Chloe,+Justin,+Spencer,+and+Andrew+Projectile+Project

Projectile Motion Through a Ring


 * Group Members:** Justin Tosi, Chloe Murtagh, Spencer Edelman, and Andrew Miller


 * Purpose:** To analyze the motion of an object moving in two dimensions under the influence of gravity.

=Calibrating the Launcher:=


 * Introduction:**

In this lab, we are trying to find the correct initial velocity, angle, and range of a launcher that will shoot a projectile through a ring hung from the ceiling. In order to accomplish the aforementioned goals, we will find the initial velocity of the launcher through many trials at different angles, and then average them. From that, we will be able to find out the angle using two of the Big Five Kinematics equations, and with the angle, we will find out how far away that we will have to place the launcher on launch day. The details of our calculations will be discussed later.


 * Materials:**

For this experiment, we used a launcher to shoot a projectile on medium speed at a variety of angles to find its average initial velocity. We used a launcher because it allowed us to set any angle to see how the angle affected initial velocity, and projected an object in 2-D motion with two clear x and y components. The launcher was about 25 cm from the ground; therefore, we stacked textbooks until they were at the same height above the ground, making the total displacement for y 0 (or, as close to 0 as humanly possible). Because of this, we would be able to use the equation:



This equation is much simpler, for it eliminates the use of trigonometric identities and complex algebra (e.g.- the quadratic formula in d = vit + .5at^2), minimizing the likelihood of an algebraic mistake. We tested the launcher to see where the ball would land, and adjusted the position of the textbooks accordingly. We placed the carbon paper on top of the printer paper, which was on top of the textbooks, so that when the ball struck the textbooks, the ball would leave a mark, allowing us to use our tape measure to insure the most accurate measurement of x displacement, thereby insuring the most accurate average initial velocity. Using Microsoft Word, we recorded the displacement for each trial at a certain angle, and then averaged them (the calculation is discussed later). Using Microsoft Excel, we created a formula using the average initial velocity that we found to find the angel at which we had to launch the ball in order for it to reach a certain maximum height (the formula and its variables are discussed in detail later).


 * Procedure To Calibrate the Launcher (Find the initial velocity) and Rationale:**

To enable us to figure out there perfect angle and distance on launch day, we needed to first find the average initial velocity of our projectile out of our launcher. To do this, we planned to record the ranges of different shots at different angles, and to do the appropriate calculations from there. First, we set up our equipment in a long, empty hallway, so our projectile would launch unhindered. For every different angle, we loaded the projectile launcher to medium – a constant in all of our launches – and shot once to find the approximate landing sight. We then placed a sheet of carbon paper with a piece of computer paper under it at the approximate landing site, so that we could record the exact location of the landing for measuring. We placed this paper on a stack of books equal to the height the ball was off the ground at the launch, to make our calculations easier. For calibration, we first shot the launcher multiple times at 45 degrees, but later decided these calculations were far from the norm, and that we would exclude this data from our calibration. We shot the launcher off 4 different times at 30 and 60 degrees and utilized this data. We chose these angles because hypothetically (without any air resistance/error) they should have the same range. We felt using these two runs could give us a relatively accurate average, and by measuring how different each of the initial velocities are from the average, we could find the amount of error/air resistance.


 * Trial Data:**

Table of Trials at Two Different Angles:


 * Angle || X-Displacement (m) || Initial Velocity (m/s) ||
 * 30˚ || 3.815 || 6.57 ||
 * || 3.730 || 6.50 ||
 * || 3.670 || 6.44 ||
 * || 3.820 || 6.58 ||
 * Average || 3.759 || 6.52 ||


 * Angle || X-Displacement (m) || Initial Velocity (m/s) ||
 * 60˚ || 3.749 || 6.51 ||
 * || 3.651 || 6.43 ||
 * || 3.635 || 6.41 ||
 * || 3.695 || 6.47 ||
 * Average || 3.685 || 6.46 ||

The following is an example of how we calculated the initial velocity, using the angle we chose and the range we measured for our first run, at 30 degrees.





