Anthony,+Jill,+Allison,+Aaron+Projectile+Project

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= ** __Projectile Project __ ** = ==== Aaron Chang, Anthony Iannetta, Allison Irwin, & Jillian Laub Period 2 11/22/10 ====

For this project, we needed to figure out a way to be able to launch a ball into a hoop at a random height. We needed to know for certain that the angle and x distance we chose will get the ball through the hoop. To do this, we needed to calibrate our launcher to figure out the best angle measurements and launcher location that would get our ball through the hoop. First, we needed to figure out the initial velocity of our launcher. Without this, we wouldn't be able to do any of our calculations to use in our final spreadsheet. This spreadsheet will consist of different manipulated kinematic equations that are already put in, so that on presentation day if we put in the launcher altitude and Y distance, it would tell us the angle that we needed to set the launcher at in order to get the ball through the hoop at maximum height.

__ **Calibration:** __

The most important part of this project was the calibration of our launcher. To complete this, we used a launcher, a ball, a loading stick, a tape measure, a meter stick, masking tape, computer paper, carbon paper, textbooks, and a computer with Microsoft Excel installed. By using the same launcher and ball each time, we were able to eliminate any discrepancies from the different launchers and balls. The loading stick was used to push the ball into the launcher (set at medium range). The masking tape was used to tape the carbon paper facing down onto the white computer paper, which was then taped onto the floor approximately where the ball was landing for each angle. We did each trial several times and took the average measurement of the cluster of dots on the paper. We used the tape measure and meter stick to measure the height of the mouth of the launcher from the floor and to measure the distance the ball traveled from the mouth of the launcher to the cluster of marks on the paper.
 * Materials and Methods:**

There were many steps that had to be completed in order to complete the calibration process. First, we picked a launcher and labeled our names on it in order to ensure that we had the same launcher everyday so that our results would be accurate. The launcher was set at medium range and we fired it at a 0 ° angle for our first trial. We taped the launcher to the floor so it would not move after each trial. After that, we taped a piece of computer paper on the floor with a piece of carbon paper taped facing down on the computer paper approximately where the ball was landing each time for the random angle we picked. Every time the ball hit the carbon paper, it would leave a small round mark on the white paper underneath. Before we launched at each angle, we would measure the distance from the floor to the crosshair located on the side where the mouth of the launcher was. After we finished launching at a certain angle, we would measure the distance from each mark made to the mouth of the launcher. We then took the average of the cluster of marks to find the distance the ball was launched. We launched the ball at different angles with at least 5 trials for each angle.
 * Procedure:**

After launching at several angles, we used Microsoft Excel on our laptop to record the data from our trials. We then measured the vertical and horizontal displacement of the ball. We knew that horizontal acceleration was 0 m/s^2 and that vertical acceleration was -9.8 m/s^2. We used the measurements and previous assumptions on acceleration to find the initial velocity of our launcher. (See Data Collection & Calculations)

We launched the ball at a variety of different angles and found the maximum height and the range for each angle. As the angle increased, the maximum height also increased with it. As the angle increased, the range increased also until 40 °, which had the greatest range. After 40 °, the range began to decrease as the angle got larger.
 * Observations:**

__ **Performance Day:** __

**Performance Day Results:** Video clip of our best trial: media type="file" key="Best AI.m4v" width="300" height="300" align="center"

**Performance Day Calculations** Using the spreadsheet from above, on Performance Day, we simply needed to measure the Altitude of the Launcher (how high the base of the launcher was from the ground) and how high the target was from the ground. It then generated the following results: * Green boxes represent quantities that were previously calculated or given. * Red boxes represent quantities that needed to be measured. * Blue boxes represent quantities that would be automatically calculated by the spread sheet. From this spreadsheet we were informed that the launcher needed to be 0.99m away from the target and set to an angle of 27.26 degrees. However, because the minimum horizontal distance that was allowed was 1.00m, we simply moved the launcher back 0.01m and made the angle slightly more towards 28 degrees.
 * __ Data Collection & Calculations: __**

We used the Excel Spreadsheet (located below) to calculate the initial velocity of our launcher. Following, is an explanation of how we derived the calculations used in the spreadsheet. We adjusted the launcher to the desired angle measure, and found the vertical displacement of the projectile (-0.262m). We also measured the horizontal displacement of the projectile. It is known that horizontal acceleration is 0 m/s2, and that vertical acceleration is -9.8 m/s2. Using these measures, and the angle measure converted in radians, we used the following equation to solve for ViT: (Example Calculation: 60°) We then used the following steps to calculate time ( t ) : (Example: 60°)  We then used the following steps to calculate the initial velocity ( vi ) of the launcher: (Example: 60°)

