Amanda,+Nicole,+and+Roshni+Projectile+Project

= = = Projectile Project - Shoot Your Grade and Sports Broadcast = = =
 * Group Members: ** Amanda Donaldson, Nicole Margulis, and Roshni Khatiwala
 * Period: ** 2
 * Dates Completed: ** Monday, October 25, 2010 - Monday, November 15, 2010
 * Date Due: ** Monday, November 22nd, 2010

__Overall Objective:__ For this two part, projectile motion project, our main goal was to analyze, through our rationales, the motion of an object due to the single force of gravity. We did this in two ways: (1) launching a projectile with a projectile launcher, and (2) examine a sports video to evaluate the trajectory and determine other values of the projectile.

__Part 1: Shoot Your Grade__
__Introduction:__ For this first portion of the project, we needed to figure out, based on trials and mathematical formulas, how to shoot a projectile (the ball) through a ring hanging at an unknown height from the ceiling. In order to successfully do this, we had to decipher the angle, initial velocity, and distance the projectile travels after having been shot from a projectile launcher. After obtaining the data from these trials, we used kinematic equations to create an Excel spreadsheet that would help us find the remaining information needed for success! Then on presentation day, we were able to use this Excel spreadsheet to find the values based on the height in which the ring was hanging.

__Materials:__ Many materials were required to complete our task for this portion. The two most important materials were the projectile launcher and the projectile, a small ball. The projectile launcher is a tool that allows students to launch a projectile at any angle between 0 and 90 degrees and at a variety of ranges (distances) to establish different measurements and values. There are two "screws" allowing users to change the angle in which the projectile launcher is being launched at. A short string hangs down over the protractor tool on the launcher itself showing the angle of elevation that it is set at. Additionally, we laid out a measuring tape flat on the floor to determine the x-distance the ball traveled. Once we had a general idea of where the ball would land at that angle, we put down a piece of carbon paper. During the next few trials of the same angle-range combo, the ball hit the carbon paper every time, and a small black dot appeared on the piece of paper that was taped underneath the carbon paper. This allowed us to measure exactly where the ball hit the ground.



__**Part 1, Section A - Trial Work:**__
__Procedure (method):__ On the first day, we obtained a projectile launcher and a ball and used these two pieces of equipment every day so that we would not have difference in the materials. We chose a place to complete our trials that had little or no obstacles. Using the two "screws" on the side of the projectile launcher we rotated the launcher so that it was at the desired angle. We knew what angle it was at, by reading the number where the string hung down, on the protractor tool on the launcher. After setting the angle, we inserted the ball into the launcher and set it to medium range. Once the angle was set, and the ball was in the launcher, we pulled the yellow string to release the ball. After this first launch, we estimated where it hit the ground. We taped a piece of carbon paper to a regular sheet of white, printer paper and then placed it at the estimated point on the measuring tape. We set the launcher back up again (checking the angle, making sure it was lined up correctly, and inserting the projectile). Now, we were able to really start our calculations! We repeated this process four times per angle to find the distance in which the ball traveled with the launcher set at medium range. After finishing all the trials, we knew the angle, the x-distance (the distance the ball traveled), the initial y-distance (the initial distance which is the distance from the floor to the middle of the launcher where the ball is released from), the acceleration for both x (which for this type of problem, is always 0) and y (which for this type of problem, is always -9.8). Based on the “given” values (given meaning, found by experiments/testing, not derived from mathematical equations and/or formulas), we were able to develop formulas acquired from the kinematic equations previously learned. We entered these equations into an Excel spreadsheet to find our remaining data including the initial velocity (both x and y components), the maximum height, the distance to the max height, and the time to the max height.