This is a sample calculation of what we did in order to find the initial velocity for each trial. After doing this, we averaged the results of all our trials at 30˚ and 60˚ in order to attain an overall average velocity. With these two angles, we hypothesized that the initial velocity would be the most accurate possible because these angles are complements--theoretically, they should have the same range. This means that our initial velocity, which we calculated using the range, will be the most accurate. Also, the difference will show us how much error we have due to air resistance (discussed later).

Average initial velocity: 6.49 m/s

Video Clip of Best Trial (watch the stack of books):

media type="file" key="AM BEST Trial.m4v" width="300" height="300"

This is our best trial, where the angle of the launcher was 60˚, and the displacement was 3.635 meters. It was our best trial because the ball landed dead center on the stack of books, making it extremely easy to measure, and this was probably our most accurate measurement.

media type="file" key="AM Our measurement.m4v" width="300" height="300"

In this video, this is how we measured the range of the launcher. We stacked books in order to make the displacement for the y-component 0 so that our calculations would be simpler. We measured as straight as we could from the place of launch to the dot on the carbon paper, pulling the measuring tape as tight as possible to insure the most accurate measurement.


 * Graph of Initial Velocity vs. Angle:**





The reason that we chose a 30˚ and 60˚ angle is that theoretically, the range of the projectile should be the same at these angles, and the max height would change. However, this would only occur in a vacuum, a place without air resistance. However, by calculating the average velocity at 30˚ and 60˚, we compared the two by graphing the average initial velocity at 30˚ and 60˚. In a vacuum, this graph would be a horizontal line: y = 6.49; however, due to air resistance, our graph has a slope (an extremely small slope) of -0.002 m. Because this slope is so close to 0, we can infer that our average initial velocity is extremely accurate, and is off by only a few millimeters.

We chose not to use this graph or the slope of the line to determine the initial velocity of the launcher at any angle because as we previously explained, there was little difference between the initial velocity of the 30˚ and 60˚ angles. Instead of using the graph, we just averaged the initial velocity of these two angles to find the initial velocity to use in our Excel Spreadsheet.


 * Excel Workbook Link and Analysis: **




 * Calculations for excel spreadsheet (General Calculations):**



Explanation of Calculation:

This is simply the literal calculation--what the excel spreadsheet does. Given this, all we had to do on launch way was input the initial velocity and dy to max height, and it gave us time, the angle (theta), and dx (distance to max height). The specifics of these calculations (the ones that we did on launch day) will be discussed later.

=**Launcher’s Margin of Error Calculation:**=

Overall Average of Percent Difference: .92%
 * Angle || Highest Initial Velocity (m/s) || Lowest Initial Velocity (m/s) || Average Initial Velocity (m/s) || Margin of Error (%) (Highest to Average) || Margin of Error (%) (Lowest to Average) ||
 * 30˚ || 6.58 || 6.44 || 6.52 || .92 || 1.23 ||
 * 60˚ || 6.51 || 6.41 || 6.46 || .77 || .78 ||
 * Average: || N/A || N/A || N/A || .84 || 1.00 ||






 * Explanation of Calculation:**

In order to see how accurate our average velocity was, we calculated percent difference comparing the average velocity to the highest and lowest velocity for each angle. The result is that our average percent difference is .92%, which translates to only a few millimeters, which further implies that our average velocity is very accurate. This difference is a result of air resistance, and in order to account for it on presentation day, we minimized the amount of time that the ball was in the air by launching the ball from the table instead of the ground. = =

=Presentation Day and Analysis:=

=**Calculations for excel spreadsheet:**=



Explanation for Calculations: Both sets of calculations, the literal and launch day calculations, solve for the amount of time it takes from the launch to the maximum height, the x-distance to the maximum height and the angle needed to shoot the ball cleanly through the hoop. Out of these three calculations, we only really needed two, the x-distance to the maximum height and the maximum height itself. These calculations are embedded in the excel spreadsheets. On launch day, all our group needed to put in the spreadsheet was the velocity initial (which we determined earlier) and the y-distance between the launcher and the middle of the ring. When we plugged in those two numbers, we were given the time it takes to get from the initial height to the max height, the x-displacement in that time to the max height, and the angle that we should orient our launcher at to shoot it straight through the hoop. We then placed our launcher 1.66 m away from the ring (the x-distance the spreadsheet gave us) and we positioned our launcher at the angle given.

