KrazitHuddlestonMcCullough

__**Introduction**__

Goal: The objective of this lab was to determine a method for getting a ball through a suspended hoop using a launcher with an unknown initial velocity.

Rationale: Whereas many groups tested their launchers at multiple angles in order to determine an average initial velocity for their launcher, we found that this method would increase the source of error within the lab. Though ideally the initial velocity of the launcher would be the same at all angles, gravity can change the launcher’s initial velocity. When shot 0º gravity plays a minimal part in affecting the power of the spring within the launcher. When shot at a greater angle however, the spring must fight against gravity in order to launch the ball upwards, therefore the initial velocity is not the same at all angles. To avoid this problem, we decided to use only one angle, which allowed us to calculate a consistent initial velocity. Since the angle and initial velocity are now set constants, after finding the height of the hoop, we would only need to calculate how high above the ground we would have to place the launcher, as well as how far away from the target. In order to further reduce error, we decided to always place the launcher 1m away from the hoop, therefore leaving only the height at which to place the launcher to calculate. By keeping the distance from the hoop minimal, we reduced the amount of time that air resistance had to act upon the projectile. This closer distance also reduced the effect of lateral change. If the launcher were to have been placed further away, it would be more difficult to aim the launcher directly at the center of the hoop.

__**Methods and Materials**__

In order to simplify our calculations, we decided to create a platform to shoot at that was equal in height to the launcher. This created a ground to ground launch, which allows us to use the simple equation R = Vi^2(sin(2theta)) / gravity. To create this platform, we measured the height that the ball was launched at with a meter stick, and then stacked various books, notebooks and folders until the height of the platform and height of the launcher were equal. We then taped a sheet of white printer paper to the platform (to ensure its stability) and atop that, placed a piece of black carbon paper. We were now ready to begin our trials. We set the launcher at 50º, placed it on the ground, and shot. We observed where the ball hit the ground, and then put the platform on that spot. To ensure that the launcher stayed in the same place throughout the experiment, we lined the edges of the launcher up with the edges of a square tile on the floor. Since the tile was stationary, we could realign the launcher in the same position every time. While shooting the ball, the launcher would jolt, so to reduce its movement, Maddy stood upon its base. The screws which held the angle in place were apt to be affected by this jolting motion, therefore after each launch, Tom would readjust the angle if necessary. Sean stood behind the platform and marked the point made on the paper after each launch. He would then roll the ball back to Tom who would use the black stick to load the ball back into the launcher at medium power. Because time was occasionally a problem, we would measure the distance from the launcher to the platform using a tape measure. We would then write this measurement on the paper, therefore allowing us to find the distance from the edge of the paper to the points using a ruler later on. Because the points were not exactly parallel to the launcher, Maddy placed a ruler alongside the paper, and would then use a straight edge held perpendicularly to the ruler. When she would arrive at the center of a point, she could more accurately read the distance of the point by looking at the ruler on the side. These were then added to the measurement that stated how far away the paper was from the launcher. Maddy would then record these total distances in the excel spreadsheet. This is a simulation measurement depicting our measurement procedure. Though the ruler says one yard, in actuality we used a meter stick.

After shooting at multiple angles, it was found that 50º generated particularly consistent results. Because of this, we decided to always shoot at this angle. Below is a chart showing our measurements for the range at 50º.
 * __Observations and Data from Calibration:__**

The initial velocity for each range point was calculated individually. These were then averaged at the end. These can be seen under the Chart to find Initial Velocity which is found in the Data Calculations portion of this lab.
 * Trials at 50º || Range (m) ||
 * 1 || 2.3835  ||
 * 2 || 2.402  ||
 * 3 || 2.3955  ||
 * 4 || 2.4045  ||
 * 5 || 2.3873  ||
 * 6 || 2.3903  ||
 * 7 || 2.3858  ||
 * 8 || 2.4054  ||
 * 9 || 2.395  ||
 * 10 || 2.4012  ||
 * 11 || 2.3915  ||
 * 12 || 2.4  ||
 * 13 || 2.3993  ||
 * 14 || 2.394  ||
 * 15 || 2.397  ||
 * 16 || 2.4098  ||

