FREE FALL TIME OF
PAPER CONES
Project Report of DOE
Jiankun SUN, Nan CHEN,
Donghui LI, Quan YUAN,
Tianyang XU
June, 2013
DIRECTORY
1
2
3
4
5
6
Introduction ............................................................................................................. 2
Lecture review ........................................................................................................ 2
Factor decomposition .............................................................................................. 4
3.1 Assumption ................................................................................................... 4
3.2 Response variable ......................................................................................... 4
3.3 Input variable ................................................................................................ 4
3.4 Control variable ............................................................................................ 4
3.5 Nuisance factor ............................................................................................. 4
Pre-experiment ........................................................................................................ 5
4.1 Design of pre-experiment ............................................................................. 5
4.2 Results of pre-experiment ............................................................................ 5
4.3 Analysis of pre-experiment .......................................................................... 7
4.3.1 Reproducibility of experiment ........................................................... 7
4.3.2 Influence degree of variables ............................................................. 7
Formal experiment .................................................................................................. 9
5.1 Experiment design ........................................................................................ 9
5.1.1 Variables level setting ........................................................................ 9
5.1.2 Full factorial design ........................................................................... 9
5.2 Experiment process ....................................................................................10
5.3 Data analysis...............................................................................................11
5.3.1 Descriptive statistics ........................................................................11
5.3.2 Scatterplot ........................................................................................12
5.3.3 Main effects .....................................................................................13
5.3.4 Interaction plot .................................................................................14
5.3.5 General linear model ........................................................................14
5.4 Result analysis and discussion ...................................................................18
Conclusion ............................................................................................................18
1 Introduction
Our project is motivated by a famous experiment in the high school physic textbook.
Originally, the experiment only studies the free fall time of the paper cone with the
same radius and different angle. We hope to do some further research of this topic by
adding more variables and creating a full factorial design, and try to establish a model
to describe the relationship between the free fall time and the variables. We think that
this more general design of the experiment will help us get a better idea of this physic
problem.
2 Literature review
We find a research paper1 relevant to what we want to study. From the paper we get
some basic design ideas:
1.
How to make the paper cone?
We use ordinary A4 size paper and cut it into a fan with different angle. Then fold
the paper into a cone, using tape to connect. Showed in the figure below.
Figure 1 The making of the paper cone
2.
How to set the variable?
According to the paper, the free fall time may be influenced by height, angle,
radius and the thickness of paper under our hypothesis. We will design the factor
and the factor level under this consideration.
3.
How to measure each variable?
We will a camera to record the whole experiment procedure. It will bring benefit
to our later data collecting and analysis. Also we make sure that every paper cone
is under the same quality. We try our best to maintain the quality of the paper cone
we make at the same level.
1陆文彬.
浅探纸锥下落快慢与纸锥大小的关系[J]. 物理实验, 2010, 30(10): 19-23
4.
Some guesses of the result and the explanations
The paper has done some simple experiment about this topic and come up with
the conclusion that the radius has nothing to do with the free fall time. Furthermore,
prove it by mathematic analysis.
Figure 2 Design with different angle
Figure 3 Design with different radius
However, the experiment only studies the free fall time of the paper cone with the same
radius and different angle. Besides, it derives a complicated equation to describe the
relationship between the acceleration and velocity of the paper cone and the impacts
from paper cone properties, which is not intuitive and practical to evaluate the free fall
time of a given paper cone. Therefore, we hope to explore the influence of more
variables on the free fall time and try to establish a relatively simple model describe
such influence.
3 Factor decomposition
3.1 Assumption
We design the experiment based on the following assumptions:


The paper of the same type has the same thickness and quality.
The wind speed of the environment is slow and we can ignore its effect.
3.2 Response variable
Response
variable (units)
Free fall time
Normal operating
level & range
0~5s
Means,
precision
Video analysis,
0.1 sec
Relationship of response
variable to objective
Estimate mean value to
reflect the objective
3.3 Input variables
Input variable
(units)
Paper thickness
Normal level
& Range
2 levels: thick/thin
Predicted effects
(For various responses)
Thick paper result in shorter free fall time
Cone radius
2 levels: small/large
Has nothing to do with the time
Central angle
10 levels: 90°~270° Smaller angle result in shorter free fall time
3.4 Control variable
We design the experiment at the same height.
