Calculus Based Assignment

Ojillius22

CoefofRestitutionWorkbook.xlsx

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Intro

initial h

final h

Simulation

	This is simulated data, showing a random initial drop height (h_i) and it's corresponding bounce height (h_f).
	On the next worksheet, you'll enter your own data.
	The letter "e" stands for the coefficient of restitution.
							Press "F9," and then you can see a new set of simulated data.
				Descriptive Statistics
	h_i	h_f	e	Average	0.8642		center	bin	freq
1	0.699	0.505	0.850	Median	0.8724	1	0.61	0.6	0			s.d. select	% select
2	0.459	0.345	0.867			2	0.63	0.62	0	s.d. off	0	0	0		s.d. display	% contains
3	0.999	0.719	0.849	Max	0.9077	3	0.65	0.64	0	1 s.d.	0.0254	0	0		0.0761550934	99.7
4	0.961	0.674	0.837	Min	0.8112	4	0.67	0.66	0	2 s.d.	0.0507700623	0	0
5	0.95	0.746	0.886			5	0.69	0.68	0	3 s.d.	0.0761550934	0.0761550934	99.7
6	0.832	0.585	0.838	Standard Deviation	0.0254	6	0.71	0.7	0
7	0.544	0.448	0.908	Standard Error	0.0051	7	0.73	0.72	0
8	0.48	0.376	0.885			8	0.75	0.74	0
9	0.738	0.568	0.877	How do these values relate to what is happening in the histogram?		9	0.77	0.76	0
10	0.389	0.305	0.885			10	0.79	0.78	0
11	0.845	0.606	0.847			11	0.81	0.8	1
12	0.321	0.224	0.836			12	0.83	0.82	6
13	0.854	0.602	0.840			13	0.85	0.84	3
14	0.461	0.359	0.882			14	0.87	0.86	7
15	0.981	0.645	0.811			15	0.89	0.88	7
16	0.776	0.592	0.873			16	0.91	0.9	1
17	0.57	0.434	0.872			17	0.93	0.92	0
18	0.818	0.650	0.892			18	0.95	0.94	0
19	0.763	0.520	0.825			19	0.97	0.96	0
20	0.837	0.667	0.893			20	0.99	0.98	0
21	0.531	0.426	0.896
22	0.59	0.454	0.877
23	0.655	0.457	0.835
24	0.931	0.706	0.871
25	0.694	0.530	0.874					Select Standard Deviation Display			Reported Value for CoF
											CoF is	0.8642	±	0.0051
								3 s.d.				(avgerage value)		(standard error)
								Contains
								99.7	%
								of data points
	Simulated Data
	© 2015 Sean M. Cordry and Walsters State Community College

Simulated Distribution of CoR Data

CoF Frequency 0.61 0.63 0.65 0.67 0.69 0.71 0.73 0.75 0.77 0.79 0.81 0.83 0.85 0.87 0.89 0.91 0.93 0.95 0.97 0.99 0 0 0 0 0 0 0 0 0 0 1 6 3 7 7 1 0 0 0 0 Average Value 0.86418827589028369 14 Standard Deviation Display 7.6155093422405173E-2 7.6155093422405173E-2 0.86418827589028369 7

Notice that more often than not, the spread in the simulated data resembles a "normal" distribution -- like the one shown here. This is the result of random errors in measurements. Pick the number of standard deviations that you would like to display on your graph, and compare them to the "normal" distribution. (Look for the black horizontal bar.)

More Stuff

To report an actual result, we give the average value and say "plus-or-minus" the standard error. The value of the standard error is equal to the value of the standard deviation divided by the square-root of the number of data points.

Your Data

	This worksheet page is for your data.
	If you are getting coefficients that are less than 0.6, you will need to find a harder surface or a bouncier ball.
	Be sure to vary your drop height.
							(Just copy this histogram into your report. You don't need the data table.)
				Descriptive Statistics
	h_i	h_f	e	Average	0.0000		center	bin	freq
1	1	0.000	0.000	Median	0.0000	1	0.61	0.6	25			s.d. select	% select
2	1	0.000	0.000			2	0.63	0.62	0	s.d. off	0	0	0	s.d. display	% contains
3	1	0.000	0.000	Max	0.0000	3	0.65	0.64	0	1 s.d.	0.0000	0	0	0	0
4	1	0.000	0.000	Min	0.0000	4	0.67	0.66	0	2 s.d.	0	0	0
5	1	0.000	0.000			5	0.69	0.68	0	3 s.d.	0	0	0
6	1	0.000	0.000	Standard Deviation	0.0000	6	0.71	0.7	0
7	1	0.000	0.000	Standard Error	0.0000	7	0.73	0.72	0
8	1	0.000	0.000			8	0.75	0.74	0
9	1	0.000	0.000	Select Standard Deviation Display		9	0.77	0.76	0
10	1	0.000	0.000			10	0.79	0.78	0
11	1	0.000	0.000	s.d. off		11	0.81	0.8	0
12	1	0.000	0.000			12	0.83	0.82	0
13	1	0.000	0.000			13	0.85	0.84	0
14	1	0.000	0.000			14	0.87	0.86	0
15	1	0.000	0.000			15	0.89	0.88	0
16	1	0.000	0.000			16	0.91	0.9	0
17	1	0.000	0.000			17	0.93	0.92	0
18	1	0.000	0.000			18	0.95	0.94	0
19	1	0.000	0.000			19	0.97	0.96	0
20	1	0.000	0.000			20	0.99	0.98	0
21	1	0.000	0.000
22	1	0.000	0.000
23	1	0.000	0.000
24	1	0.000	0.000
25	1	0.000	0.000
	© 2015 Sean M. Cordry and Walsters State Community College

