APSC 100 Module 2 Tutorial Lab #1

Name: Ian Ip

Student Number: 10011223

Engineering Section: 205

Jan/20th/2012

1)

Table #1: Elasticity values

Single Trial Statistical Analysis

Graphical Analysis

Big Ball, diameter = 4.260 +/- 0.005cm 0.671+/-0.030 0.622+/-0.054 0.636+/-0.005

Small Ball, diameter = 2.970 +/- 0.005cm 0.850+/-0.030 0.805+/-0.036 0.817+/-0.007

2)

The uncertainty of single trial was significantly higher when compared to that of the multiple trials based graphical analysis. The result was expected because multiple trials are able to identify random errors and find a line of best fit among all data points. Whereas single trial make use of only one single data point, which can sometime be an error itself. The single trial uncertainty should be bigger because there are no other data points to cross reference to. In multiple trials, the use of linear regression allow a more accurate determination of the line of best fit (and subsequently the slope, which is the elasticity value) and significantly lowers the uncertainty by 83% of the big ball single trial uncertainty and 77% of the small ball.

3)

The data obtained from both balls demonstrated that elasticity values did not vary by height. Based on the residual plot of both balls (Refer to appendix 2), the data points of both balls are close to the horizontal axis. This was expected because of the following rearrangement of the elasticity formula:

Elasticity=H 1

H 0

The Elasticity value of any object is given as the rebound height (H 1 ¿divided by the initial height (H 0 ¿

H 1=Elasticity∗H 0

By rearranging the original equation to the familiar y=mx+b form (Elasticity =0 at (0, 0)), we can now plot the data recorded where the rebound height is plotted in the y axis and the dropped height is plotted in the x axis (Seeing as the H 0 is common in the small and big ball, and that the both sets of y values will

be plotted against the common x valuesH 0). The new rearranged equation is a linear equation, which means that the slope (Elasticity) has to be constant and that the dropped height and rebound height have no effects on the elasticity because they are proportional to each other by a common proportion, elasticity.

4)

The elasticity values of the two balls tested in the lab are not the same. After calculating elasticity from the three trial methods, all of which confirmed that the elasticity of the small ball is greater than that of the big ball. Furthermore, the uncertainty obtained from graphical analysis (Most accurate method) show that the small ball has a bigger uncertainty than that of the big ball. The reason for this is because the elasticity value is bigger.

SE=( Sh1h1 + Shoho )E The Equation shows that if all the variables in the bracket are constant, then the

uncertainty should be directly proportional to elasticity itself.

5)

During the lab, systematic errors are constantly being eliminated with careful setups. For example, one member of the pair made sure that she was eyelevel with where the ball rebounded to. Rather than looking and approximating from above, she made sure that parallax was eliminated from the possible list of systematic errors. In addition, it was ensured that the actual drop height did not vary much from the measuring taped stand by lining up a ruler from the stand and out approximately 40cm away from the stand where the ball was dropped, so the ball would not hit the stand and affect the results.

The lines of best fit of both balls have the intercept of zero, verifying the effectiveness in systematic error reduction.

6)

The sources of random errors are listed in order of most to least significant:

The accuracy of the measurement of rebound height (Eyeballing) The accuracy of the measurement of dropped height (Eyeballing) The inconsistency of each ball drop (the release of the ball)

7)

After going through the original pot and regression line, the error estimates made reflected the measurement errors that were made during the lab. Although it was initially expected that the measurement errors would have been higher before the data sets were plotted. The data points remained very closely with the regression line and the error bars on each point consistently crosses with the line of best fit.

8)

In conclusion, I discovered many other things during this lab (particular Newtonian mechanics based topics). First, I was able to see the effect of inertia first hand that it does not matter where the ball strikes because projectile motion states that all objects descend at a rate of 9.8m/s^2 (And of course, rebounding at the same speed in the y direction). Secondly, I was able to see a great example of energy-mass conservation from the ball dropped at a certain height, and coming back up to a height close to the dropped height. This is particularly true when the ball was dropped close to the ground where air friction and energy lost in the strike are in small amount where energy conservation does become more apparent. The super balls were most likely to be produced from the polymerization of certain material. Because of this, the material is uniform throughout the ball, which implies that it does not matter which side of the ball hit the ground because the elasticity will be the same. At the same time, the material of the floor will cause the elasticity to differ slightly. The elasticity will differ on the ability for the floor material to absorb the momentum and energy generated.

