Top angle and the maximum speed of falling cones
Guiding student investigations
J.M. Wooning, A.H. Mooldijk and A.E. van der Valk
Centre for Science and Mathematics Education, Utrecht University, Netherlands
Email: mailto:[email protected]
As a research assignment, students
investigated the movement of falling
paper cones. In their guiding, teachers
could not adequately answer two
questions students asked them. One
about the point from what falling
distance onwards the speed of the cone is
surely at its maximum and one about
the relation between the top angle and
the maximum speed. To answer these
questions, the relation between the top
edge and the drag coefficient is needed.
In this article a formula is constructed
using adapted Newton theory about the
cone. The relation is validated
experimentally using a CBL
environment and relevant coefficients
are determined. Results are used for
suggesting teachers how to guide
students to make a ‘reasonable guess’
for the ‘maximum speed distance’ and
to investigate the relation between the
top angle and the maximum speed of
cones with equal masses.
Osborne (1996) has made a plea for a shift
from practical work to research and
discussion activities. In a recent curriculum
reform for upper secondary education in
the Netherlands this shift is promoted by
introducing a final research assignment as
a part of the examination. As preparing for
it crosses the boundaries of the single
subjects (Millar et al. 1994), there is a need
for cooperation between school
departments. In 1997 our institute started
the BPS-project, which aims at guiding the
Biology, Chemistry, Physics and
Mathematics departments in schools to
cooperate (Broekman and Van der Valk,
1999).
The departments of four project schools
agreed to conduct a research assignment
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incorporating two subjects. The one on a
Physics and Mathematics topic is reported
in this study: Falling Cones. The goal was
to have students experience the research
process from the start to the end.
The falling cones research
assignment
The falling cone research assignment is
about the construction of paper cones and
their movement when they are released. In
groups, students explore the cones and
their motion, orient themselves to relevant
variables and formulate a research
question. They have to plan and do
experiments or mathematics activities in
order to get an answer to that question.
The teacher starts with the introduction of
the paper cone. Its movement is studied,
showing a rapidly diminishing
acceleration, until its maximum speed is
reached. Then pairs of cones are studied,
among others two cones of different sizes,
having the same top angle and made of the
same paper (see figure 1). To students’
surprise, these two cones fall at equal
speed. Students are challenged to make
themselves a couple of cones and study
their movements. Next, working in groups,
they have to formulate a research question,
to study some aspect of the cone or of its
Figure 2. Scheme of a cone with variables
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movement in depth, theoretically or
experimentally. For this, an investigation
design has to be made and implemented.
At the end, groups present the answer to
their research question to each other on a
poster.
In figure 2, a scheme of the cone is drawn
and relevant variables are shown. Using
Physics and Mathematics theory, students
can understand how the maximum speed
(vmax) of the cone is related to variables
investigation. No complete answers should
be given, as it would make students’ own
contributions unnecessary. Instead, the
teacher can ask for reasons, explanations,
can answer questions by asking questions
in return, can offer suggestions etc., in
order to stimulate them to make concrete
what they want to know and investigate.
To provide adequate guidance, teachers
have to have a thorough understanding of
the cone issue and how to investigate
(Tamir 1989).
The issue of this study
Figure 1. Paper cones falling down.
such as its mass (m), the radius (r) of the
ground circle, the air density (ρ) and the
drag coefficient (C):
Fgrav = Fdrag
m⋅ g =
v
2
max
1
2
2
CAρv max
mg 2
= 2⋅
πr Cρ
(1)
(2)
(3)
Using these formulae and the drawings of
figure 2, students can design an
experimental or a theoretical piece of
research.
The teacher guides the student groups
when they are trying to formulate a
research question, making a research
design, doing experiments or mathematical
activities. In order to learn how to conduct
a research, teachers have to give students
many opportunities to determine on their
own what to investigate, how to do and
why to do it this way. At the same time,
teachers have to show students some
direction, in order to get a fruitful
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The falling cones investigation assignment
was done in four classes (form 11). The
materials for doing the experiments were
simple: rulers, stopwatches, scales etc. No
advanced equipment, such as a position
sensor, was available.
Experiences have shown that students were
challenged to ask questions about the
motion, about variables being relevant for
the maximum speed, to probe relationships
and to do experiments. However, students
met two content matter problems that may
hinder their learning how to conduct a
research, as the teachers appeared to have
no means for guiding them to a solution.
This study was done to solve this.
First problem
Experiments about how the maximum
speed of the cone depends on the mass, the
shape of the cone, its radius etc. appeared
to be favourite with students. However,
trying to measure the maximum speed, a
problem was met: what distance does the
cone have to fall for its speed to become
constant?