Distribution of CoR Data -- Fall 2021

CoF Frequency 0.61 0.63 0.65 0.67 0.69 0.71 0.73 0.75 0.77 0.79 0.81 0.83 0.85 0.87 0.89 0.91 0.93 0.95 0.97 0.99 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Average Value 0 14 Standard Deviation Display 0 0 0 7

Lavf54.59.106

Ball drop edit v1.mp4

The Coefficient of Restitution

The coefficient of restitution is a quantity that relates before- and after- speeds regarding a collision. It’s related to the word “restore,” and it works like this.

Suppose that a racquetball hits a wall going 50 m/s, but when it bounces away from the wall, it’s only going 40 m/s. The ball’s speed does not experience full “restitution” after the impact, but only an 80% restitution level. (40/50 = 80%) If your racquet ball partner steps in front of your shot, blocking the ball with his left butt-cheek, then the restitution of the collision might only be 20%, so the ball would glance from the gluteal muscle at 10 m/s.

The amount of restitution in any collision depends on the types of objects involved. Today, we’ll be measuring the coefficient of restitution of bouncy rubber ball, which will be done by letting the ball bounce and measuring how high it goes.

Right now, you don’t need to know the theory behind this – this Excel program will do the math for us. We’re interested in seeing how data “smears” out in the real world. Each time you drop the ball from a certain height ( hi )and record the bounce height ( hf ), the program will calculate a value of the coefficient of restitution ( e ). When you measure your height values, make sure that you always measure to the bottom of the ball.

Check out the video (above and to the right) so that you have an idea of what’s going on.

Since this is a real experiment, the value of e will not be consistent: there will be a variety of numbers around some average value. An important number to tell us how “blurry” or how “spread-out” our data is a number called the standard deviation of the mean – or just “standard deviation” – or just “s.d.” – or just “.”

What to do next:

1. Take a look at the simulation tab. It’s “job” is to help you understand the randomness that can occur in actual experiments. Make sure you understand the relationship between the average and the standard deviation.

2. Construct a similar apparatus with your ball and meterstick. Take two pictures of your apparatus: One with your face in the shot and one with the apparatus in action. (The pics are for security purposes – to make sure that everyone does their own work.)

3. Drop the ball from various heights twenty-five times and record the corresponding bounce heights.

4. Answer the questions on the Lab Report about your results and turn them into the Dropbox in eLearn.

Note: You can click on the cells in the next two tabs, but you can only alter some of the info. On the Simulation tab, you can only select the number of standard deviations for the graph to illustrate. On the Your Data tab, you can also enter you data.

The Coefficient of Restitution

The coefficient of restitution is a quantity that relates before - and after- speeds regarding a collision. It’s related to the

word “restore,” and it works like this.

Suppose that a racquetball hits a wall going 50 m/s, but when it bounces away from the wall, it’s only

going 40 m/s. The ball’s speed does not experience full “restitution” after the impact, but only an 80%

restitution level. (40/50 = 80%) If your racquet ball partner steps in front of your shot, blocking the ball

with his left butt-cheek, then the restitution of the col lision might only be 20%, so the ball would glance

from the gluteal muscle at 10 m/s.

The amount of restitution in any collision depends on the types of objects involved. Today, we’ll be measuring the

coefficient of restitution of bouncy rubber ball, which will be done by letting the ball bounce and measuring how high it

goes.

Right now, you don’t need to know the theory behind this – this Excel program will do the math for us. We’re interested

in seeing how data “smears” out in the real world. Each time yo u drop the ball from a certain height ( h

)and record the

bounce height (h

), the program will calculate a value of the coefficient of restitution ( e). When you measure your height

values, make sure that you always measure to the bottom of the ball.

Check out the video (above and to the right) so that you have an idea of what’s going on.

Since this is a real experiment, the value of e will not be consistent: there will be a variety of numbers around some

average value. An important number to tell us how “bl urry” or how “spread-out” our data is a number called the

standard deviation of the mean – or just “standard deviation” – or just “s.d.” – or just “.”

What to do next:

1. Take a look at the simulation tab. It’s “job” is to help you understand the randomness that can occur in actual

experiments. Make sure you understand the relationship between the average and the standard deviation.

2. Construct a similar apparatus with your ball and meterstick. Take two pictures of your apparatus: One with your face in

the shot and one with the apparatus in action. (The pics are for security p urposes – to make sure that everyone does

their own work.)

3. Drop the ball from various heights twenty-five times and record the corresponding bounce heights.

4. Answer the questions on the Lab Report about your results and turn them in to the Dropbox in eLearn.

Note: You can click on the cells in the next two tabs, but you can only alter so me of the info. On the Simulation tab, you can

only select the number of standard deviations for the graph to illustrate. On the Your Data tab, you can also enter you data.

Lavf54.59.106

Ball drop edit v1.mp4