Sam Garst &
Kyle Fuller
Background
Since the development of the gun, archery and the use of bows and arrows have become mostly recreational in nature. Today, people generally only use a bow and arrow for sport, not for war. However, originally, the bow and arrow was developed as a weapon for hunting and for warfare. It is possible that the first use of the bow and arrow was as early as 20,000 B.C., but it is widely accepted that the Egyptians were the first people group to use the bow and arrow about 5000 years ago (Barton). The Assyrians were the first to develop the recurve bow, which is shorter and more powerful than a longbow. A recurve bow curves away from the archer at the ends, allowing it to store more power within a shorter bow, which is more convenient on horseback (Strickland). While the design of the bow is relatively consistent, the elasticity (A determinant factor in a bow’s performance) is often particular to a specific bow. Each particular bow can vary in its general size, construction, and material composition. Given the numerous factors that contribute to a bow’s performance, we have chosen to analyze simply one recurve bow.
Statement of the Problem
The purpose of this investigation is to determine the efficiency of a singular recurve bow by comparing its kinetic and potential energies.
Hypothesis
Due to the numerous factors that contribute to the bow’s ability to generate and produce force, we estimate the calculated the efficiency to lie around 60%.
Materials
1) A 45 lb recurve bow along with an accompanying bow string.
2) Precautionary wrist strap, finger release, and safety glasses.
3) A standard 28” aluminum arrow with a field point tip (40.6 grams).
4) Cardboard, 2x4s, 4x4s, assorted lumber, bolts, nuts, screws, and necessary machinery to assemble contraption.
5) A 1x1 foot square dry erase board, with an accompanying pen.
6) A ruler, measured in centimeters.
7) All electronics necessary to perform mathematical calculations, computer, calculator, etc.
Procedures, Methodology
1) The primary procedure is to separately calculate the kinetic and potential energies.
2) The methods we used to calculate each one of these numbers were independent of one another.
3) Figure A. To calculate the kinetic energy, we decided to measure the vertical displacement of a pendulum when struck by an arrow. This required the construction of a device to hold a two armed swinging mass, along with a device to hold a dry erase board to measure the pendulum’s displacement. The frontal area of the pendulum is lined with cardboard to reduce the possibility of a piercing arrow and the side supported a pen. The white paper is the designated area for the pen and dry erase board.
Figure B (Potential
Energy) Figure A (Kinetic Energy)
Figure B. In order to calculate the potential energy we had to
analyze the displacement of the bow string when definite intervals of weight
were applied. We added 1Kg of weight between each measurement, from 0 to 19 Kg.
This permitted an acquisition of data that illustrated the distance as a
function of weight in (Kg). This will be discussed in the following section.
 |
 |
Graphs, Calculations, and Measurements
Potential Energy:
|
Distance (CM) |
Distance (m) |
Weight (KG) |
Force(N) |
|
2.2 |
0.022 |
1 |
9.81 |
|
3.4 |
0.034 |
2 |
19.62 |
|
4.7 |
0.047 |
3 |
29.43 |
|
6.2 |
0.062 |
4 |
39.24 |
|
8.1 |
0.081 |
5 |
49.05 |
|
10.2 |
0.102 |
6 |
58.86 |
|
12.5 |
0.125 |
7 |
68.67 |
|
15.1 |
0.151 |
8 |
78.48 |
|
17.8 |
0.178 |
9 |
88.29 |
|
20.6 |
0.206 |
10 |
98.1 |
|
23.8 |
0.238 |
11 |
107.91 |
|
27.5 |
0.275 |
12 |
117.72 |
|
31 |
0.31 |
13 |
127.53 |
|
34.7 |
0.347 |
14 |
137.34 |
|
38.3 |
0.383 |
15 |
147.15 |
|
41.8 |
0.418 |
16 |
156.96 |
|
45.1 |
0.451 |
17 |
166.77 |
|
47.9 |
0.479 |
18 |
176.58 |
|
50.6 |
0.506 |
19 |
186.39 |
In order to calculate the potential energy of the arrow, we used a graph of the force in Newtons vs. the distance the string was pulled down in meters. This gave us a force curve, which we could use to calculate the area underneath, which would give us the potential energy. Here is the graph we came up with.
Data File (*To access the excel spreadsheet click here)
Data File (*Text version of the data)
We found a line of best fit for the scatter plot. We found the line of best fit to be a 3rd degree polynomial, and the equation for that line was: y = 1558.2x3 - 1504.2x2 + 738.16x + 3.1906
We then found the definite integral of that equation, which was: Y = 389.55x4 – 501.4x3 + 369.08x2 + 3.1906x
In order to find the area, we substituted the highest and lowest x values into the integral equation and subtracted them. That is how we arrived at 53.6 J of potential energy.