From equation (3) it can be seen that one
needs to have the value of the drag
coefficient C for calculating that distance.
The drag value depends on the shape of the
cone, but no formula is available. So,
teachers suggested to make a reasonable
guess or said: just assume that after
falling1 meter the speed is at its maximum.
The students accepted these suggestions,
though reluctantly. One group argued
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rightly: The start phase of the fall may be
comparable to a cone, that are found in the
literature (table 1).
Flat circle
Open half sphere
Drop of water
1.11
0.34
0.06
Table 1. Drag coefficient of some different
forms (Vademecum 1995)
Figure 3. Transcription of the group’s
drawing quoted in the text, where 1 meter is
used for “start phase”.
dependent on the mass of the cone. In a
drawing (see figure 3), they showed what
they meant by ‘start phase’.
Second problem
One group studied how the maximum
speed depends on the top angle. They
found that the bigger the top angle is, the
slower the maximum speed. For getting a
formula for this relation, they studied the
theory and concluded: If you insert all
known values into formula (3), only the
drag coefficient and the speed are left as
variables. So you can insert vmax. Then, the
drag coefficient only depends on the shape
of the cone, so we ‘ll get three different
values of C. They asked their teacher what
formula to try. He suggested
−1
2
v max
∝ (sin (γ )) but in fact had no basis
for it.
Both issues can be solved by getting an
answer to the question: how does the drag
value depend on the top angle?
Design of the study
The drag coefficient C is dependent on the
top angle of a cone. This can be illustrated
by looking at the C values of some forms
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The flat circle equals a cone with a top
angle of 180 degrees. Cones with sharper
top angles will have smaller drag
coefficients than the flat circle. The cone C
values can be expected to equal the open
half sphere at minimum, as the half sphere
equals the bottom shape of the ‘ideal’
water drop and has an open side, like the
cone. So the cone drag coefficients will
have values between 0.34 and 1.1.
In the theory part of the study, the relation
between the drag coefficient and the top
angle is explored by using Newton’s model
of a cone falling through a uniform
medium, as it is elaborated by Edwards’
internet site (see references). This resulted
in a formula for the drag coefficient,
predicting a sinus square relation with the
top angle.
In the experimental part, a set of different
cones was made, the only variable being
the top angle. Using a TI position sensor
and Coach 5 (CMA 2000) environment,
the drag coefficient of all cones was
determined and results were compared
with the values theoretically predicted by
the Newton/Edwards model. In that way, a
relationship was found.
The Newton/Edwards model
Newton already studied the resistance that
falling objects experience when falling
through a homogeneous medium. On his
website, C.H. Edwards presents a
simplified version of the complicated
Principia Mathematica theory. Here, we
use the Newton/Edwards model to find a
relationship between the cone drag
coefficient and the top angle.
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Newton assumed that the medium - air in
our case - consists of tiny elastic particles
with mass m, uniformly distributed in
space, having no speed. So, Newton (and
in the vertically upward direction. The
number of particles that strike a piece dA
of the cone surface in time ∆ t is
N dA = Nv ⋅ cos(ϕ ) ⋅ dA ⋅ ∆t
(7)
Using the equality of change of impulse
and momentum, ρ being the density of the
medium the cone is falling in, with
ρ = Nm
(8)
the vertical force on the surface piece dA
from air drag is
dFdrag = 2 ρv 2 ⋅ cos 3 (ϕ ) ⋅ dA
(9)
The air drag on the total cone surface S is
given by the surface integral:
(10)
F w = ∫∫ 2 ρ v 2 ⋅ cos 3 (ϕ ) ⋅ dA
S
Figure 4. A particle colliding elastically
with the cone.
Edwards) only account for collisions of
particles on the bottom surface of the cone,
which is moving vertically downward
having speed v. If the cone is supposed to
be at rest, the air particles are moving
vertically upward with speed v. As the
directions of the velocities of the actual air
molecules are uniformly distributed in
space, this model can be applied here as a
first order approximation. It has to be
noted that turbulence is not accounted for.