3.5 Nuisance factor
There are several noises in this experiment. First, the performance of the operator may
vary from person to person, but we do some training before the experiment. Therefore,
we assume that they will not affect the experiment. Also the error of the time recorded
will be neglected.
4 Pre-experiment
4.1 Design of pre-experiment
Before the formal experiment, we want to check the repeatability and reproducibility
of our experiment and have a general idea of the influence degree of factors, paper
radius, cone radius and central angle, on the free fall time. Here, repeatability means
that the experiment results are stable among different operations by the same operator,
and reproducibility means that the experiment results are stable among operations
among different operators,
Then we design a pre-experiment to check the free fall time of paper cone at different
factor levels by operator Jiankun Sun and Vincent Chen. The variables process
parameter and their selected range is shown in table 1.
Table 1 Actual and code values of the variable parameters in pre-expriment
Code
-1
1
Paper Thickness
Thin
Thick
Cone Radius
5 cm
10 cm
Central Angle
150°
270°
The code of the two operators are -1 and 1 for Jiankun Sun and Vincent Chen,
respectively. Both operators run the experiment for three times, coded 1, 2 and 3.
4.2 Results of pre-experiment
After analyzing the video and we get the following data.
Table 2 Results of Pre-experiment
Operator
Group No.
Cone
Radius
Paper
Thickness
Central
Angle
Free Fall
Time
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
2
1
1
1
1
-1
-1
-1
-1
1
1
1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
1
-1
1
-1
1
2.77
1.80
3.87
2.43
2.73
1.80
3.70
2.10
2.80
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1.77
4.17
2.40
2.63
1.70
3.80
2.07
2.80
1.77
3.87
2.40
2.63
1.73
3.77
2.17
2.87
1.80
4.00
2.43
2.70
1.70
3.73
2.13
2.83
1.77
4.00
2.43
2.73
1.73
3.77
2.13
2.73
1.80
4.03
2.47
2.70
1.77
3.90
2.13
4.3 Analysis of pre-experiment
4.3.1 Reproducibility of experiment
First we check whether this experiment can be repeated, that is to say, whether there is
significant difference between data of different operators. We show the descriptive
statistics of two groups of data, apply pair t-test and get the following results.
Figure 4 Descriptive Statistics
The descriptive statistics shows that the two groups of data have very similar mean and
standard deviation. The minimum, median and maximum are also very close.
Figure 5 Pair t-test results
From the pair t-test, we can get that the 95% confidence interval for difference is
(−0.0585, 0.0058), which includes 0. The p-value is 0.103, we cannot reject the null
hypothesis at a 0.95 significance level.
Hence, we can find that there is no significant difference between different operators
and this experiment can be repeated.
4.3.2 Influence degree of variables
From the results of pre-experiment, we can also get a general idea of the influence
degree of variables on the free fall time of paper cone.
Then we check the main effects plot and interaction plot of the free fall time as follows:
Figure 6 Main Effects Plot
From the main effects plot, it seems that cone radius has no influence or a very weak
positive influence on the free fall time, the paper thickness has a negative influence,
while central angle has a significant positive influence.
Figure 7 Interaction Plot
According to interaction plot, these three variables seem to have no significant
interaction since the lines are nearly parallel of different levels.
5 Formal experiment
5.1 Experiment design
5.1.1 Variables level setting
Cone radius is coded as -1 and 1; -1 for 5cm and 1 for 10cm. Paper thickness is also
coded in two levels; the actual value, however, is defined indirectly: -1 is for the A4
paper with the weight of 70 g/m2 and 1 for the A3 paper with a relatively heavier weight
of 200 g/m2. Central angle is defined as a continuous variable with the range from 90°
to 270°. In the experiment, we take the angle value with 20° as an interval to
approximate it with discrete value. All in all, the variables we set are shown below. In
the following analysis, we use radius, thickness, angle and time to represent the three
variables respectively and the response variable for simplicity.
Table 3 Actual and code values of the variable parameters in formal expriment
Cone Radius
Paper Thickness
Central Angle
-1
1
-1
1
continuous variable
5 cm
10 cm
70 g/m2
200g/m2
(90+20k) ° (k=0,1,2…9)
5.1.2 Full factorial design
A full-factorial design is taken considering radius and thickness. Due to the ten value
levels of angle, 40 pilots are required.
