DeGisi,+Dember,+Rabin,+Samani,+Projectile

= Projectile Projects = **Spreadsheet Link**: (Please open in Excel)
 * Group Members: ** Rebecca Rabin, Ross Dember, Nikki DeGisi, Tyler Samani
 * Period 4 **
 * Dates Completed: October 25th - November 15th, 2010 **
 * Date Due: ** November 22nd, 2010

** Part A: Shoot Your Grade **
__** Theory and Rationale: **__  Objective: Analyze the motion of an object moving in two-dimensions under the influence of gravity.  The projectile project provided us with task to calculate the angle necessary to shoot a ball through a ring hanging at any given height from the ceiling. To successfully complete this objective, we thoroughly studied the trajectory of a projectile though many tests and trails. To begin, we split up our projectile into two components, the x and y, with which we would be able to make further calculations. Each component would have a specific initial velocity, acceleration, and distance. Both components would have equivalent times due to the fact that both the x and y components are moving simultaneously thus creating the projectile. At this time we know that the acceleration of the x-component will be 0 because following the launch there is no horizontal force acting upon the ball. The acceleration for the y-component is -9.8 m/s, due to the force acted upon by gravity. Air resistance could have a small effect on the acceleration; however, due to the diminutive size of the ball, it is unlikely that this will be a huge factor and therefore was ignored. After performing our trial launches, for each specific angle launched at, the distance it travels becomes the x-component distance. Recording all results into Excel, we were able to use this program to calculate the relationship between the angle at which the launcher is placed at and the remaining variables of the projectile's two components. This being said, come presentation day we were quickly able to input our given height and calculate the angle and the x-distance at which to set our launcher.

__** Methods and Materials: **__ As there can only be a certain amount of controlled variables, it was integral to consider the certain measurements that could be set before the experiment began. Since the angle could be determined before the launch, it served as the starting point of the process, and books and folders were used to equal the launcher's vertical height in order to be able to use the equation, which was the simpler choice. Also, paper and carbon paper were used to mark the spot the ball landed, and they were placed over the stack of books and folders.

The launcher would go off at a variety of angles, and when the ball hit the "island", the horizontal distance was recorded using a tape measure. This process would be repeated at the same angle multiple times to ensure that the launcher was accurate. Although the tested angles would not be used for the presentation, they were still necessary, as they could provide an accurate relationship between the variables that we could use to create graphs. Those graphs (below) would then be used for the performance.

__** Observations and Date from Calibration: **__

The first chart, columns A through H, shows our known information when solving for initial velocity. We set the launcher to a certain degree angle, and in Excel, converted the degrees to radians for use later. We launched the ball and measured the x-distance of where it left a mark on the carbon paper. We knew the y-distance was zero because we set the carbon paper on a height of .26 cm so that we would have a ground to ground launch. The x-acceleration is always zero and y-acceleration of a projectile is -9.8 m/s due to acceleration of gravity. The x-distance and angle degree were used in the following equation to solve for the initial velocity of each angle.

The initial velocities that were solved for both the x and y components are seen in the second chart. Using the x-distance from the first chart divided by the initial x-velocity from the second chart, we were also able to solve for time in the second chart.

The final chart shows our calculations for trials we conducted before performance day. We plugged in the max height of y and through physics calculations explained below, solved for the angle and x-distance of the launcher.

__** Observations and Date from Performance: **__

media type="file" key="Physics Launch.m4v" width="300" height="300" ** * ** The bold numbers are the measurements that were calculated on performance day. Equations in these rows obtained from the following graphs calculated the angle and x-distance for the launcher to be set at. All distances are in meters in this chart.

__ ** Graphs: ** __ The first graph we needed was one made in order to show us what angle to set the launcher at depending on the max height we were given. To create the graph, we first used the initial velocities we had solved for and plugged them into the equation Vf^2 = Vi^2 + 2ad in order to solve for the max height at each initial velocity, knowing the Vf at max height is 0m/s and acceleration is -9.8m/s. Those max heights were the x-coordinates on the graph. The y-coordinates for each point were the angles (in radians) that we had tested the launcher at. Once these points were graphed, we created a best fit line. The r^2 value of the line was .99, indicating that our graph would be able to predict the trend between max height and angle. By using the equation of the best fit line, we can plug in any x-value, the max height, to solve for the matching y-value, the angle in radians. (All angles from the graph were solved in radians, and subsequently converted into degrees in the spreadsheet.) On performance day, we measured the height of the hoop and subtracted .26 cm, the height of our launcher. By plugging that height into the equation from this graph, we were able to solve for the angle to set the launcher at.