Be N particles per unit of volume. All
particles hit the cone surface at angle φ
with the surface normal and bouncing of
with the same angle. From figure 2 and 4 it
can be seen that φ equals π/2 - γ/2. If a
particle collides elastically with the cone,
its momentum p changes by
∆p particle = 2mv ⋅ cos(ϕ )
(4)
along the normal on the outside cone
surface. This results in the opposite change
of momentum of the cone:
∆p cone = 2mv ⋅ cos(ϕ )
(5)
As the horizontal components of the
momentum of the particles that bounce on
the cone are cancelled out, only the vertical
components count. So one collision
contributes:
∆p cone = 2mv ⋅ cos 2 (ϕ )
(6)
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This equation can be written as follows:
Fw = 2 ρv 2 R
(11)
R equalling:
R = ∫∫ cos 3 (ϕ ) ⋅ dA
(12)
S
According to Edwards, you can write for
the coefficient of resistance:
r
R
2x
=∫
⋅ dx
(13)
π 0 1 + y ' ( x) 2
with y(x) the function that determines the
shape of the falling body. In the case of the
cone, you can write for that function
y ( x ) = ax
(14)
a being
sin ϕ
a = tan ϕ =
(15)
cos ϕ
So y’ = a.
Substitution of the requirement (14) for the
cone into (13) gives:
(16)
R
π
r
r
2
2x
2x
r
⋅ dx = ∫
⋅ dx =
2
2
1 + a2
0 1 + y ' ( x)
0 1+ a
=∫
Insert
ing (15) into (16), using ϕ = π − γ and
2
2
A = π r2
one gets:
R = A ⋅ cos 2 (π 2 − γ 2 ) = A ⋅ sin 2 (γ 2) (17)
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So we can write the drag force:
Fdrag = 2 Aρv 2 ⋅ sin 2 (γ 2 )
(18)
Assuming that (18) and (2) are identical,
one finds for the drag coefficient:
C (γ ) = 4 ⋅ sin 2 (γ 2)
(19)
Critical evaluation of this result: it predicts
that C = 4 when the top angle is 180˚ and it
tends to 0 when the top angle tends to 0˚.
So theory based on the Newton/Edwards
model predicts 0 < C < 4.
However, using experimental data, it is
expected 0.34 < C < 1.1. This shows that
the model is not adequate. It only accounts
for drag at the bottom end of the falling
cone. At the upper side turbulence effects
are to be expected, that are not dependent
on the top angle (if the upper side area is
not too small, conf. the open half sphere).
This suggests that the drag coefficient
depends on the square sinus of the half top
angle:
C (γ ) = a + b ⋅ sin 2 (γ 2)
position of the falling cones at different
moments of time. Using an interface
(Coachlab II, CMA 2000), the sensor
signal was sent to the computer and
processed by the program Coach 5 (CMA
2000) to produce distance-time and speedtime graphs. From the graphs, maximum
speed of every cone was calculated and the
average of three measurements was used in
the calculations.
Formula (3) was used to calculate the drag
coefficients. Results were plotted in a C,
sin2 (γ/2) diagram.
Measurement difficulties
Before becoming a definitive series of
data, several problems had to be overcome.
First, it was experienced that the cone
tended to deviate towards the wall when
released less than 1 meter from the wall of
the room. This is due to Bernouilli effects.
Second, the height chosen (2 m) appeared
to be too little for the sharper cones to
leave a sufficient distance for measuring
(20)
a and b being coefficients.
The experiments
To study the relation between the drag
coefficient and the top angle, a series of
cones were constructed. All cones had the
same ground circle area (A = 150 ± 7 cm2)
but different top angles (between 60o and
120o).
To keep A constant, the radius r of the
ground circle has to be kept constant.
Therefore, an increase of top angle γ means
a decrease of the length l of the cone. This
makes the amount of paper used for the
cones, and thus the mass, vary.
The mass of each cone was made 3.00 +
0.05 g by putting some sand in.
A position sensor (CBR of Texas
Instruments) was used to measure the
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Figure 5. A diagram from Coach5
the maximum speed. Due to the height of
the laboratory room (4,5 m), the falling
height was limited to 4 m. Therefore, no
reliable data could be gathered for cones
with a top angle less than 60o.
Third, problems appeared because of
fluttering, in particular for the larger top
angles. By careful release and repetition,
from a large series of measurements of one
cone, three movements without fluttering
could be selected. This was not possible
for cones with γ > 120o.
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Results
In figure 5, a typical diagram produced by
Coach5 is shown. It illustrates that, some
time after release, the (x, t) graph shows a
straight line, the (v, t) graph reaching a
maximum.
The drag coefficient and sin2(γ/2) are
plotted in figure 7. This diagram confirms
formula (20), the coefficients being
a = 0.40 + 0.04
b = 0.57 + 0.11
Discussion
The linear relation between the drag value
and sin2(γ/2) is confirmed within the realm
of 60o < γ < 120o . The range of the drag
coefficient is:
0.40 + 0.04 < C < 0.97 + 0.15
This is within the range expected
(0.34 < C<1.1) determined by the drag
coefficients of a flat circle and a half open
sphere.