__**Observations and Data from Performance:**__ Trial #5: media type="file" key="Physics - Medium.m4v" width="316" height="316"
 * __Trial__ || __Result__ ||
 * 1 || Cleanly through ||
 * 2 || Hit and went through ||
 * 3 || Hit and went through ||
 * 4 || Cleanly through ||
 * 5 || Cleanly through ||

__**Calculations:**__ __Collected Data Calculations:__

Collected Data: This chart shows the collected data from our experiment. For each trial we left our angle constant in an effort to gain the most accurate results knowing that for each angle, gravity’s effect on the spring was slightly different. We converted angles in degrees to angles in radians because radians are required to execute equations on excel. The equation to convert degrees to radians was:
 * angles (degrees) || angles (radians) || range(m) ||
 * 50 ||  0.872222222  ||  2.3835  ||
 * 50 ||  0.872222222  ||  2.402  ||
 * 50 ||  0.872222222  ||  2.3955  ||
 * 50 ||  0.872222222  ||  2.4045  ||
 * 50 ||  0.872222222  ||  2.3873  ||
 * 50 ||  0.872222222  ||  2.3903  ||
 * 50 ||  0.872222222  ||  2.3858  ||
 * 50 ||  0.872222222  ||  2.4054  ||
 * 50 ||  0.872222222  ||  2.395  ||
 * 50 ||  0.872222222  ||  2.4012  ||
 * 50 ||  0.872222222  ||  2.3915  ||
 * 50 ||  0.872222222  ||  2.4  ||
 * 50 ||  0.872222222  ||  2.3993  ||
 * 50 ||  0.872222222  ||  2.394  ||
 * 50 ||  0.872222222  ||  2.397  ||
 * 50 ||  0.872222222  ||  2.4098  ||

Radians = (degrees X 3.14) / 180 Radians = (50 X 3.14) / 180 Radians = 0.872222222

Chart to find initial velocity:
 * Angles (radians) || Range(m) || vi^2 (m/s) || Vi (m/s) ||
 * 0.872222222 ||  2.3835  ||  23.71494881  ||  4.869799669  ||
 * 0.872222222 ||  2.402  ||  23.89901701  ||  4.888662088  ||
 * 0.872222222 ||  2.3955  ||  23.8343444  ||  4.882043056  ||
 * 0.872222222 ||  2.4045  ||  23.92389109  ||  4.891205485  ||
 * 0.872222222 ||  2.3873  ||  23.75275742  ||  4.873680069  ||
 * 0.872222222 ||  2.3903  ||  23.78260631  ||  4.876741362  ||
 * 0.872222222 ||  2.3858  ||  23.73783297  ||  4.872148701  ||
 * 0.872222222 ||  2.4054  ||  23.93284576  ||  4.892120784  ||
 * 0.872222222 ||  2.395  ||  23.82936959  ||  4.881533528  ||
 * 0.872222222 ||  2.4012  ||  23.89105731  ||  4.887847922  ||
 * 0.872222222 ||  2.3915  ||  23.79454587  ||  4.877965341  ||
 * 0.872222222 ||  2.4  ||  23.87911775  ||  4.886626418  ||
 * 0.872222222 ||  2.3993  ||  23.87215301  ||  4.885913733  ||
 * 0.872222222 ||  2.394  ||  23.81941995  ||  4.880514312  ||
 * 0.872222222 ||  2.397  ||  23.84926885  ||  4.883571321  ||
 * 0.872222222 ||  2.4098  ||  23.97662415  ||  4.896593116  ||
 * ||  ||   || average: ||
 * ||  ||   ||  4.882935432  ||

Next we had to find the initial velocity for each trial. We first used an equation to find the initial velocity squared and then used the square root to find the initial velocity as seen below.

__Sample Calculation to find Initial Velocity Using Measured Range:__ Range = Vi^2 X sin(2 X radians) / gravity Vi^2 = (gravity X range) / (sin(2 X radians)) Vi^2 = (9.8 X 2.3835) / (sin(2 X 0.87222222)) Vi^2 = 23.7194881 m/s

Vi = (Vi^2) ^(1/2) Vi = 23.7194881^(1/2) Vi = 4.869799669 m/s

We then found the average initial velocity by finding the sum of each initial velocity divided by the number of trials.