5.2 Experiment process
The experiment is undertaken in the hall of Shunde Building at 8:00 pm on Friday, with
few people in the hall, thus the experiment moves smoothly. 40 paper cones are made
previously for the experiment.
Figure 8 Paper cones for experiment
For each pilot, 3 replications are made and there are altogether 120 times of free falls
of the paper cones. The free fall is taken place at the platform of the stairs in the hall,
and each free fall is taken as follows: the operator stretch out the arm horizontally and
hold the paper cone with thumb and index finger only, and then open two fingers at the
same time to release the cone, after a short interval, the experimenter holds another
cone and undertakes it again. A camera is used to record the whole process so as to find
out the time needed.
Figure 9 Free-fall place
Figure 10 Free-fall posture
5.3 Data analysis
5.3.1 Descriptive statistics
First we check the descriptive statistics for our data.
Figure 11 Descriptive statistics
From the result above, we can see that for variable thickness, it may influence the free
fall time; while for the variable radius, it may not be that obvious to conclude that the
paper cones with different radius will spend different time for the free-fall. To find the
conclusion, further analysis is needed.
5.3.2 Scatterplot
Then we can use scatterplot to get a general impression of the data.
Scatterplot of Time vs Angle
5
Radius Thickness
-1
-1
-1
1
1
-1
1
1
Time
4
3
2
1
100
125
150
175
200
Angle
225
250
275
Figure 12 scatter plot
From the scatterplot, we can find there is a linear relationship between time and angel.
Also, the thickness will affect the time a lot, but the radius has little effect on time.
This scatter plot can also help find outliers. We can see for small angle (90 degree and
110 degree), the variance is very large because these paper cones cannot fall smoothly.
The paper cones on the black line (thin paper cones with small radius) also cannot fall
smoothly. They may tilt in the air and cannot fall along a straight line perpendicular to
the ground. We can remove these outliers and get the following scatterplot.
Scatterplot of Time vs Angle
5.0
Radius Thickness
-1
-1
-1
1
1
-1
1
1
4.5
Time
4.0
3.5
3.0
2.5
2.0
120
140
160
180
200
Angle
220
240
260
280
Figure 13 Scatter plot of modified data
This time the data fits the line better and there are relatively nice linear relation
representation.
5.3.3 Main effects
It is necessary to have a general idea of the effect on the response variable of each
variable we choose. Thus a main effect graph is plotted, shown below.
Main Effects Plot for Time
Data Means
Radius
Thickness
3.6
3.2
2.8
Mean
2.4
2.0
-1
1
-1
Angle
1
Replication
3.6
3.2
2.8
2.4
2.0
130
150
170
190
210
230
250
270
1
Figure 14 Main effects plot
2
3
From the graph, one thing that should be pointed out is that we add the replication in
this graph to check whether the experiments are stable. Form the main effects plot, we
can get that angle and thickness will affect the time significantly. But radius and
replication number have little effect. The replication number, in our common sense,
does have little effect on the experiment results if the noise is controlled well. As to the
variable radius, we want to study further before jumping to the conclusion of wiping it
out now.
5.3.4 Interaction plot
Interaction Plot for Time
Data Means
-1
1
130 150 170 190 210 230 250 270
4
Radius
-1
1
3
Radius
2
4
Thickness
-1
1
3
Thickness
2
Angle
Figure 15 Interaction plot
From the interaction plot, we can get there is not significant interaction between radius
and thickness or between angle and radius.
But there are interaction between thickness and angle. For thin paper, when the angle
increases, the time will increase faster than thick paper.
We also notice that the two lines, in both the radius versus thickness and that versus
angle, almost coincide. This may suggest that the variable radius have little effect on
the free-fall time.