The Newton/Edwards model does not
account for two phenomena:
- particles not only bouncing on the
downward, outside of the cone, but
also on the inner, upward side
- turbulence at the rim and the
upward side of the cone.
It is likely that these two phenomena are
the important factors in explaining the
differences between the Newton/Edwards
model and experimental findings.
Consequences for guiding
students
Having found the formula (20) and the
values of its coefficients, the issue of our
research has been solved. Now, another
question arises: how can a teacher guide a
group of students that ask one of the two
questions that were at the start of our
study?
The first problem described in section 3
was: what distance does the cone have to
fall for its speed to become constant?
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First, an approximation is constructed and
checked and then it is discussed how to
guide the students.
In a first ‘reasonable guess’ prior
knowledge is used: the distance a free
falling body has to fall to reach the
maximum speed is given by formula (21):
2
v max
(21)
2g
This guess is checked by inserting in (21)
the maximum speed from the figure 5
graphs, reproduced in figure 7. The result:
s=
Figure 6. Diagram with the drag coefficient as a
function of the square of the sine of the top
angle
s = 0,3 m, producing the lower horizontal
line in figure 7. Using the crossing point of
this line with the (x, t) graph shows that the
cone is not yet at its maximum speed as it
is not on the linear part of the graph.
The second guess is suggested by figure 7:
double the distance! The upper line at
double distance does cross the linear part.
The same is the case for the other cones we
used. Our conclusion: the cone will surely
be at its maximum speed having passed
distance smax:
v2
(22)
s max = max
g
When students meet problem 1, they
cannot use formula (22), because they
don’t know the maximum speed.
As a solution, the teacher can suggest to
guess from what point onwards the speed
is constant. With it, the approximated
maximum speed can be measured. Using
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that speed, the distance smax can be
calculated with formula (22). Only if this
distance is about the same or more than the
distance of the guess, the measurement has
to be repeated. This procedure takes less
make a reasonable guess using what you
already know; check the result and if
needed, adapt your guess. This method is
an important one in the light of the main
goal of the investigation assignments:
learning about doing investigations.
References
Figure 7. Estimate of distance of falling before
reaching maximum speed
time and is easier to students than the
alternative: calculating vmax theoretically,
using the formulae (3) and (20)!
The second question we started with was:
how does the maximum speed depend on
the top angle? From a guidance point of
view, it is better to transform it into: what
graph is the best to make, to process the
data gathered when studying how the
maximum speed depends on the top angle?
The teacher can suggest the following
procedure to the students:
- calculate the drag coefficient of all
cones from experimental data, using
formula (3)
- try a graph of the drag coefficient and
sin(γ/2) or sin2(γ/2).
When trying to measure the maximum
speed, they will meet problem 1. As that
can be solved now without using formula
20, they can find the formula themselves!
Vademecum van de natuurkunde (1995), blz.
142, Utrecht: Het Spectrum,
Andereck B S (1999) measurement of air
resistance on an airtrack American Journal of
Physics 67 528-533.
Broekman H and Valk van der A (1999)
Autonomous learning in the Upper secondary
science and mathematics curriculum: How to
guide the Teachers? Proceedings of the 24th
annual ATEE conference Leipzig
CMA (2000) Coach5 and Coachlab II at
http://www.cma.science.uva.nl/english/index.h
tml
Edwards, C.H. (1997), Newtons’s Nose-Cone
Problem; The Mathematica Journal 7(1), 64–
71
Millar R, Lubben F, Gott R and Duggan S
(1994) Investigating in the school science
laboratory: conceptual and procedural
knowledge and their influence on performance.
Research Papers in Education, 9 (2) 207 – 249
Osborne J (1996) Untying the Gordian Knot:
diminishing the role of practical work. Phys.
Ed. 31 (5), 271 - 278.
Polya G (1954) Mathematics and plausible
reasoning Princeton Princeton UP
Tamir P 1989 Training teachers to teach
effectively in the laboratory Science
Education, 73 (1) 59 – 69.
TI for information about the used motion
sensor
http://education.ti.com/product/tech/cbr/feature
s/features.html
In this study, the relation between the drag
coefficient and the top angle of a cone is
found: formula (20). Ways of using this
formula (without giving it to students!) to
solve problems students meet, are
suggested. One important, more general
aspect of our suggestions has to be
stressed: the Polya method we described:
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