-End

Appendix

Appendix 1: Graph and data set (Statistical analysis included)

Initial Height

Rebound Height

Elasticity

30 14.2 0.47333347 27.4 0.58297964 41.7 0.65156381 51.6 0.63703798 63.5 0.647959115 76.3 0.663478132 85.3 0.646212149 97.2 0.652349166 107.3 0.646386183 113.6 0.620765200 124.4 0.622

Mean: Stardard Error:0.622187 0.054057

Elasticity 0.6362

Initial Height

Rebound Height

Elasticity

30 21.6 0.7247 37.4 0.79574564 51.6 0.8062581 66.1 0.81604998 76.4 0.779592115 91.5 0.795652132 111.2 0.842424149 124.3 0.834228166 141.1 0.85183 150.2 0.820765200 158.2 0.791

Mean: Standard Error:0.804701 0.035904

Elasticity 0.817

Table 2: The data set of big ball Table 3: The data set of small ball

20 40 60 80 100 120 140 160 180 200 2200

20

40

60

80

100

120

140

160

180

f(x) = 0.816970637181621 x

f(x) = 0.636237836008236 x

The Elasticity of Rubber Balls

Big BallLinear (Big Ball)Small BallLinear (Small Ball)

Initial Drop Height (cm)

Rebo

und

Heig

ht(c

m)

Figure 1: The plot of Dropped height vs. rebound height of the big ball and the small ball

Appendix 2: Residual Graphs

20 40 60 80 100 120 140 160 180 200 220

-4

-3

-2

-1

0

1

2

3

4

Big Ball Residual Plot

Elasticity

Resid

uals

Figure 2: The residual plot of the big ball

20 40 60 80 100 120 140 160 180 200 220

-6

-4

-2

0

2

4

6Small Ball Residual Plot

Elasticity

Resid

uals

Figure 3: The residual plot of the small ball

Appendix 3: Linear regression data

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.999688R Square 0.999376Adjusted R Square 0.888265Standard Error 2.228678Observations 10

ANOVA

df SS MS FSignifican

ce F

Regression 171578.9

9 71578.9914410.

9 2.59E-14

Residual 944.7030

4 4.967005

Total 1071623.6

9

Coefficients

Standard Error t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 0 #N/A #N/A #N/A #N/A #N/A #N/A #N/A

Elasticity 0.6370690.00530

7 120.04549.81E-

16 0.6250640.64907

40.62506

40.64907

4

RESIDUAL OUTPUT

Observation

Predicted 14.2

Residuals

Standard Residuals

1 29.94225-

2.54225 -1.2024

2 40.772430.92757

5 0.4387133 51.6026 -0.0026 -0.00123

4 62.432781.06722

4 0.504762

5 73.262953.03704

8 1.436426

6 84.093131.20687

3 0.570812

7 94.92332.27669

7 1.076804

8 105.75351.54652

2 0.731455

9 116.5837-

2.98365 -1.41117

10 127.4138-

3.01383 -1.42544

Table 4: The linear regression data of big ball

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.999592R Square 0.999184Adjusted R Square 0.888073Standard Error 3.270764Observations 10

ANOVA

df SS MS FSignifican

ce F

Regression 1117855.

9 117855.911016.

73 7.58E-14

Residual 996.2810

7 10.6979

Total 10117952.

2

Coefficients

Standard Error t Stat P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 0 #N/A #N/A #N/A #N/A #N/A #N/A #N/A

Elasticity 0.8174650.00778

8 104.96063.28E-

15 0.7998470.8350

840.7998

470.8350

84

RESIDUAL OUTPUT

Observation

Predicted 21.6

Residuals

Standard Residuals

1 38.42088-

1.02088 -0.32901

2 52.31779-

0.71779 -0.231333 66.2147 -0.1147 -0.03697

4 80.11162-

3.71162 -1.19617

5 94.00853-

2.50853 -0.80844

6 107.90543.29455

6 1.06176

7 121.80242.49764

3 0.8049338 135.6993 5.40073 1.740532

9 149.59620.60381

7 0.19459610 163.4931 -5.2931 -1.70584

Table 5: The linear regression data of small ball

Appendix 4: The single trial error calculation

Big ball:

SE=( S H 1

H 1

+S H 0

H 0)E

Sub in values…

¿( .03.671 + .00051 )(0.671 )

¿0.03

Small ball:

SE=( S H 1

H 1

+S H 0

H 0)E

Sub in values…

¿( .03.850 + .00051 ) (0.850 )

¿0.03