Range versus Height PDF

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The TA's name was Michael Schott. This is a full lab report....

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Woods

Range versus Height Lab Report

Dana Woods

Lab Partner: Katherine Andersh Course: PHYS181-015 TA: Michael Schott Due Date: 5:00 PM on 9/15/16

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Abstract The overall goal of this lab was to determine the best method to drop a ball from a drop plate so it would hit a bounce plate and eventually hit a piece of paper with carbon paper on top. Then, this method would be used to derive data points and graphs for range versus height. The best method was determined by the smallest standard deviation. The best method, which gave a standard deviation of 1.1 cm, was found to be putting a ruler under the hole in the drop plate, and removing it so the ball would drop through the hole. Using this method, the ball was dropped at various heights and the general trend was that the higher the drop plate, the further the range and the higher the standard deviation.

Introduction For the first part of the experiment, we were simply dropping a ball from a drop plate 20 cm above the table so it would hit a bounce plate 10 cm above the table and make marks on a piece of paper covered with carbon paper. This allowed us to get a sense of how the lab would be performed, and how to calculate standard deviation. In the second portion of the lab, we had to choose two methods for dropping the ball, other than just using our fingers to drop it through the hole in the drop plate. Standard deviation was measured again, this time for the purpose of finding the better method between the two chosen. This determination was then used to carry out the third part of the experiment. Using the better method, the ball was dropped from varying heights relative to the bounce plate. This lab utilizes the concept of free fall, which is when the acceleration of an object is only due to gravity. Because gravity is a constant acceleration, you can relate it back to the kinematics equations. However, the ball is undergoing both horizontal and vertical motion. Therefore, the equations have to be split into the horizontal and vertical components. There is no acceleration in the horizontal direction, only the vertical direction, so that can be excluded from the equations. Gravity is also negative because the way the units have been defined, down is in the negative direction. The horizontal distance, known as range, can be determined, and was determined by the end of the lab.

Procedure The range versus height was measured using the apparatus shown Figure 1. The drop plate is seen at the top of the pole, and the bounce plate is located closer to the bottom the pole (would always be 20 cm above the table and at a 45 angle). The hole located in the drop plate allowed the ball to through the drop plate and, when lined up correctly, bounce on the bounce plate. The drop plate and bounce plate were lined up according to the piece of tape on the table. This corresponded to a range of zero, and all measurements were taken from there. The ball would then land on a piece of regular paper covered with carbon paper. The carbon paper would create a carbon dot on the paper where the ball would hit it.

Drop plate of go Bounce plate

“Zero” point

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Woods For the first part, the drop plate was placed 20 cm above the bounce plate, and therefore 40 cm above the table. We used only our fingers to hold the ball above the hole in the drop plate and then to release it through the hole. Once the ball was released, the carbon paper was lifted and we placed a number next to the dot that drop made. This was done 18 times. After the ball was dropped 18 times, a mark was made on the paper corresponding to how far away that mark was from the zero point (each mark was different for each trial). All of our measurements were done relative to that mark, and then those measurements were added to the original measurement the mark was made at. The sum of the ranges was calculated by adding the ranges together, and the averages were determined by taking sum and dividing it by the drop number. Then, the standard deviation was determined for the 18 measurements. This was done by drawing a line through the fourth point from the top and the fourth point from the bottom. The distance between those two points was measured, and then divided by two. During the second part of the experiment, two different methods were used for dropping the ball. The drop plate remained 20 cm above the bounce plate (40 cm above the table). For the first method, the ball was rolled into the hole in the drop plate and then bounced on the bounce plate. We realized that the drop plate was slightly tilted towards the table. We were able to place the ball next to the hole and have it roll towards the hole and it would go through the hole. This was also done 18 times, and both the range measurements and standard deviation were calculated in the same way as part one. The second method we used was placing a ruler under the hole to block the ball from going through. Once the ruler was removed, the ball was able to go through the hole to bounce from the bounce plate onto the paper. Again, the range and standard deviation were calculated in the same way. The final part of the lab was determining range versus height. The method with the lowest standard deviation was used, because a lower standard deviation indicates more accurate and precise measurements. In our case, the method where we removed the ruler from under the hole had the lowest standard deviation. The drop plate was moved to 5 cm above the bounce plate, and the ball was dropped using the ruler method 12 times. This was repeated for 10 cm, 15 cm, 20 cm, 25 cm, and 30 cm above the bounce plate. The range measurements were made in the same manner as previously described. The standard deviation was calculated the same, except only two points were excluded on either side rather than three due to having fewer total data points.