Kinetic Energy:
In order to calculate the kinetic energy of the arrow, we shot the arrow into a swinging mass and applied the principle of the law of conservation of momentum. With the equation: MaVa = (Mt + Ma)(2 Mt gh)1/2 , where Ma is the mass of the arrow, Va is the velocity of the arrow, Mt is the mass of the target, g is 9.81, and h is the height that the target (swinging mass) moved. We solved this equation, and calculated a range of 38J to 77J for kinetic energy
Analysis of Results:
We discovered that our hypothesis was almost correct, if a little too low. The recurve bow was more efficient than we had expected, with an efficiency ranging from 72% to 140%. This wide range is due to the inaccuracy and error we encountered in measuring the kinetic energy of the arrow. We found the potential energy of the bow when fully drawn to be approximately 53.6 Joules, which we thought to be fairly accurate. As for the kinetic energy, it was more difficult to obtain an accurate estimation. We found that the kinetic energy ranged from 38 to 77 Joules. The potential energy fits nicely in between the range of the kinetic energy, exemplifying that while there was error in finding the kinetic energy, the principle of the conservation of momentum held true. The most likely efficiency for the bow would be somewhere in the lower end of the range, around 75%, due to the energy that is lost to the movement of the bow itself as well as the air friction that the arrow experiences.
Error Analysis:
There were multiple errors that were involved in the process of determining the potential and kinetic energy of a bow and arrow. First, I will discuss the errors in the process of discovering the potential energy. There was error in the structure that we used to hang the bow from because the bow was bending from two points, rather than the natural point in the center of the bow. This may or may not have changed the way the bow flexed; if it did change the flexion, the change would have been minimal. There was also error in measuring the distance the string of the bow traveled each time. It was especially difficult due to the fact that we had to put blocks of wood under the bow in order to raise it up so that all the weights could fit underneath it. We attempted to put the string back in the exact same spot, and I believe we did a fairly decent job, but there could have been some error in the process. There were also obvious errors in measuring the distance between the marks on the string due to the limits in accuracy of measuring devices. Second, I will discuss the error in calculating the kinetic energy. There are only a few ways to calculate the velocity of a high-speed projectile such as an arrow. One is to shoot it straight up and time it until it hits the ground. We did not do this for obvious safety reasons. Another is to use a tachometer, which we do not have. A third way is to shoot the projectile into a swinging mass in order to measure the momentum of the object, from which one can calculate the velocity. We chose the third option, which coincidentally also involves the most error. First, there was error shooting the arrow itself, because it was difficult to pull it back to the same spot each time. Second, there was error in the swinging mass due to the friction in the arms that the mass swung from. We tried to reduce this friction with graphite and washers, but the force still affected the target. Another source of error was the inability to hit the target in the direct center each time. Hitting the mass in the center would have transferred the most energy to the mass so the mass wouldn’t swing slightly sideways. The final source of error was in measuring the curve the mass made, which was very slight. Because it was rather heavy, the swinging mass did not go up very much at all, making it very difficult to obtain an accurate measurement. We measured the height as being .25cm to .5cm, which are very small measurements. There were obvious human limitations in measuring, as well as the limitations of the measuring device itself. In conclusion, despite our error, the law of the conservation of mass held true and we were able to estimate the surprisingly high efficiency of a bow and arrow.
Barton, Linda. History of Bow and Arrow. Articlesbase 2008.
< http://www.articlesbase.com/extreme-sports-articles/history-of-the-bow-and-arrow-403289.html>
"This site was an interesting account of the history revolving around the use of the bow and arrow"
Metcalfe, Luke.
"Encyclopedia > SUVAT Equations." NationMaster. 2003.
<http://www.nationmaster.com/about_us.php>
"This site demonstrated how to use physics equations which allowed the calculations of bow efficiency"
Tunis, Edwin. "Bow and Arrow." Encyclopedia Americana. 2008. Grolier Online. 31
Oct. 2008 <http://ea.grolier.com/cgi-bin/article?assetid=0057040-00>
"This encyclopedic site was a very informative source for the development of the bow and arrow"
Wilson, Tracy V. "How Crossbows
Work." How Stuff Works. 2003.
<http://science.howstuffworks.com/crossbow1.htm>
"This site was an excellent source for understanding how bows and arrows work"
<http://www.huntersfriend.com/bow-review-400-fps-bow/400-fps-compound-bow.htm>