Phil,+Danielle,+Sam,+Eric


 * Shoot Your Grade**



Vi = initial velocity X (m): range in meters YMax: Maximum height XRange to Max: Horizontal distance to max height

__Introduction__ In this project, we tried to apply our understanding of projectiles. Given a ring, we could easily derive the height of the ring with a measurement tool. Our job was to calculate what distance and at what angle to set a launcher, so that a ball could pass through the ring. Because all projectiles shot at an upward angle have a maximum height, we solved for the variables above using the height of the ring as the maximum height.

We sought to create a graph, in order to predict the motion of our projectile at different ranges and launching angles. We performed launches at several angles, then measured the total horizontal distance traveled at each launch. Using that information, we then calculated the maximum height the projectiles would reach with each angle, and the horizontal x distance the ball would travel until reaching the maximum height. We graphed our results and set a trend line in each graph.

__The Calculations__ All the information derived from our test launches is compiled in the excel document. Because the task requires us to launch a ball that will reach a certain maximum height at a certain range, we first had to determine the initial velocities. The initial velocities vary depending on the launching angle. To find the initial velocities, we used the two equations of the x and y components that often appear in projectile problems: dx = Vixt and dy = Viyt + 1/2at2

For example, in one of our test launches, we launched the ball at a 38 °. Using carbon paper, we found that the total range or horizontal distance of the ball was 5.063m. We also measured that the ball began at a height of 0.26m above the ground, when it is in the launcher.

In order to find the initial velocity, we first solve for the time it takes for the ball to reach the ground, 0.26 meters below its initial y position. By finding the time of the ball in the air, we can then solve for the initial velocity of the ball.

dx = Vixt 5.063 = Vi(cos38 ° )t 5.063 = .788Vit 6.425 = Vit

dy = Viyt + 1/2at2 -0.26 = Vi(sin38 ° )t + ½(-9.8)t2 -0.26 = .616Vit - 4.9t2 -0.26 = .616(6.425) – 4.9t2 t = .928 seconds

6.425 = Vi(.928) Vi = 6.926

After solving for the initial velocity, we have the information to the find the maximum height the ball travels to at 38 °.

Vfy2 = Viy2 + 2ady 0 = (6.926sin38 ° )2 + 2(-9.8)dy 0 = 18.18 -19.6dy dy = 0.927 m

Next, we found out the distance horizontally, the ball traveled until it was at maximum height. We did this by first solving for the time to the maximum height. Then, we plugged the time into the x-equation, dx = Vixt to find the horizontal (x) distance to the maximum height.

dx = Vixt dx = (6.926cos38 ° )t dx = 5.458t

Vfy = Viy + at 0 = (6.926sin38 ° ) + (-9.8)t 0 = 4.27 -9.8t t = 0.4357

dx = 5.458(0.4357) dx = 2.37 m

Using excel, we performed these calculations many times for the many angles and ranges we measured. Above is a sample of the calculations for one of our test launches. In addition, to write the equations in excel, we would condense the calculations into one equation. For examples, the last calculation of horizontal distance to the maximum height is essentially input into excel as follows: = (Vicos38 ° )*((Visin38 ° )/-9.8) = ( Vix) (time to max height)



By calculating the maximum height and horizontal distance to the maximum height for many angles, we can graph trend lines to predict the angle and horizontal distance needed to launch a ball to a certain maximum height at a certain distance away from the launcher.

To increase our precision, we often performed several test launches from the same angle. For one angle, the maximum heights and horizontal distances to maximum height were averaged. Therefore, on the graphs, each x value (launching angle) had only one y value (whether it be the maximum height or the range to maximum height.

Example of average height calculation:



__The Graphs__ All of our graphs are made up of the following information: The above chart is shown to contain Theta as a starting point and the information derived from the respective angles. As stated in the calculations section above, we were able to obtain theta and height at the start. After the launches (the above is calculated using averages for each launch angle), we measured the distance that the ball traveled. That left us with values for acceleration, distance, and theta. From this, we derived the respective numbers for average velocity, Y Max, and range. Throughout our practice runs, as well as the day of testing, we used this information in order to obtain a certain target height.
 * Theta |||| YMax average ||  || Angle || XRangeMax Average ||
 * 38 ||  || 0.9345399 ||   ||   || 38 || 2.3923058 ||
 * 55 ||  || 1.620129931 ||   ||   || 55 || 2.2688583 ||
 * 45 ||  || 1.239590567 ||   ||   || 45 || 2.47918442 ||
 * 20 ||  || 0.302155683 ||   ||   || 20 || 1.66033336 ||
 * 0 ||  ||   ||   ||   || 0 || 0 ||
 * 60 ||  || 1.884570358 ||   ||   || 60 || 2.17611885 ||