Theory In this lab, we had to measure the distance of dots relative to a “zero” point. This horizontal distance is known as the range. The distance was 1. Apparatus for experiment, measured simply by starting our ruler at the “zero” point, and Figure showing pole, drop plate, “zero” point, and bounce plate.

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Woods measuring to where the dot was. Next, the sum of the ranges needed to be found. This was found using the equation (1) where Rsum is the sum of the ranges, n is the number of times the ball was dropped, and Ri is the range for drop i. Next, the current average needed to be found, and can be found by (2) where Rave is the current average range, Rsum is the current sum of the ranges, and n is the number of times the ball has been dropped. This method was done for all three parts of the lab. Next, standard deviation had to be measured in order to figure out how accurate and precise our measurements were. There is an equation for standard deviation, but we used a different method, as shown in Figure 2. We wanted to calculate where 66.7% of the ranges were. Because there were 18 measurements, we needed to determine where the middle 12 measurements were. To do this, three points on either side were excluded, which got rid of the 33.3% we did not want (the outliers). A line was drawn through the fourth point up and the fourth point down. During the third part of the lab, when there were only 12 drops, this same method was done, except only two Figure 2. Sample calculation of points were excluded from either side because 66.7% of 12 is standard deviation. eight. The length between these two lines was measured using a ruler. This length indicates two standard deviations. Therefore, the length was divided by two to get the standard deviation for that set of data points. After the range versus height data points were graphed and a line of best fit was added, the slope needed to be determined. The equation (3) can be used, where y is the vertical position on the graph, m is the slope, x is the horizontal position on the graph, and b is the y-intercept. The slope of the graph gives us the range as a function of the height.

Sample Calculation and Results

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Woods The raw data for part one is shown in the table on the worksheet attached to the end of the report. The raw data for the second part of the experiment can be seen in Table 1 and Table 2. The raw data for section three of the lab can be seen in Table 3, Table 4, Table 5, Table 6, Table 7, and Table 8. All of the values for these tables were calculated in the same manner. As an example, we can look at Table 1. The range of the first dot was found to be 28.8 cm from the origin. From here, it was easy to determine the sum and the average because there was only one dot. However, once more drops were completed, actual calculations can be done to determine the sum and the average. The range of the second 2 was 28.5 cm. From here, we can use Equation 1 above to determine the sum. 28.8cm

57.3cm

(4)

Next, the average range was determined using the sum we just found. Equation 2 can now be used to determine the average range. 57.3cm 2

28.7cm

(5)

This process was repeating 18 times after measuring each carbon dot on the paper. This process was also repeated for the raw data seen in Tables 2-8. As the ball was dropped more times, the average fluctuated less. Standard deviation was also calculated for each of the trials in the lab. As previously explained, and as can be seen in Figure 2, standard deviation was calculated based on measurements, not an equation. The “middle” 66.7% of the points were used to calculate the standard deviation, while the “outside” 33.3% were considered outliers. The length between the the upper- and lower-most dots within the 66.7% was measured. Because this length is two standard deviations, it was divided by two to get the final standard deviation. After these numbers were calculated for the different heights, we could get our numbers for range versus height, as seen in Table 9. For 5 cm, the average range was 15.6 cm. For 10 cm, the average range was 21.5 cm. For 15 cm, the average range was 25.7 cm. For 20 cm, the average range was 28.9 cm. For 25 cm, the average range was 29.0 cm. For 30 cm, the average range was 33.4 cm. By looking at Graph 1, it can be seen that as the height of the drop plate increases, the range also increases. This indicates that range is directly proportional to height. Finally, the slope of the best fit line for range versus the square root of height needed to be calculated. This was graphically calculated using Excel, and can be seen in Graph 2. The equation for the slope was used. 5.1926cm

1/2

x + 4.7221cm

(6)

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The slope was determined to be 5.1926 cm1/2. This tells us that for every 1 cm1/2 the drop plate was raised, the range increased by 5.1926 cm.