The next graph we created was one to show the angle we solved for in the first graph in relation to the x-distance we needed to place the launcher away from the hoop. To create this graph, we used the angles we had tested at as the x-coordinates. We measured the x-distance that the ball traveled during each trial and used that number as the y-coordinate. Once plotted, we created a best fit line for the data series. The r^2 value was once again .99, a very strong number. By using the equation of the best fit line, we were able to plug in the angle we solved for in the first graph as x to solve for the x-distance, the y-coordinate. On performance day, we plugged the angle into the equation and found the distance away from the hoop that we needed to place the launcher at in order for the ball to go through at max height.

__** Excel Spreadsheets: **__ Since the vertical height was set at zero, the equation could be utilized. As the horizontal distance was recorded, the angle was predetermined and g was a constant, so solving for v i was possible by changing the equation to read. After doing this, it was clear to see that all the attempts had similar velocities, and it could be assumed that the relationship between the variables would be similar.

For the max height at // y //, it was required to use the equation v f ﻿2 =v¡ 2 +2ad, as it is known that the final velocity is equal to 0 and the initial velocity is only the // y // component of the overall velocity, as this is referring to only the vertical components. Therefore, the height can be determined. Since the // x // distance is also necessary for the presentation, it is important to know time, as that is the only identical variable in finding the distance of both the // x // and // y // components. By using the equation v f= v ** i + at, time can be determined, and plugged into the equation d x = v ix t in order to find the // x // distance at max height. **

Now that every necessary variable can be solved, it may seem that the work is done, yet that is not so. Since all the variables that were solved for were based on the initial angle, instead of the max height, which is the only given during the presentation, a precise estimate of the initial angle and horizontal distance cannot be made. Thus, the graphs above become necessary. The first relationship needed is between max vertical height to angle, so after graphing it and making a trend line, the equation of that trend line serves as the relationship between the two variables, and the angle can be found. Now that the angle is known, the second step is finding the relationship of the angle to the // x // distance at max height, and repeating the process. Then, plugging in the max height will give the angle, which will in turn give the // x // distance, and the device can be properly set up.

__** Physics Calculations: **__ Sample Spreadsheet Calculation: R=2.42 m Angle=15º (.26 radians, needed for excel) v ﻿i = 6.89 m/s v f= v ** i + at ** 0=6.89sin(15)-9.8t t=.18 sec d y =v iy t + .5*a*t 2 d=6.89sin(15)*.18-4.9*.18 2 d y =.16 m (Max y) ** d  x =   v ix t  ** ** d x = 6.89cos(15)*.18 ** ** d x = 1.21 m (x at max y) ** Presentation Day: ** y ** ** ﻿max ** ** = .86﻿9 m ** y ﻿max = θ(radians)*.4476+.2245 .869= θ(radians)*.4476+.2245 θ(radians)=.613 θ(radians)*(180/π)= θ(degrees) .613*(180/π)=35.15º x max = -3.6483*( θ 2 )+5.8572* θ x max = -3.6483*(.613 2 )+5.8572*.613 ** x ** ** max ** ** = 2.22 ** ** m **

__** Error Analysis: **__ Margin of Error: As we were solving for velocity, that was the source of error, which led to the inexactness of the results. There were plenty of sources of error, such as the spring becoming looser as more trials were done, differing forces on the spring when set at different angles, and difficulty in getting the exact measurements. Using the margin of error equation, the difference of the highest and lowest results are taken, and divided by their average to see the range of fluctuation of the velocities. As shown here, the margin of error was only 3.59%, so although the results did not come out uniformly, the error was minimal.

__** Conclusions: **__ Between the setup and the five shots we took, we finished the project within a time of five minutes and three seconds (5:03). Out of the five shots we were given, we made one shot in perfectly. The others only hit the side then went through. So 100% of our shots were on target, but only 20% went through as we had hoped. This is because of the many possible areas where errors could have been made. To begin with, the tape measures did not start exactly at 0 centimeters. We did not realize this until after measurements had been taken in the trials, which possibly made some of the measurements inaccurate from the very beginning, in turn affecting the entire project. We could have used the meter sticks instead because they gave much more accurate results. Next, instead of solving for the initial velocity for our launch angle to use in the equation on performance day, we averaged all the initial velocities of the trial angles. If we had actually solved for the initial velocity for each angle, our shots could have been more precise. Another source for error was the accuracy of the launcher. The launcher, if working correctly, should launch the ball straight out of the barrel of the launcher, but our launcher seemed to curve to the right after every shot. There is not much that can be done about that except for moving the launcher to the left to make up for the curve, which we did. Lastly, after every shot, the launcher angle would drop many degrees and the entire launcher would hop slightly due to the power. To avoid this, we tightened the bolts on the launcher and held the launcher down before every shot. Despite these precautions, the launcher angle and position would change slightly.