=**Procedure for Launch Day:**=

**Procedure:**

First, measure from the center of the ring until the ground, to find the maximum height. After subtracting the height of the launcher and table from the height of the ring to the floor, plug in the result into the excel worksheet. (By doing this it will give you the range from the launcher to the ring and the angle, which the launcher should be inclined at.) We found the result to equal 0.166m with an angle of 25.3 ° .Second, put the launcher at the range and angle provided by the excel sheet, and place the ball in the launcher, until medium range. After that, release the launcher. If it does not go in, depending on the height the ball reaches or the direction, adjust the launcher and try again. (Move left if it goes too much to the right, lower the angle if it goes too high, etc.)Then do multiple trials, adjusting in between until the ball goes through the hoop. Because the first trial was just above the launcher, we adjusted the launcher to have a smaller angle, 25 °. The second trial went in but hit the side of the hoop, so we adjusted the launcher to be perfectly centered with the hoop. The third, fourth, fifth trial went perfectly through the center of the hoop. This took 4 minutes and 49 seconds.



= = =**Presentation Day Trials:**= media type="file" key="PRESENTATIONDAYSE!.m4v" width="300" height="300" = = =**Results:**=

We performed 5 trials on Presentation Day which are shown in the following table:


 * Trial || Angle ( q ) in degrees || Maximum Height (m) || Range (Dx from the ring) (m) || Go through hoop? ||
 * Trial 1 || 26.0 || 0.392 || 1.660 || no ||
 * Trial 2 || 25.0 || 0.392 || 1.660 || yes ||
 * Trial 3 || 25.0 || 0.392 || 1.660 || yes ||
 * Trial 4 || 25.0 || 0.392 || 1.660 || yes ||
 * Trial 5 || 25.0 || 0.392 || 1.660 || yes ||

The angle that our excel worksheet gave us was 25.3˚. On our first trial, the ball went just above the target. Because of this, we realized that we made the angle on the launcher closer to 26˚; therefore, we adjusted the angle to be closer to 25.3˚. We had thought that the range to the max height was 1.660 m because we thought that the angle that we used was closer to 25˚; however, the range to max height was slightly higher than 1.660 m, which was shown by the fact that the ball went slightly above the ring. This small changed enabled us to launch the ball through the target; however, in our second trial, the ball hit the right edge of the ring. To adjust for this, we moved our launcher millimeters to the left, and the next three launches were perfect.

= = = = =Overall Conclusion:=

By measuring the range of a launch and the angle, it is possible to solve for the initial velocity. It is possible to create a formula to produce an angle necessary to reach a certain max height. Because of our formula, we were able to launch the ball through the target four out of five times.

In our trials, we found that the initial velocity varied about on average 5.07%, meaning that the velocity would change about a few millimeters per second. Also, air resistance factored into the total range of the ball, which we accounted for on launch day by launching the ball from table level, instead of ground level. Because of this, we minimized the amount of time that the ball was in the air, and therefore minimized the resistance that would have resulted in the ball not going as far on the “x-axis,” and not reaching its appropriate max height.

In the future, we could raise the launcher’s height even above the table by stacking textbooks (like we did when we were trying to calibrate the launcher’s initial velocity) in order to minimize the ball’s time in the air, further decreasing air resistance.

For our initial velocity calculation, we will do more trials in the future at a variety of different angles, find the x-displacement, and then find the initial velocities, and then comprehensively average them, creating a more accurate initial velocity.

For the actual launch, we assumed that the velocity was our average velocity of 6.49 m/s. We assumed that we were aiming for the center of the hoop, and that when we launched the ball, the pulling back of the strings did not change the angle, nor affect the velocity of the ball. We also assumed that there was no air resistance for the purpose of our calculations; therefore, as stated before, there is error.