We mainly use two of the graphs on test day. The first graph shows the maximum height the ball will reach based on the launching angle (theta). This will allow us to determine what launching angle to set the launcher at. The next graph shows the horizontal distance to the maximum height based on the launching angle.

Because the ball starts 0.26 meters above the ground, we must subtract 0.26 from the height of hoop to the ground. Using the difference, we can determine what angle the launcher must be at to reach the appropriate maximum height. This is achieved by using the graph showing maximum height based on launching angle. Having then determined the necessary launching angle from the prior graph, we can predict the horizontal distance we must place the launcher from the hoop. The angle that satisfies the maximum height (height of the hoop) should also be the angle that causes the maximum height to be reached at the horizontal distance of the hoop from the launcher.

In our test launches, we noticed that the graph of max height based on launching angle was more accurate if we excluded the assumable point of (0,0) (zero launching angle, zero initial velocity). This allowed us to have an R^2 value closer to 1. The graph of range to max height based on launching angle was very close to one with the value of (0,0) incorporated in the best fit line. Therefore, we decided to leave that point in.







We assumed the speed for our launcher was consistent, which it is was not. This had an effect on our launches because it affected the other calculations of max height and range to max height. We assumed that we could average the variables calculated from tests launches of the same angle, because the diameter of the hoop would allow us room for slight errors in the distance we placed the launcher away from the hoop, and the launching angle.
 * Assumptions**

Setup Time (minutes): 3:05 Total Time (minutes): 4:29
 * Presentation Day**


 * Height of Hoop || 1.67m ||
 * Max Height || 1.41m ||
 * Angle || 49 degrees ||
 * Range || 2.41 m ||
 * Trial 1 || To the side ||
 * Trial 2 || Went below ||
 * Trial 3 || Hit the bottom ||
 * Trial 4 || Went through and hit the side ||
 * Trial 5 || Hit the edge ||

As stated above in the graphs section, we mainly used two of our three graphs. Our entire launch plan was based off of the idea that the ball would travel through a certain height at a certain distance away, all at the apex of its trajectory. For the most part, we were successful in calculating the correct measurements. Previously though, we had been experiencing some difficulties and malfunctions with our launcher, causing it to launch in an unexpected manner or break altogether. In our launches our ball exhibited some of these tendencies. This was especially prevalent on the first launch. Directly after the ball was loaded into the launcher, it unexpectedly shot out of the launcher, the excess spin and friction presumably caused from incorrect ball movement inside of the launcher forced it far short and to the side of the target. This launch likely had altered some of the parts of the launcher, such as its original placement and possibly the steadiness of it (it had a tendency to loosen on such launches). Personally, we blame this launch as the cause for any subsequent problems. our theory is partially proven by the fact that the launcher from then on, consistently shot the ball slightly below the hoop. This leads us to believe that the launcher was either moved backwards or the angle was tilted slightly downwards relative to its original placement.

**__Error Analysis__** **Margin of Error** For an angle of 20 degrees we calculated velocities ranging from 6.97 m/s to 7.22 m/s.

6.97 m/s
 * || x || y ||
 * vi || 6.97 X cos20 = 6.55 || 6.97 X sin20 = 2.39 ||
 * a || 0 || -9.8 ||
 * d ||  || -.26* ||
 * t || .58 || .58 ||
 * initial height of launcher = .26 m

dx = vit dx = 6.55(.58) dx = 3.80

vf^2 =vi^2 + 2ad 0 = (2.39)^2 + 2(-9.8)d d = .29 m

7.22 m/s


 * || x || y ||
 * vi || 7.22X cos20 = 6.78 || 7.22 X sin20 = 2.47 ||
 * a || 0 || -9.8 ||
 * d ||  || -.26 ||
 * t || .59 || .59 ||

dx = vit dx = 6.78 (.59) dx =4.00 m

vf^2 =vi^2 + 2ad 0 = (2.47)^2 + 2(-9.8)d d = .31 m

Average Velocity for 20 degree angle: 7.12 m/s Average Range: 3.90 m Average Max-Height: .30 m