__Data Tables:__ The table below represents the trials at varying angles.

This next table below was used to calculate the initial velocity of our launcher: This table shows for the angles we tested, the range, the time it took, initial velocity, maximum height and distance to maximum height. Below is a sample calculation of what was done to receive these values.
 * Trial || Distance (x) (m) || Distance (y) (m) || Angle (°) || Radians || Acceleration (x) || Acceleration (y) || ViT || Time (s) || .5at2 || Vi (x) || Vi (y) || max height (m) || Distance to MH (m) ||
 * 1 || 1.509 || -0.26 || 0 || 0 || 0 || -9.8 || 1.509 || 0.2304 || -0.260 || 6.551 || #DIV/0! || 0.2600 || 0 ||
 * 2 || 1.504 || -0.26 || 0 || 0 || 0 || -9.8 || 1.504 || 0.2304 || -0.260 || 6.529 || #DIV/0! || 0.2600 || 0 ||
 * 3 || 1.566 || -0.26 || 0 || 0 || 0 || -9.8 || 1.566 || 0.2304 || -0.260 || 6.798 || #DIV/0! || 0.2600 || 0 ||
 * 4 || 1.636 || -0.26 || 0 || 0 || 0 || -9.8 || 1.636 || 0.2304 || -0.260 || 7.102 || #DIV/0! || 0.2600 || 0 ||
 * 5 || 3.142 || -0.26 || 15 || 0.2618 || 0 || -9.8 || 3.253 || 0.4742 || -1.102 || 6.860 || 6.860 || 0.4208 || 1.2096 ||
 * 6 || 3.149 || -0.26 || 15 || 0.2618 || 0 || -9.8 || 3.260 || 0.4746 || -1.104 || 6.868 || 6.868 || 0.4212 || 1.2096 ||
 * 7 || 3.149 || -0.26 || 15 || 0.2618 || 0 || -9.8 || 3.260 || 0.4746 || -1.104 || 6.868 || 6.868 || 0.4212 || 1.2096 ||
 * 8 || 3.148 || -0.26 || 15 || 0.2618 || 0 || -9.8 || 3.259 || 0.4745 || -1.103 || 6.867 || 6.867 || 0.4212 || 1.2096 ||
 * 9 || 4.661 || -0.26 || 30 || 0.5236 || 0 || -9.8 || 5.382 || 0.7760 || -2.951 || 6.935 || 6.935 || 0.8735 || 2.0951 ||
 * 10 || 4.660 || -0.26 || 30 || 0.5236 || 0 || -9.8 || 5.381 || 0.7760 || -2.950 || 6.934 || 6.934 || 0.8733 || 2.0951 ||
 * 11 || 4.663 || -0.26 || 30 || 0.5236 || 0 || -9.8 || 5.384 || 0.7762 || -2.952 || 6.937 || 6.937 || 0.8738 || 2.0951 ||
 * 12 || 4.654 || -0.26 || 30 || 0.5236 || 0 || -9.8 || 5.374 || 0.7755 || -2.947 || 6.930 || 6.930 || 0.8725 || 2.0951 ||
 * 13 || 5.036 || -0.26 || 45 || 0.7854 || 0 || -9.8 || 7.122 || 1.0396 || -5.296 || 6.851 || 6.851 || 1.4572 || 2.4192 ||
 * 14 || 5.057 || -0.26 || 45 || 0.7854 || 0 || -9.8 || 7.152 || 1.0417 || -5.317 || 6.866 || 6.866 || 1.4624 || 2.4192 ||
 * 15 || 5.062 || -0.26 || 45 || 0.7854 || 0 || -9.8 || 7.159 || 1.0422 || -5.322 || 6.869 || 6.869 || 1.4637 || 2.4192 ||
 * 16 || 5.063 || -0.26 || 45 || 0.7854 || 0 || -9.8 || 7.160 || 1.0423 || -5.323 || 6.870 || 6.870 || 1.4639 || 2.4192 ||
 * 17 || 4.367 || -0.26 || 60 || 1.0472 || 0 || -9.8 || 8.734 || 1.2636 || -7.824 || 6.912 || 6.912 || 2.0881 || 2.0951 ||
 * 18 || 4.412 || -0.26 || 60 || 1.0472 || 0 || -9.8 || 8.824 || 1.2699 || -7.902 || 6.949 || 6.949 || 2.1076 || 2.0951 ||
 * 19 || 4.443 || -0.26 || 60 || 1.0472 || 0 || -9.8 || 8.886 || 1.2742 || -7.955 || 6.974 || 6.974 || 2.1210 || 2.0951 ||
 * 20 || 4.397 || -0.26 || 60 || 1.0472 || 0 || -9.8 || 8.794 || 1.2678 || -7.876 || 6.936 || 6.936 || 2.1011 || 2.0951 ||
 * ||  ||   ||   ||   ||   ||   ||   ||   ||   || Ave Vi (x) || Ave Vi (y) ||   ||   ||
 * ||  ||   ||   ||   ||   ||   ||   ||   ||   || 6.870 || 6.902 ||   ||   ||
 * ||  ||   ||   ||   ||   ||   ||   ||   ||   || 6.870 || 6.902 ||   ||   ||