Average = (sum of trials) / number of trials Average = 78.12696691 / 16 Average = 4.882935432 m/s

Chart to find X components, Y components and Time: Next we found the initial velocity of the x and y components and time. We also found the averages of each component and time.
 * vi || vi of X || vi of Y || time (s) ||
 * 4.869799669 ||  3.131896962  ||  3.729097777  ||  0.761040363  ||
 * 4.888662088 ||  3.144027883  ||  3.743541864  ||  0.763988135  ||
 * 4.882043056 ||  3.139771008  ||  3.738473274  ||  0.762953729  ||
 * 4.891205485 ||  3.145663608  ||  3.745489495  ||  0.764385611  ||
 * 4.873680069 ||  3.13439255  ||  3.732069233  ||  0.761646782  ||
 * 4.876741362 ||  3.136361349  ||  3.734413448  ||  0.762125194  ||
 * 4.872148701 ||  3.133407687  ||  3.730896572  ||  0.761407464  ||
 * 4.892120784 ||  3.146252262  ||  3.746190394  ||  0.764528652  ||
 * 4.881533528 ||  3.139443317  ||  3.738083097  ||  0.762874102  ||
 * 4.887847922 ||  3.143504271  ||  3.742918408  ||  0.763860899  ||
 * 4.877965341 ||  3.137148522  ||  3.735350723  ||  0.762316474  ||
 * 4.886626418 ||  3.14271869  ||  3.741983029  ||  0.763670006  ||
 * 4.885913733 ||  3.142260343  ||  3.741437283  ||  0.763558629  ||
 * 4.880514312 ||  3.138787833  ||  3.737302623  ||  0.762714821  ||
 * 4.883571321 ||  3.140753876  ||  3.739643558  ||  0.763192563  ||
 * 4.896593116 ||  3.149128537  ||  3.749615128  ||  0.765227577  ||
 * Average: || Average: || Average: || Average: ||
 * 4.882935432 ||  3.140344919  ||  3.739156619  ||  0.763093188  ||
 * Average: || Average: || Average: || Average: ||
 * 4.882935432 ||  3.140344919  ||  3.739156619  ||  0.763093188  ||

To find the x component:

Vi of x = Vi X Cos(radians) Vi of x = 4.869799669 X Cos(0.87222222) Vi of x = 3.131896962 m/s

We used excel for the calculations so we used cosine of radians rather than cosine of degrees. All of our angle measurements were constant at 0.8722222 radians.

To find the y component:

Vi of y = Vi X Sin(radians) Vi of y = 4.869799669 X Sin(0.872222222) Vi of y = 3.729097777 m/s

Time (seconds):

Time = (Vi of y) / (1/2) X gravity Time = 3.729097777 / 4.9 Time = 0.761040363 seconds

To find the average of each we took the sum of each trial divided by the number of trials the same way we found the average initial velocity earlier.

__ Calculations on Performance Day: __


 * range || time || distance of y meters || distance of y cm || height of ball above books (cm) || height of launcher (cm) || height of hoop (cm) ||
 * 1 ||  0.318436358  ||  0.693815017  ||  69.38150169  ||  26.25  ||  86.56849831  ||  182.2  ||

Above is the chart we used to find at what height we needed to put the launcher. First we found the time for an x distance of 1 meter. We chose to set the x displacement at 1 meter as a constant. The equation for the time was: Distance of x = Vi of x X time Time = (Displacement of x) / (Vi of x) Time = 1 / (3.14034492) Time = 0.3184363 seconds

Next we found how high from the ground the ball would reach. The time for the x component and y component are the same. We used this time in our equation to find the height of the launcher. The equation we used was:

Displacement of y = Vi of y X time + (1/2) X (acceleration of y) X time^2 Displacement of y = (3.73915662 X 0.3184363) + ((1/2) X -9.8) X (0.3184363)^2) Displacement of y = .693815017 meters

We then converted meters to cm making the distance of y 69.3815017 cm

We then took the height of the hoop and subtracted the distance of y and the height of the ball inside the launcher:

Height of launcher (cm) = height of hoop– displacement of y – height of ball inside of launcher Height of launcher = 182.2 cm – 69.3184363 – 26.25 Height of launcher = 86.56849831 cm

These equations were all done simultaneously in our excel chart, so all we had to do was enter the height we measured for the hoop (cm) in the "height of hoop box" and it would tell us the height to put the launcher at. It was automatically accounting for the 26.25cm between the base of the launcher and the point at which the ball was fired.