5.3.5 General linear model
To study in detail, a general linear model is built. Firstly we put radius, thickness and
angle in our model (radius is covariate)
Figure 16 GLM regression
Residual Plots for Time
Normal Probability Plot
Versus Fits
99
0.50
90
0.25
Re s id u al
P er c en t
99.9
50
10
0.00
-0.25
1
-0.50
0.1
-0.50
-0.25
0.00
Residual
0.25
0.50
2
24
0.50
18
0.25
12
0.00
-0.25
6
0
4
Versus Order
R e si d ua l
F re q ue n cy
Histogram
3
Fitted Value
-0.50
-0.4
-0.2
0.0
0.2
Residual
0.4
1
10
20
30 40 50 60 70
Observation Order
80
90
Figure 67 Residual plots of GLM regression
From the result above we can find that considering the ̅𝑅̅̅2̅, the regression leads us to
believe that the time of free-fall of paper cones have strong linear relationship with the
factors we choose. Notice that at the 95% confidence level, the correlation of the radius
and the response variable, time, is not significant, which proves what we guessed
previously. From the residual plot, we find that the residual fits normal distribution
quite well. But we can also find that the residual versus fits is abnormal. A bowl-like
curve has been observed and we guess this may be caused by the interaction between
angle and thickness.
5.3.5.1 Add the interaction
According to the regression result above, we add the interaction of angle and thickness
and do the regression again.
Figure 78 GLM regression with interaction
Residual Plots for Time
Normal Probability Plot
Versus Fits
99.9
0.4
90
Re s id u al
P er c en t
99
50
10
0.2
0.0
-0.2
1
-0.4
0.1
-0.4
-0.2
0.0
Residual
0.2
0.4
2
3
Fitted Value
Histogram
4
Versus Order
0.4
R e si d ua l
F re q ue n cy
20
15
10
5
0
0.2
0.0
-0.2
-0.4
-0.3 -0.2 -0.1 0.0
0.1
Residual
0.2
0.3
1
10
20
30 40 50
60 70
Observation Order
Figure 89 Residual plot of GLM regression (adding interaction)
80
90
After adding the interaction of radius and thickness, we get a relatively higher ̅̅
𝑅̅2̅, and
more importantly, we can find that the residual versus fits shows no obvious regulation.
The effect of the interaction is significant under the 95% confidence level. We can
conclude that there does exist the interaction between angle and thickness.
5.4 Result analysis and discussion
From the experiment result, we can find that the thickness and the angle of the paper
cone has significant correlation with the time taken for free-fall. With a larger angle
(smaller than 270), the free-fall time will increase; this is due to the influence of the air
resistance. The air resistance on the paper cone is related to the velocity and the
horizontally projected area: a higher velocity and a larger horizontally projected area
will lead to a larger air resistance. Here we deduce the process of two paper cones’ freefall with same radius, same thickness, at the same height and only with different angles
(denote 𝑠 as the one with smaller angle, 𝑙 the larger). At the beginning, both cones
fall at zero speed, and 𝑠 will face a smaller air resistance thus a larger acceleration
speed (notice that 𝑠 is also slightly lighter in mass), and gets a larger speed, and finally
falls to the ground earlier.
For paper cones made by different types of paper (denote t as the one made by thin
paper, T as the one made by thick paper), if the angle decrease from α to β, the air
resistance will decrease by similar amount, or we can say the total force on the paper
cones will increase a similar amount. For t, because it has smaller mass, the time will
increase more than T. This can explain why angle and thickness have interaction and
the free-fall time of paper cones made by thin paper will increase more when angle
increases.
Also we can notice the coefficient of thickness has wrong sign, thin paper has negative
coefficient which means it will drop faster. But we can see the coefficient of thickness
fail the t-test, which means it is around zero. This may because the interaction between
thickness and angle is much more significant than the main effect of thickness.
6 Conclusion
Using general regression of Minitab, we can get our final model.
If made by thin paper
Time =
0.461408 + 0.0143581 Angle
Time =
0.461408 + 0.00927134 Angle
If made by thick paper
Figure 20 Final results
Note that the above two equations are only applicable for the specific conditions in
this experiment, including the specific height, paper thickness and other paper
properties and environmental conditions. However, we’ve proved that there does exist
some simple way to evaluate the free fall time of a paper cone. As long as we know
the central angle and some other parameters, a linear model can help us finish this
task. We have to admit that there are some problems remained to be solved for this
experiment because of our limited experimental conditions, and thus we cannot
explore the detailed impact of possible variables on these parameters (the coefficients
got from the model). If conditions permit, it can be explored further in the future.