//(Angle used 30°=.5236 radians)// //The initial height of the launcher was .26 and it is known that the horizontal acceleration is 0 m/s2 and the vertical acceleration is -9.8 m/s2. Using these constants and the angle in radians, we found what Vit was equal to://



//We then did the following calculations to find the time it took.//

//Using this new information, we found the initial velocity .// //Lastly, we calculated the maximum height of the projectile and then added the height of the launcher to calculate the total maximum height .//

__Graphs:__

The first graph we made was to show what angle was needed given the maximum height. Using the data from our tests, we were able to find the maximum height due to the angle. We created a best fitting line with an r^2 value of .9604 showing that our graph was okay to be used to predict the trend between the two values. All that was needed to be done was to measure the height of the hoop, subtract .26 m (height of the launcher) from it and plug that into the y value of the equation to solve for the angle needed.

The next graph we made was to show the angle (which we solved for in the other graph) in relation to the x-distance to maximum height (which we solved for with the information from the trials). The best fit line for this data was better with .9972 as the r^2 value. By plugging in the angle found in the first graph into the best fit line equation, we were able to solve for the y coordinate of distance to maximum height.

On performance day all that would be needed to be done was to plug the numbers into the equations. We did not use these graphs because the r^2 value for the first graph was not the best. Instead we used the Excel spreadsheets to give us the angle and x-distance automatically. By doing it this way, we had a .9795 average percent error.

__ Margin of Error: __ Sample Calculation: Explanation of Margin of Error Calculation: To see how accurate our average velocity was, we found the percent difference by comparing the highest and lowest velocity at each angle and the average velocity. Our average velocity was off by 5.156% the most, which shows it was accurate enough to use.
 * Angle || Highest Vi (m/s) || lowest Vi (m/s) || Average Vi (m/s) || Margin of Error (%) Highest to Average) || Margin of Error (%) (Lowest to Average) ||
 * 0 || 7.102 || 6.529 || 6.745 || 5.156 || 3.254 ||
 * 15 || 6.868 || 6.860 || 6.866 || 0.029 || 0.087 ||
 * 30 || 6.937 || 6.930 || 6.934 || 0.043 || 0.058 ||
 * 45 || 6.870 || 6.851 || 6.864 || 0.087 || 0.190 ||
 * 60 || 6.974 || 6.912 || 6.943 || 0.445 || 0.447 ||
 * Average: ||  ||   ||   || 1.152 || 0.807 ||

__Part 1, Section B: Presentation Day__
__Calculations for the launch:__ Using the spread sheet shown above, on the performance day all we needed to do was to measure how high the target was from the ground and then it generated these results: We already knew the launcher's height, the acceleration, and initial velocity. By measuring the maximum height, we found the what angle was needed, the x-distance, and the time it would take. We placed the launcher on 5 textbooks to make the launcher have a higher initial height which would make the angle smaller. We wanted a smaller angle because we got better results for smaller angles.
 * Max Height (m) || Acceleration || Average Vi (x) || Average vi (y) || launcher height (m) || added height (m) || (Vi(sinR))^2 || Radians || Degrees || Time to Max Height (s) || Dist to MH (x) (m) ||
 * 0.88 || -9.8 || 6.870 || 6.902 || -0.26 || 0.1875 || 8.477 || 0.438 || 25.075 || 0.2985 || 1.8573 ||

__Presentation Video:__ media type="file" key="edited video presentation NM.mov" width="300" height="300"

__Presentation Result Table:__ This table below indicates our presentation results with the check marks showing the result of each trail.
 * Trial || -Went through Center of Target- || -Hit Target, but Still Went Through- || -Missed Target- ||
 * #1 ||  ||   || [[image:Check_Mark width="22" height="19"]] ||
 * #2 ||  || [[image:Check_Mark width="22" height="19"]] ||   ||
 * #3 ||  || [[image:Check_Mark width="22" height="19"]] ||   ||
 * #4 ||  ||   || [[image:Check_Mark width="22" height="19"]] ||
 * #5 ||  ||   || [[image:Check_Mark width="22" height="19"]] ||

__Result Analysis:__ After completing the experiments and data collection for this project, it was time for the real thing. After missing the ring on the first attempt, we were successful on the next two, but missed again on the last two. Part of the reason as to why we missed on some of the presentation trials, was due to the fact the launcher was not exactly lined up with the ring; therefore, the ball went above or below or to the side of the ring. When the launcher was more precisely lined up, the ball tended to bounce off the edge of the ring. This is a human placement error, rather than a mathematical calculation inaccuracy. We then ensured the launcher was correctly lined up, and changed our angle ever so slightly to ensure that it was close, down to the tenth at 25.075 and launched it again. Although we did miss the target on the first and last two attempts, we did not change the distances. It is likely that after the second and/or third trials, the projectile launcher slightly shifted which resulted in a miss on the last two tries.

