course 201
Mr Smith,Beth and I had very little time to work together that day, but I did go back and check my calculations and everything seemed fine. I will also email this assignment to you just to be safe.
I chose experiment number nine, which asks how the horizontal range of a projectile which leaves the end of a straight ramp at a constant height above the floor, after accelerating the full length of the ramp, depends on the slope of the ramp. Assuming slope and acceleration are proportional; how then would the horizontal range depend on the velocity at the end of the ramp?
The materials used in conducting this experiment includes; a steel metal track (width 13.5mm +- .25mm) used as a ramp (length 57.7cm) attached to plywood by a screw at the lower end of its slope to ensure its position stays the same, a screw adjustment mechanism placed at the other end of the ramp to be used for slope adjustment, one steel ball (diameter 21mm) with directional/positioning marks placed on its top and lateral side to ensure proper ball placement on the ramp for each trial, a piece of white paper taped to the floor with its end corresponding to the end of the ramp from which the ball leaves 94.2cm above that spot on the floor, a piece of carbon paper laying on top of the white paper to ensure the exact landing spot of the ball can be marked, a metal ball, dominoes (used for ramp elevation above 1.5in or 3.81cm), and a meter stick for measuring.
Twenty trials were conducted in this experiment. The first trial began with a ramp slope of .006cm. The ball was placed on the ramp in the position corresponding to the markings placed on it and the ramp and allowed to accelerate down the ramp from rest through a distance of 57.7cm before reaching the end of the ramp. The ball was then allowed to land on the carbon paper on the floor (94.2 cm below), which caused an ink dot to be left on the white paper. The dot was then marked with its corresponding slope and measured to obtain its horizontal range of travel after leaving the ramp. Each of the 20 trials was conducted in this manner and the data obtained is listed below. (Note that each trial was conducted at an increased slope from the previous trial. This was done in the first 12 trials by adjusting the screw mechanism (4 turns which corresponds to 1/8 of an inch in rise of the ramp after each trial). Each rise was converted to centimeters in order to determine what percent of a domino with the thickness of .95cm it was. The remaining 8 trials were conducted by removing the screw mechanism and using whole dominos to adjust the ramp slope.
The following pages contain a table consisting of the data collected.
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Dominoes Rise
(cm) Run
(cm) Slope Ramp Length
(cm) Vertical
Drop
(cm) Horiz.
Range
(cm)
.3342 .3175 57.7 .006 57.7 94.2 10.85
.6684 .635 57.7 .011 57.7 94.2 13.4
1.0026 .9525 57.7 .017 57.7 94.2 15.7
1.3368 1.27 57.7 .022 57.7 94.2 17.75
1.6711 1.5875 57.7 .028 57.7 94.2 19.4
2.0053 1.905 57.7 .033 57.7 94.2 21.1
2.3395 2.2225 57.7 .039 57.7 94.2 22.5
2.6737 2.54 57.6 .044 57.7 94.2 24.25
3.0079 2.8575 57.6 .050 57.7 94.2 25.5
3.3421 3.175 57.6 .055 57.7 94.2 26.9
3.6763 3.4925 57.6 .061 57.7 94.2 28.0
4.0105 3.81 57.6 .066 57.7 94.2 29.1
5.0 4.75 57.5 .083 57.7 94.2 31.6
6.0 5.7 57.4 .099 57.7 94.2 34.4
7.0 6.65 57.3 .116 57.7 94.2 37.0
8.0 7.6 57.2 .133 57.7 94.2 39.3
9.0 8.55 57.1 .150 57.7 94.2 41.4
10.0 9.5 56.9 .167 57.7 94.2 43.45
11.0 10.45 56.7 .184 57.7 94.2 44.85
12.0 11.4 56.6 .201 57.7 94.2 46.75
Final
Velocity
(cm/s) Change
Velocity
(cm/s) Ave.
Velocity
(cm/s) Ramp
Time
(s) Accel.
(ramp)
(cm/s2) Sin
Theta
24.71 24.71 12.36 4.67 2.65 0.006
30.53 30.53 15.27 3.78 4.04 0.011
35.81 35.81 17.91 3.22 5.56 0.017
40.55 40.55 20.28 2.85 7.12 0.022
44.40 44.40 22.20 2.6 8.54 0.028
48.42 48.42 24.21 2.38 10.17 0.033
51.78 51.78 25.89 2.23 11.61 0.039
56.01 56.01 28.01 2.06 13.6 0.044
59.12 59.12 29.56 1.95 15.16 0.05
62.64 62.64 31.32 1.84 17.02 0.055
65.60 65.60 32.8 1.76 18.64 0.061
68.42 68.42 34.21 1.69 20.24 0.066
75.55 75.55 37.78 1.53 24.7 0.083
83.94 83.94 41.97 1.37 30.64 0.099
92.45 92.45 46.23 1.25 37 0.115
100.8 100.8 50.4 1.14 44.21 0.132
109.4 109.4 54.7 1.05 52.1 0.148
118.6 118.6 59.3 0.97 61.13 0.165
126.5 126.5 63.25 0.91 69.51 0.181
136.9 136.9 68.45 0.84 81.49 0.197
Graph 1: Slope vs. Horizontal Range
Graph 2: Final Velocity vs. Horizontal Range
Graph 3: Acceleration vs. Slope
Graph 4: Acceleration vs. Sin Theta
The first graph (Slope vs. Horizontal Range) shows that that horizontal range increases as the slope increases, however I did notice that the range increase by a smaller amount after each increase of the slope. From this, I formed the belief that at some point in the elevation of slope the horizontal range will begin to decrease.
This should be the case; if the ramp slope increases until the ramp is vertical, the horizontal range will of course decrease to zero.
The second graph (Final Velocity vs. Horizontal Range) and the third (Acceleration vs. Slope) produced results that I anticipated. First that the greater the velocity when the ball left the end of the ramp the greater the horizontal range it would travel its fall to the floor and second that the greater the slope the greater the acceleration. However, I wasn’t sure why this second point was so. The fourth graph (Acceleration vs. Sin Theta) helped answer this question.
If the ball roles down the ramp without slipping, then the graph of acceleration vs. sin theta should be a straight line. It is easy to see in the graph above that this was not the case in this experiment. Using a strait edge to project a line through the data points on this graph it is clear that slipping is definitely present at a slope of .15 (a = 51.1 cm/s2 and sin theta =.148). We know this because this is where the data points deviate from our straight line. It seems that the acceleration increases as the slope increases due to increased slipping of the ball. As the slope approaches vertical (90 degrees) the acceleration approaches that of gravity (980 cm/s2) due to the absents of friction. From this I also feel that further analysis of this experiment would confirm my suspicion that a continued increase in slope would eventually begin to decrease horizontal range (obviously with track at a 90 degree angle with the floor, friction would be equal to 0 and the ball would fall straight down to the floor a 980 cm/s2 with a horizontal
range = 0).
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Excellent data and very good work in reporting your results.