Richie,+Jae,+Chris+Projectile

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__//"Shoot Your Grade"//__
 * Group Members:** Richie, Jae, Chris
 * Date Due:** Monday, Nov. 22nd


 * Introduction **

In this lab we were given a spring-operated launcher with three settings: low range, medium range, and high range. As a class we were instructed to use the medium range setting, however our launcher was very weak (probably due to overuse) and thus our trials were performed on the high range setting. Our job was to attempt to find the initial velocity of the projectile (in this case a hard yellow ball), so that we could eventually set up our launcher at any reasonable distance and shoot through a hoop with a diameter of 7cm suspended at a given height from the ceiling. This would be done through a series of trials by finding the range (x-component) of the projectile at given angles and using the projectile motion equations to solve for initial velocity. Our given information was the y distance (height of the launcher), as well as the angle of launch which was found using a weighted string attached to the launcher. Acceleration we know is 0 in the x direction and -9.8 m/s2 in the y direction.

** Procedure **
The first step was to conduct the trials, which would test for range. This would give us the x value needed to solve the projectile motion equations for initial velocity. The first step was to set the launcher up at a given angle. We began with 45º. We took multiple shots, placing a sheet of carbon paper over a sheet of white computer paper under where the projectile was landing. This served as an efficient way to mark the exact landing locations of the shots, which we then measured with a tape measure and averaged together to get one x-value. Before averaging the points together, we eliminated any outliers that appeared due to the sporadic nature of the launcher. These tests were done at a variety of angles including both 15º and 60º angles. The x-value was then plugged into the projectile motion equations to find the initial launch velocity. We also conducted the trials in another fashion so that the projectile would land on a stack of textbooks at the same height as its launch height. This would allow us to use the x-value in the max range equation because the y-velocity component would be unchanged. This is the max range equation:

**Materials**
Obviously two of the most important materials were the launcher and projectile. The black plunger was used to push back the spring inside the launcher, which was locked into place by an internal mechanism until ready to fire. The launcher had three settings, low, medium and high, which could be adjusted by pushing the spring further inside the launcher. During the trials, we needed to use carbon marking paper over white computer paper to record the landing locations of the projectile. The carbon paper left a small black mark on the white paper, which we could measure with both meter sticks and tape measures. Throughout the lab, we used metric measurements because gravity was already given to us in m/s2. When performing the ground-to-ground max range trial, we needed to use a combination of textbooks and notebooks to reach the desired landing zone height, the same as the launcher, 25cm. We stacked the textbooks and notebooks, and placed the two sheets of paper on top. All of the information we found during the trials was recorded on an excel spreadsheet on our Mac computers. Lastly, to prepare for launch day we needed to run a few practice runs shooting at a hoop. So, we hung a roll of tape from the ceiling by tying two pieces of string to two ceiling clamps and the other ends to the tape roll.

** Data: **
Sample Tables of ranges at two different angles


 * Angle || Range (m) ||
 * 15º || 2.34 ||
 * || 2.36 ||
 * || 2.32 ||
 * || 2.36 ||
 * || 2.37 ||
 * Average || 2.35 ||


 * Angle || Range (m) ||
 * 60º || 3.81 ||
 * || 3.82 ||
 * || 3.81 ||
 * || 3.79 ||
 * || 3.79 ||
 * Average || 3.80 ||

[[image:excelchart-jae.png width="800" height="94"]]
These were all the calculations that we needed to do to figure out how far we needed to put our launcher away from the target on performance day. Initial velocity was found using the average range and the max range equation. Sample Calculation for finding initial velocity:

The y component of the velocity was then found using the initial velocity. Sample Calculation for finding y component:

Finally, the max height was found by solving for time and using that value to find the y component of velocity. Sample Calculation for finding max height: After max height was found we could then make our two graphs that would produce the equations that we would use on performance day.