We calculated the initial velocity for a variety of angles (0, 10, 20, 30, 40, 50, 60, 70 80). We then averaged these results to reach the initial velocity that we used in further calculations. We then used these calculations to generate the following two graphs: We calculated the maximum height of the projectile from the launch point, and then added that to the height of the launcher in order to get the maximum height of the projectile from the base of the the launcher: (Example: 60°)

__** Results: **__
We performed practice calculations in the following Excel Spreadsheet in order to simulate possible scenarios for Performance Day. This spread sheet, after inserting the height of the launcher (Launcher Altitude) and the height of the target (Y Distance), will generate the desired angle and horizontal distance from the target that are needed in order to shoot the projectile through the target at the maximum height of its trajectory. We first measured the vertical distance from the ground to the center of the hoop ( Y ). From this number we subtracted the height from the base of the launcher to the initial launch height (0.262m) and the vertical altitude of the launcher ( A ). This number represents the vertical displacement ( delta D ) of the projectile from its initial height to its maximum height. (Example Calculation: 1st Row of Chart) After plugging in these two variables( Y & A ), our spreadsheet automatically generated the angle in radians that we should set our launcher to. It then converted that numbers into degrees so we could set the launcher to the appropriate angle. We also needed to know exactly how far away from the hoop our launcher needed to be. So using the following calculations, our spreadsheet also calculated this variable ( delta D ): 1. We first needed to calculate the time (t) it took for the projectile to reach maximum height: 2. We were then able to solve for what the horizontal distance should be from the launcher to the target ( delta D ):
 * Practice Scenarios**

__** Error Analysis: **__


 * Margin of Error:**



Sample Calculation (10 degree angle/Highest to Average Margin of Error):

By doing all of these calculations we can see that the margin of error is rather low. This does not mean that we can ignore this fact though. So when we plug in our values for the presentations we have to know that what our excel sheet gives us is not 100%. This is seen in our presentation because our 1st trial did not go through the hoop. Even the smallest margin of error can affect the results. This should affect the way we went about the presentation because by knowing that we have some sort of margin of error, we could have tested the launcher on presentation day and view our margin of error on that specific day.
 * Explanation:**

We set up our launcher on presentation day and it was supposed to reach a height of 0.52, but it reached a height of 0.58 and went over the hoop. This is the percent error of this trial:
 * Percent Error:**

The reasons for error are explained in the conclusion. This error affected our presentation by making us miss 2 out of the 5 times we shot the ball.

Using the two graphs above, we were able to use the R^2 valued in order to calculate percent error. If the spreadsheet was 100% accurate in predicting the maximum height of the projectile due to the angle and the the range of the projectile due to the angle, then the R^2 value should have been 1.00. The following calculations reveal that our spreadsheet had a 1.49% error when calculating the maximum height of the projectile due to a given angle.

The following calculations reveal that our spreadsheet had a 2.00% error when calculating the range of the projectile due to a given angle. These percent errors could possibly have had a significant impact on our results. For example, a 2.00% difference in the range (that was supposed to be exactly 1.00m) means that the projectile falls 2 cm. before or after reaching the target, causing it to hit the edge or miss entirely. This margin of error could have been the reason that we missed the target twice, and hit the edge twice also. __** Conclusion **__

On presentation day, 3 out of the 5 trials were successful (1 actually going through the center, while 2 hit the side and went in). The first trial was the biggest surprise to us because it showed that our calculations were off. To get the ball to go through on the other trials we had to adjust the angle of the launcher. Here is a video showing the 5 trials, and it also shows how we had to adjust the angle of the launcher to achieve our objectives.

media type="file" key="AI, AI, AC, JL.mov" width="270" height="270"

On launch day, something that we could have done to improve our results would be to test out the launcher and get some readings before actually doing the presentation. As we have figured out during this project, the spring can give us different results (by a little) each day. Air resistance can also affect this because we have learned to not take air resistance into consideration. While that seems not to affect the data, even a small change can make a difference. So by testing out the launcher on the day of the presentation we would be able to tell how the launcher is for that day, and adjust our calculations. Since we did not do that, we completely relied on our calculations and the ball went over the hoop.

During our calibrating, we should have tried different methods. After we figured out how to succeed in our objective we only tried one method. This method was to test each angle and make sure that the actual results matched our theoretical results. We felt that we did not test out the many situations that might be available to us. This includes but is not limited to: using textbooks underneath the launcher, getting the ball through the hoop on its way down, and analyzing the error of the launcher closely. By examining our options we could have chosen the best way possible to be able to achieve our objectives.

__**Spread Sheet Attachments:**__
 * 3 Calculation Spread Sheets
 * 2 Graphs
 * 1 Result Chart