__Percent Differences__ For Max-Height smallest velocity to largest (.31 - .29/ .29 +.31) X 100 = 3.33 %

smallest velocity to average (.30-.29 / .29 +.30) X 100 = 1.69 %

largest velocity of average (.31-.30/ .30 +.31) X 100 = 1.64 %

For Range smallest velocity to largest (4.00 - 3.80 / 3.80 +4.00) X 100 = 2.56 %

smallest velocity to average (3.90 - 3.80/ 3.90+ 3.80) X 100 = 1.29%

largest velocity to average (4.00 - 3.90/ 3.90 +4.00) X 100 = 1.27%

These percent differences describe our margin of error, which luckily were not too high. The percent difference of these two extremes velocities to the averages for this angle we tested were not even 2% and the differences of these percent difference (between low and high to the average) were less than one percent. This further describes how our margin of error was relatively low. This makes sense with what we experienced when using our launcher, for we did not experience large differences in ranges of maximum heights when we did several tests of our launcher at a given angle. The margin of error for the range also corresponds with what occurred when we tested our launcher because we had the most trouble (yet still very little) with getting the correct range at a given angle, which would affect our ability to obtain the correct maximum height. We were aware of this upon going into the launch day, however, we never stopped to calculate the exact margin of error in order to compensate for it more exactly on launch day. We did not see it make a big impact on our launches when we tested, so we just knew going into launch day that we might need to adjust the launcher's horizontal distance from the hoop as we began launching.

**Percent Error** Expected Maximum Height: 1.41 m Achieved Max-Height (trial 2): 1.38 m

% Error = |Theoretical - Experimental| / |Theoretical| x 100

% Error = |1.41m - 1.38m| / |1.41m| x 100

% Error = .03 / 1.41 x 100

% Error = 2.13%

This is the percent error of one of our launches on launch day. This error was most likely due (as discussed above under results from launch day) to our initial setup of our launcher. Although we obtained the correct range for the hoop given to be the apex of our launch we most likely did not place it at the correct distance. Also, more importantly we made the mistake of loading our ball at the end of our setup so it was left loaded in the launcher for several minutes before it was our turn to go. This affected most obviously our first launch in which the ball shot out very far to the right of the target. From this point on all of our other launches were off because we tried to compensate for this mistake by dusting our launcher laterally as well as adjusting its horizontal distance from the hoop. These adjustments only caused us further error as shown in how our ball continually hit the edges of the hoop in the other four launches.


 * Video**

We do not have a video of our launch day due to issues with the person who was assigned to record it.

**Conclusion**

Although not likely a topic we will readily apply when throwing a ball or other such things, it may be an applicable topic in terms of the causality that it produces. Many projects consist of not just a single goal, but a multitude of goals. Often times one of these targeted achievements effects the entirety of the others. Therefore, if one small variable is even slightly off, the entire goal can not be achieved. In relation to projectiles, for example, if a ball were to be fired out of a cannon over one mile's distance, and it needed to be placed on a zero degree horizontal angle, even the slightest fraction of a difference change in this angle would cause a significant horizontal variation. The cannonball would likely end up tens or hundreds of meters sideways of the intended target. That can relate to a possible error in our project. We likely did not have the launcher perfectly aligned with the hoop. Since all of our calculations were coordinated so assuming that the hoop was directly lined up, we would not achieve the expected outcome.

Aside from the alignment, there were a myriad of other factors accounting for our failure to attain our goal. The gravitational pressure from the ball being left in the launcher prior to launch likely contributed to a minute difference in distance and height. The swinging of the hoop and the air resistance/movement also were possible contributors to our deficiencies. These components all in compilation with one another start to add up. First the difference is irrelevant, but soon enough there is enough of a contribution to cause us to miss the target, which is what ultimately happened. All in all though, this projected helped us discern the idea of strict attention to detail, especially when dealing with a project like this requiring near perfection. Next time, we will certainly be more careful and considerate of all of the possible variables that can possibly cause even a minor mistake.