** Graph of X-distance (Range) vs. Angle: **


As you can see the R2 value on this graph is not as high as we would have liked it to be. Ideally R2 would be equal to 1. Although it is bad, we were able to work around it. We knew that the R2 value told us how consistent our ratios between angle and range were, so because it was low did not mean that the launcher would not launch the ball around 3.8 meters at 60º. It merely meant that it would be difficult for us to guess what range we would need to put the launcher at to get the ball through the hoop if the angle produced from the Max Height vs. Angle graph was not one of the angles we tested. When first looking at the time low R2 value we decided to do some calculations to see how different the highest and lowest ranges were form the average range at any angle. To do this we used the max range equation and our max, minimum, and average velocities that were found. Sample calculation for finding Range Sample Calculations As you can see the most the ranges in High compared to Average were under 10%. The ranges differed greatly in the Low compared to Average because our low velocity was a lot lower then many of the other numbers, leading us to believe it was an outlier and something had gone wrong when testing. We had to keep the difference in mind, but we did not need to fret over it.
 * Angle || Highest Range || Lowest Range || Average Range || Margin of Error: Highest to Average (%) || Margin of Error: Lowest to Average (%) ||
 * 15º || 2.35 || 1.70 || 2.12 || 10.29 || 21.99 ||
 * 60º || 4.07 || 2.94 || 3.68 || 7.18 || 22.36 ||
 * Average || N/A || N/A || N/A || 8.74 || 22.18 ||

To figure out the angle and range we needed to set on launch day we first needed to measure the length from the center of the hoop to the floor. We would then plug that number into the Max Height vs Angle function equation as Y and solve for X (angle) with the quadratic formula. When we found that X (angle) we plugged it in as X in the Range vs. Angle function equation and solved for Y (max range). We then divided this max range by 2 because we knew the max height would happen half way though the max range. When testing this approach on our launcher we figured out that each angle needed to be lowered 10º for the ball to go through the hoop. This worked for every angle and y height that we tested. At first we thought that it was due to the low R2 value on our Range vs. Angle graph but then we ruled that out due to the fact that the range was not a problem. We then figured out that it was due to the fact that when measuring the height of the hoop, we were measuring to the ground and not to the height of the launcher. Because we did not notice this until the day before testing we decided to stick with lowering the launcher 10º as we knew it worked well with each angle, and we got our ball through the hoop 4 times in a row.



Launcher's Margin of Error Calculations:


 * Angle || Highest Range || Lowest Range || Average Range ||< Margin of Error: Highest to Average(%) || Margin of Error: Lowest to Average(%) ||
 * 15º || 2.37 || 2.34 || 2.35 ||< 0.85 || 0.45 ||
 * 60º || 3.82 || 3.79 || 3.80 ||< 0.52 || 0.26 ||
 * Average || N/A || N/A || N/A ||< 0.69 || 0.36 ||

Sample Calcutions As you can see the margins of error in the launcher's ability to get the ball to go a certain distance is very low. This was very good for our group. Due to the fact that the error was less then one percent, we could be confident that if we put the launcher at a certain spot and at a certain angle we could have the ball travel the distance we needed it to travel. However, as seen above we did have a very bad R2 value for our Range vs. Angle graph. This does not mean that the launcher was inconsistent in launching the ball a certain range at a given angle but that the angle to range ratio was not consistent between the tested angles. This meant that it would be harder to guess what angle would be needed if we needed to go a certain distance. We compensated for that as seen above.



Overall Assumptions: We assumed that the path of the ball would be along the same line if launched at the same angle each time. By this, we mean if the angle was 60º and launched 5 times in a row without moving the launcher to the left or right, then the ball would hit the same spot at the same height on each shot. (Essentially, it would go through the center of the hoop every time and not off to the left or right.) The reasoning behind this assumption was that the ball small so it minimized air resistance, there was no wind or outside force other than gravity acting on the ball, and the ball was not launched with a curve or spin. These three components led us to believe that the left and right difference of the ball when it went through the hoop would be very minimal.

