before the rate of distance began to plummet at the sixth hole with a ratio of 0.7:1. For example, the first hole had a 9:1 ratio that shot a consistent 9 meters, the second hole shot 17 meters with a ratio of 4:1, and the third hole had a ratio of 2.4:1 to get the clay ball to reach an average of 22.3 meters. We believe this was due to the balance created by the 1:1 ratio, allowing the majority of potential energy to be converted into kinetic energy. With this experiment, we know exactly where the fulcrum should be to reach the optimal fire of our trebuchet.
CALCULATIONS
- Horizontal Distance: 26.6 meters
- Vertical Distance: 2.47 meters
We calculated with this formula because it found how high the ball went while gravity’s effect on the mass (acceleration) and how fast the projectile moved (time) were also found. Our trebuchet kept the ball rather close to the ground due to the arm’s slight angle of release and the projectile’s rather small mass, furthermore affected by gravity’s strong pull in particular.
- Total Time in Air: 1.423 seconds
- Horizontal Velocity: 18.69 meters/second
With an average horizontal distance of 26.6 meters, we needed to find the rate of distance covered in an increment of time. Using the time in air of 1.423 seconds, we found the ball travels at a velocity of about 41.8 miles per hour.
- Vertical Velocity: 6.17 meters/second
Because our group had already found the time rising/falling by dividing the total time in air by two to get 0.7115 seconds and knew that the acceleration due to gravity was 9.8 meters/second squared, we used the acceleration formula to isolate the velocity variable. By multiplying 9.8 meters/second squared and 0.7115 seconds, we found that the projectile moves at about 13.8 miles per hour vertically, showing how little our ball rises.
- Total Velocity: 19.68 meters/second
After finding the vertical and horizontal velocity vectors, we created a triangle with the vertical velocity as the height and the horizontal as the base to calculate the hypotenuse using the Pythagorean Theorem to find that the ball moves at a total velocity of 44 miles per hour.
- Release Point: 18°
- Spring Constant: 70 Newton/meters
The spring constant is a measurement of how long and strong the rubber bands we used to fire the arm of the trebuchet are. We found that the three rubber bands’ force is 28 Newtons, or 6.29 pounds, by stretching them as far as they could go on a spring scale to measure the magnitude of the force through weight. We then pulled the rubber bands to their highest point next to a meter stick to find that the bands can stretch to 0.4 meters. Dividing the force of 28 Newtons by 0.4 meters, we measured that the rubber bands have a spring constant of 70 Newton/meters, or about 51.63 foot-pounds.
- Potential Energy of the Spring: 10.02 Joules
To find how much energy the trebuchet has due to its positioning height wise in the gravitational field, we substituted the spring constant of 70 Newton/meters and found that the spring (rubber bands) compressed and expanded 0.535 meters, resulting in a potential energy of 10.02 Newton(meters), or Joules.
- Kinetic Energy of the Ball: 1.35 Joules
With a 7 gram ball and a total velocity of 19.67 meters/second, we calculated that the ball has 1.35 Joules of energy due to motion.
- Percent of Energy Converted: 13.47%
Having calculated the potential energy of 10.02 Joules and the kinetic energy of 1.35 Joules, our group knew that energy was lost throughout the process of launching the ball due to mainly air resistance and friction. However, our percentage of energy transferred from potential to kinetic energy was much greater than anticipated and rather larger than some of the other groups.
ADVANTAGES OF OUR TREBUCHET
- Consistent in horizontal distance and fires extremely far
- Simple and easy to work
- Total velocity is high
- Energy transfer is larger than other groups
- Modified to create ideal machine