Discussion and Conclusions We found that as the height of the drop plate increased, the horizontal distance that the ball traveled also increased. According to the trend line in Graph 2, the range increased by 5.1926 cm for every 1 cm1/2 that the drop plate was raised. This is not exact, as the trend line does not go straight through the points, and there was some error, as explained later. The average range changed less as more points were generated. This is because the more data points taken, the more likely the errors are random. When the average is calculated with a larger amount of points, the calculations are statistically closer to what the actual value should be. So, the average not only fluctuates less, but it also makes the data more reliable. Another trend that was noticed was that as the height increased, the standard deviation increased. The standard deviation values can be seen in Tables 3-8 under the Standard Deviation column. There was one “bad” calculation that did not fit this trend, which was the standard deviation for 25 cm (0.7 cm). Otherwise, the calculations followed this trend. This could be due to the fact that the closer the ball is to the ground, the less time there is for other variables to affect it and cause an error in the measurements. Given the information found, we are able to derive an expression for range as a function of height. When the ball is first dropped, the ball is only moving in the vertical direction due to gravity (known as free fall). The time it takes to reach the bounce plate at 5 cm can be 2

0 m = -0.05m + 0m/s(t) – ½(-9.8m/s )t

2

0.1s

(7)

determined with the equation where y is the current vertical position, y0 is the initial vertical position, v0y is the initial vertical velocity, t is time, and g is acceleration due to gravity. Now that we have the time, we can use the vertical velocity equation to find the final vertical position, vy which becomes the initial horizontal velocity. 2 0m/s – (9.8m/s )(0.1s)

0 99m/s

(8)

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Woods The next step is calculating the time it takes for the ball to reach the ground after it hits the bounce plate. This can be found with Equation 7. This time, the end distance will be from the 2

0m = -0.25m + 0m/s(t) – ½(-9.8m/s )t

2

drop plate to the table. Then, the drop plate to the bounce plate will be subtracted.

0 23s

0 1s

time

0.13s (9) from the

Finally, this time can be put into the horizontal position equation to determine how far the ball will travel until it hits the ground. The equation, 0 + (0.99m/s)(0.13s)

0.1287m

12.87cm

(10)

where x is the final horizontal position, x0 is the initial horizontal position, and v0x is the initial horizontal velocity, tells us that, when the drop plate starts at 5 cm above the bounce plate, the ball should travel 12.87 cm before hitting the ground. Our actual value was 15.6, which is most likely due to some of the errors mentioned shortly. These calculations can be applied to any of the drop plate heights to find the expected range. A large source of error in our measurements would be the ruler we used. The ruler only measured to the nearest millimeter, so it had an uncertainty of 0.05 mm. Because of how large the dots were, it was difficult to accurately measure their distance with the ruler given. The error bars seen in Graph 1 also indicate error in our measurements according to the standard deviation at each height. This was because the ball did not land in the same place for each drop, so the area where most of the dots were had to be measured. Other errors for this lab included slight differences in how the ball was released into the hole during a trial, the fact that the drop plate could have been rotated slightly as the height was changed, and occasionally the ball would get stuck in the hole in the drop plate before falling.

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Table #1 Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 28.8 28.8 28.8 1.2 2 28.5 57.3 28.7 3 27.6 84.9 28.3 4 28 112.9 28.2 5 28.4 141.3 28.3 6 28.1 169.4 28.2 7 29.4 198.8 28.4 8 28.1 226.9 28.4 9 31.1 258 28.7 10 29.7 287.7 28.8 11 30.2 317.9 28.9 12 28.4 346.3 28.9 13 31.3 377.6 29 14 27.7 405.3 29 15 28.3 433.6 28.9 16 30.4 464 29 17 31.8 495.8 29.2 Table 1. Ranges, range 18 sums, and average 29.3 ranges for the rolling method. 525.1 29.2

Dot Number Range (cm) 1 30.3 2 31 3 30 4 30.3 5 27.7 6 29.6 7 29.2 8 29.8 9 29.7 10 28.2 11 28.5 12 28.7 13 28.3 14 30.3 15 30 16 29 17 28.1 18 29.2

Table #2 Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 30.3 30.3 1.1 61.3 30.7 91.3 30.4 121.6 30.4 149.3 29.9 178.9 29.8 208.1 29.7 237.9 29.7 267.1 29.7 295.8 29.6 324.3 29.5 353 29.4 381.3 29.3 411.6 29.4 441.6 29.4 470.6 29.4 498.7 29.3 527.9 29.3

Table 2. Ranges, range sums, and average ranges for the ruler method.