__ Trial Table __
 * Results of Presentation Day. **
 * Trial Number || 1 || 2 || 3 || 4 || 5 ||
 * = Y distance ||= 1.4m ||= 1.4m ||= 1.4m || 1.4 m || 1.4m ||
 * = Angle ||= 45º ||= 45.4º ||= 45.4º || 45.4º || 45.4º ||
 * = X distance ||= 1.8m ||= 1.8m ||= 1.8m || 1.8m || 1.8m ||
 * = Did it go through? ||= no ||= yes ||= yes || yes || yes ||

media type="file" key="pres day2.m4v" width="300" height="300"

Analysis:

First we measured the distance from the middle of the hoop to the floor which was 1.4 meters. . With the measurement that we we found we used the height of the hoop in the equation from the fist graph as our y-distance and calculated x which we got to be 55.4º. Then we used this angle measurement in the second graph to figure out the x- distance which was 1.8 meters. We then placed the launcher 1.8 meters away from the hoop. Due to our previous knowledge of our spreadsheet, we knew that the angle that we got was exactly 10 degrees more than what the correct angle should be, so we adjusted it accordingly. When we launched the first ball it missed the hoop and hit the bottom. We figured that this was due to the fact that we could not put the angle at exactly 45.4º because the angles went by 1º intervals, so we ever so slightly put the angle up and shot again. When we shot we got the ball through the middle of the hoop. We did not change our angle at all and shot through the middle of the hoop 3 more times in a row.

As previously mentioned we only got the ball through the hoop 4 out of 5 times. On the first attempt the ball went under the hoop but hit the hoop. Due to the miss we adjusted the angle slightly from 45º to 45.4º. After the angle adjustment, on the second shot the ball went through the hoop perfectly. After the perfect shot we kept the same angle and kept our feet on the bottom of the launcher so that we did not accidentally move it, and shot it again. As expected the ball went through the hoop perfectly again. We repeated this process another two times and all the shots went through the middle of the hoop. To improve this performance we could have measured the angle much more precisely to ensure that the ball would go through the hoop.

Conclusion
Our efforts in this lab were largely very successful. On launch day we were able to shoot through the hoop 4 out 5 times with only a slight miss on the first shot, which was easily compensated for. One of the most glaringly obvious mistakes we made during this lab was that when we got the height (y-value) from the Excel spreadsheet, we measured from the hoop to the floor, instead of from the hoop to the height of the launcher. This caused the angle to be higher than necessary. We ran multiple trials with this layout and each time we shot over our target. We figured out that all we needed to do was lower the launch angle of our launcher by 10º, and each trial after that resulted in a perfect shot. Overall it turned out not to be such a bad mistake. Another factor for error in this lab was found when we were conducting the original trials to find the range vs. angle. When we averaged the numbers together, we realized that something was wrong: we had a 22% error when comparing the lowest range recorded to the average. At first we were nervous that this would greatly influence our results, but then we realized that it was probably just the result of an outlier and that even though the computer said 22% error, we knew there was nothing wrong with the launcher itself. The outlier could have come about from a number of things, largely as the result of human error. If when we were shooting the base was not anchored, the launcher could move, resulting in inconsistent shots. Also, we quickly noticed that the spring inside the launcher needed to be loosened before getting consistent results, requiring us to warm it up with a few shots. Another way human error could factor in to these test shots is that if the screw that held the launcher in place was not adequately tightened prior to shooting, the launcher had a tendency to move, changing the launch angle. If this went unnoticed, the data would be corrupted. All of these factors play into the large margin of error we observed in both the comparisons of highest range to average and lowest to average. In the future, if we were to perform these trials again, all of these mistakes would be easy to correct by slowing down and checking for correctness in angle and launcher position before shooting. It turns out that in this example, quality of the shot was much more important than quantity, especially if you had to sacrifice accuracy to get more results.