__**Graphs:**__ Whereas most groups used graphs to determine trendlines and relationships between angle/initial velocity or angle/range, our group had no need for them due to our set angle. We decided that setting the angle would give us the most consistent results when it came to initial velocity because gravity's effect on the spring would be the same every time. We also decided to set the range at 1m, the minimum distance allowed. This further decreased the margin of error by reducing the amount of time air resistance had to act upon the ball (which was not accounted for in our calculations) and by making it easier to aim the launcher directly at the middle of the hoop, which at a further distance would be more difficult. Because we chose to take both of these actions, our project leaves no possible graphs that would show anything of interest. They would only be linear averages which are already depicted through our calculations.

For our data collection, we found that the points were very close together, therefore calculating an average distance was appropriate. If our points were vastly spread out, or even if there were a few outliers, an average may not have been a good choice, for these outliers could distort the average distance. When it came to measurement, the data we found with the green measuring tapes has a +/- .01 margin of accuracy. We used this type of measurement for finding the height of the hoop. For all other measurements, we used meters sticks, which have a +/- .001 margin of accuracy.
 * __Error Analysis:__**

__**Conclusion:**__

__Results on Performance Day:__ Our method for getting the ball through the hoop proved to work rather well, for the ball went cleanly through three times. On the other two, it hit and went through as opposed to hitting and missing or missing altogether. Our method minimized error by keeping the range as short as possible and standardizing the angle therefore making Vi a constant. We feel the only reason it didn't go through all five times was due to hoop's swaying movement. When practiced in the hall, where there was less movement and where we used a heavier hoop, our results were perfect every time at multiple hoop heights.
 * __ Trial __ || __ Result __ ||
 * 1 || Cleanly through ||
 * 2 || Hit and went through ||
 * 3 || Hit and went through ||
 * 4 || Cleanly through ||
 * 5 || Cleanly through ||

__Error:__ Although our method was designed to minimize error, there were still some aspects of our execution that contributed to the overall possibilities of error.

__ Errors in our procedure: __ The first source of error we encountered when taking our measurements was the fact that the string which hangs down to measure the angle would sometimes stick to the hole through which it was tied. This, at times, made it difficult to get an accurate angle reading. Another source of error was that when taking our ground-to-ground measurements, the height of our platform may not have been exactly the same height as the launcher. Also, although Maddy stood on the launcher, there was still a slight jolting action which could’ve affected the ball’s distance.

__Error on Performance Day:__ Whereas in our procedure, the launcher was located on the ground, on performance day we had to elevate it using a stack of books, notebooks and folders. This stack may not have been exactly the height that was determined by our spreadsheet, so it may have been slightly higher or lower than it was supposed to be. Another source of error was that in order to stabilize the launcher, Tom had to press it down against the platform, which compressed said platform slightly. To account for this, we made the stack a few milimeters higher than was determined by the spreadsheet. Atop this, though the base of the stack was measured at 1m away, given the height of the stack, it may have leaned slightly away from the hoop, therefore making the launcher further than 1m exactly away. We also had to measure the height of the hoop, which if not measured in a perfect vertical line, could both throw off the determined height, as well as the point from which one must measure away from the hoop to find where the stack should be placed. The largest source of error on launch day was one that we couldn’t avoid. Our hoop was rather light, therefore the air-currents in the room both from students walking and from the heater caused a fair amount of movement in our hoop. This is what most likely caused two of our shots to hit the rim and go through.

__Ways to Address Error:__ If we were to do this experiment again, in our procedure we would be sure to use a thinner string, both for its more accurate measurements as well as the lesser likelihood of it getting caught on the hole. In order to make all of our length measurements more accurate, we could use a tool with smaller increments. To address the compression of the platform, instead of books we could use something that would not compress as easily, such as block of wood, or cinderblocks. As stated in the introduction, our method was designed to minimize error. By setting the angle, we had a consistent initial velocity. By setting the distance, we left ourselves with only one variable for which to account. By choosing the shortest possible distance, we also minimized the effects of air resistance which were not accounted for in our calculations, and we also reduced the amount of lateral change that could occur from not aiming perfectly at the target. To address the hoop's movement in the future, we should use a heavier tape roll, turn off the heating system, and ask students to remain still throughout the launching process.