Table #3 8

Woods Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 15.3 15.3 15.3 0.3 2 15.2 30.5 15.2 3 15.5 46 15.3 4 14.9 60.9 15.2 5 15.5 76.4 15.3 6 15.7 92.1 15.4 7 16.1 108.2 15.5 8 15.4 123.6 15.5 9 16.1 139.7 15.5 10 15.7 155.4 15.5 11 15.5 170.9 15.5 12 15.7 186.6 15.6

Table 3. Ranges, range sums, and average ranges when dropping from 5 cm above the bounce plate.

Table #4 Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 21.7 21.7 21.7 0.4 2 20.5 42.2 21.1 3 21.4 62.7 20.9 4 21.2 83.9 21 5 21.2 105.1 21 6 21.7 126.8 21.1 7 22.2 149 21.3 8 22 171 21.4 9 21.6 192.6 21.4 10 21.9 214.5 21.5 11 22 236.5 21.5 12 22 258.5 21.5 Table 4. Ranges, range sums, and average ranges when dropping from 10 cm from the bounce plate.

Table #5 9

Woods Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 24.7 24.7 24.7 0.9 2 26.1 50.8 25.4 3 23.3 74.1 24.7 4 25.9 100 25 5 26.8 126.8 25.4 6 24.7 151.5 25.3 7 26.4 177.9 25.4 8 25.5 203.4 25.4 9 25.1 228.5 25.4 10 26.9 255.4 25.5 11 26.2 281.6 25.6 12 26.3 307.9 25.7 Table 5. Ranges, range sums, and average ranges when dropping from 15 cm above the bounce plate.

Table #6 Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 27.9 27.9 27.9 1.7 2 30.9 58.8 29.4 3 26.6 85.4 28.5 4 29.1 114.5 28.6 5 27.2 141.7 28.3 6 29.4 171.1 28.5 7 29.9 201 28.7 8 26.9 227.9 28.5 9 27.7 255.8 28.4 10 31 286.8 28.7 11 30.3 317.1 28.8 12 30.1 347.2 28.9

Table #7 Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 28 28 28 0.7 10 T bl 6 R

d

h

d

f

20

b

h b

l

Woods 2 3 4 5 6 7 8 9 10 11 12

28.5 27.5 29.4 29.1 29.7 30.4 28.9 28.9 30.2 28.5 29.4

56.5 84 113.4 142.5 172.2 202.6 231.5 260.4 290.6 319.1 348.5

28.3 28 28.4 28.5 28.7 28.9 28.9 28.9 29 29 29

Table #8 Dot Number Range (cm) Sum of Ranges (cm) Average Range (cm) Standard Deviation (cm) 1 33.5 33.5 33.5 1.2 2 33.8 67.3 33.7 3 32.9 100.2 33.4 4 31.8 132 33 Table 7. Ranges, range sums, and average ranges when dropping from 25 cm above the bounce plate. 5 34.3 166.3 33.3 6 32.8 199.1 33.2 7 33.9 233 33.3 8 34.7 267.7 33.5 9 32.4 300.1 33.3 10 37 337.1 33.7 11 32 369.1 33.6 12 31.5 400.6 33.4 Table 8. Ranges, range sums, and average ranges when dropping from 30 cm above the bounce plate.

Table #9 11

Woods Height (cm) Range (cm) Standard Deviation (cm) Square Root of Height (cm^1/2) 5 15.6 0.3 2.236067977 10 21.5 0.4 3.16227766 15 25.7 0.9 3.872983346 20 28.9 1.7 4.472135955 25 29 0.7 5 30 33.4 1.2 5.477225575 Table 9. Heights and ranges of ball when dropped using the ruler method. Standard deviation used to create error bars for Graph #1. Square root of height used to find line of best fit for Graph #2.

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Graph #1 40 35

Range (cm)

30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

Height (cm) Graph 1. Range versus height data for the ball when dropped using the ruler method. The black bars indicate error bars, which were added according to the standard deviation of each point. The higher the ball was dropped from, the further its horizontal distance.

Graph #2 40 35

f(x) = 5.19 x + 4.72 R² = 0.97

Range (cm)

30 25 20 15 10 5 0

2

2.5

3

3.5

4

4.5

5

5.5

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Square Root of Height (cm^1/2) Graph 2. Range versus square root of height data for the ball when dropped using the ruler method. Graphing with the square root of height gave a straighter line to determine the slope. The red line indicates the best fit line for the data. The higher the bal...