On this spreadsheet there are multiple tabs: spreadsheets (trial work and presentation day), graphs, and error work

__Part 1, Section D: Overall Error, Analysis, and Conclusion:__
Prior to presentation day, we had created a formula (during trial work) that would give us the exact degree to angle the projectile in order for the ball to reach a certain maximum height. Therefore, after measuring the maximum height (the distance from the ground to the center of the ring), we were able to plug in the height of the ring and quickly set up our launcher in terms of the x-distance (distance from the ring), the initial height (distance from the ground to center of the launcher), and the angle using the values our Excel formulas gave us from the equations we entered.

One of the decisions we made was to factor in air resistance. Because we do not normally factor that into our calculations, and because we had no way of making it negligible, air resistance was the reason why some of our trial launches and some of our launches on presentation day were slightly different than those predicted by our calculations and formulas. In this lab, there were not many assumptions we could have made, since everything was primarily based on calculations. However, as stated before, we did assume that air resistance was negligible, simply because we did not know how to account for it.

Some other significant assumptions that contributed to our margin of error included velocity miscalculations, as well as incorrect angle speculation. We assumed that our velocity is always constant, when that is clearly not the case. There were different factors affecting it, such as the pull of Amanda's hand on the launching string, the position of the ball inside the launcher itself, etc. We should have found the velocities of all of our trials, average them together, and then use those two newly averaged numbers for our x- and y-velocities. Also, we assumed that the angle was always the same; however, sometimes it would shift as we reset the ball. Therefore, we can logically conclude that the launcher did slightly move, and that the angle did slightly change after each launch.

Our margin of error ended up being a little less than 1% (precisely, 0.9795%). This is shown by the fact that during our different trials (per angle), our resultant range was always around the same number. One of the reasons our 5 launches did not go perfectly in the target was because of the fact that we did not account for our percent error. Perhaps if we had included that in our calculations, the launches would have been more precise. Though the difference was not significant, it did affect our results, becoming the reason why only two balls out of five successfully went through the hoop.

Overall, the objective for this projectile project, to analyze the motion of a projectile, was satisfied. In the Shoot Your Grade section, we physically launched the projectile using a previously created formula that enabled us to find what angle was ideal to ensure that our ball would go into the target. Also, these formulas helped us find varying values to evaluate its trajectory. This project proved to us that just calculations and formulas are not the only things that can lead to errors. We must be certain all the time that all "human factors" (such as the exact angle, the distances, and the positioning - lining up) are accounted for. If we were to do this lab again, we would likely need to think more about air resistance, the assumptions we have made, and our percent error.

**__Part 2: Sports Broadcast__**
__Introduction:__ For the second part of this projectile project, we had to create a broadcast to illustrate the projectile motion in sports. After creating the video of the sport, we had to "broadcast" the projectile motion shown and the physics behind the demonstration. Finally, based on our knowledge of the kinematic equations, we had to decipher estimates for the values of initial velocity, the maximum range, and the maximum height.

__Calculations:__

__Analysis and Error:__ After completing our project, we found a serious mistake involving all of our calculations. We accidentally miscalculated the vertical y-distance (distance between where Roshni released the ball and Amanda's racket made contact with the ball). Roshni released the ball 1.29 meters above the ground and Amanda hit the ball 1.6256 meters above the ground, thus the y-distance should be 0.3356 meters and not 0.94 meters. Also, after using logical judgement, we believe that our x-distance (horizontal distance between Amanda and Roshni) was off. It turns out, that this distance should be 2.14 meters instead of 3.7 meters. Due to these human errors, our new calculations are below. How did we find the angle? By drawing the trajectory of the projectile, we were able to measure the angle to be 52. By using the height of our reference height, the net, we were able to find many values to help with the above calculations.