1. Zaid Shammas
IB Physics Hl 1: Period 1
November 16, 2010
Freefalling Cupcakes Varied by Formation
Introduction:
In this investigation, the relationship between the amount of cupcakes in a formation and the
change they have on the coefficient of drag as they free fall through the air will be studied. Before the
investigation can be discussed, an understanding of the concepts behind this investigation must be
attained. The most prominent concept in this investigation is terminal velocity. Terminal velocity is a
state achieved when the force of drag on an object is equal to the force of gravity
while free falling, as shown in figure 1.01. Terminal velocity is governed by the equation [1].
[1]
Where:
Vt is terminal velocity
g is the force of gravity
m is the mass of the object
A is cross sectional surface area
ρ is the density of the fluid which the objects is falling through
Cd is the coefficient of drag
Also, it is important to understand that when an object reaches terminal velocity, it is no longer
accelerating nor decelerating (negative acceleration) but in fact, its acceleration is equal to zero. Other
factors to consider are the forces that affect the terminal velocity of an object. The driving force of the
object in terminal velocity is the force that tries to cause the motion of the object to accelerate, such as
gravity, while the resistive force of the object is the force that attempts to decelerate the object and when
these two forces balance out, terminal velocity is achieved.
In this investigation mass and cross sectional surface area, two factors that influence terminal
velocity, will change. In addition, by adding cupcakes to the free falling formation of cupcakes, the
coefficient of drag, or the dimensionless number that is used to quantify the amount of resistance an
object has in a fluid environment such as air, will change1. As more cupcakes are added, the mass of the
system of cupcakes will stay proportional to the cross sectional area. This is because each cupcake being
added is similar. As cupcakes are added, it is expected that the coefficient of drag will increase in a square
power function. If equation [1] is manipulated to give a function of coefficient of drag, it will appear as
shown below.

2. This equation is expected to yield a linear relationship if a graph is plotted, where the y axis will be
defined as the (coefficient of drag)2., and the x axis will be defined as the number of cupcakes in a
formation.
Design:
The research question of this investigation is to understand what relationship a falling formation
of cupcake has to the system’s coefficient of drag, and what happens to the coefficient of drag when
cupcakes are added to the formation. The independent variable of this investigation the amount of
cupcakes there are in a formation. The dependent variable is the coefficient of drag. There are multiple
control variables. Mass must stay proportional to cross sectional surface area to remain as a control, so
similar cupcakes must always be added to the formation, shown in figures 1.0 and 1.1
Figure 1.0 Figure 1.1
Control variables such as air disturbances and air temperature will be kept constant by turning off
the air conditioners in the room. This allows for less temperature change, and less changes in the flow of
air around the room. The placement of tape connecting the cupcakes together must be consistent. Each
cupcake must be gently handled, and tape connecting the cupcakes together must be applied consistently,
where it does not obstruct airflow through the ridges and holes around the cupcakes. A height that the
cupcakes are dropped from must be established, to allow for the same time of free fall for every
formation. The cupcakes are only viable for a number of trials before their structure breaks down.
Therefore, if the same formation of cupcakes is used for an extended amount of trials, the data becomes
invalid; the structure of the cupcakes must be maintained.

3. Around 30 cupcakes, a motion detector system, and tape are required for this investigation.
Cupcake formations should be made first, as the cupcakes are connected gently with tape. For this
investigation, cupcake formations of one, three, four, seven and thirteen were created. A set height of 1.4
meters was established, and the investigation was ready. Each formation of cupcake was dropped from
the set height right above the motion detector three times. This provides three trials for five data points, to
increase confidence in the research. The wide range number of cupcakes in the formation is to ensure a
wide range of applicability, and to increase confidence within in research. The research setup should
resemble figure 2.0.
Figure 2.0
Data Processing and Analysis:
Table 1: Trial data as a cupcake formation free falls, with mass measurements.
Terminal Velocity (m/s) (± 0.01)
Number of Total Mass Total Mass (kg) Average Speed
Trial 1 Trial 2 Trial 3
Cupcake Holders (g) (±0.01) (±0.00001) (m/s) (± 0.03)
1 0.51 0.00051 1.75 1.73 1.69 1.72
3 1.62 0.00162 1.56 1.52 1.52 1.53
4 2.18 0.00218 1.46 1.53 1.43 1.47
7 4.01 0.00401 1.39 1.33 1.41 1.38
13 7.65 0.00765 1.27 1.39 1.39 1.35
Table 1: Table of values that shows the total mass of each cupcake formation, along with trial data for
each free fall. The terminal velocity was calculated with a motion detector, as the slope of the position
time graph of the falling cupcake formation. The average speed is an average calculated column that
reads the mean of the three trials of each data point.

4. Graph 1: Sample position-time graph of trial one, where a formation of one cupcake free falls.
Graph 1: The linear fit is placed where the cupcake seems to experience terminal velocity. The slope of
the linear fit determines terminal velocity. In this sample, the magnitude of terminal velocity is noted as
1.75 m/s. Also, a starting height of 1.4 m is visible on the y axis, where the cupcake does not seem to fall
yet.
Table 2: Cross sectional surface area of each formation of cupcakes.
Number of Cross-sect Surface Area Cross-sect Surface
Cupcake (cm2) (± 1 cm2) Area (m2)
Holders (± 0.0001m2)
1 50 0.0050
3 151 0.0151
4 201 0.0201
7 352 0.0352
13 654 0.0654
Table 2: The cupcake holders used have an exposed radius on the bottom of the cupcake of 4 cm. When
used to attain the surface area that is exposed while free falling, 50.3 cm2 is the result, from the
equation . For multiple cupcakes in the formation, the cross sectional surface area for one
cupcake is multiplied by however many cupcakes are present.

5. Table 3: Coefficient of Drag and (Coefficient of Drag)2 for each formation of cupcake holder
# Cupcake Holders Coefficient of (Coefficient of Drag)2
in Formation Drag (±0.04) (± 0.04)
1 0.56 0.32
3 0.75 0.57
4 0.82 0.68
7 0.99 0.99
13 1.06 1.11
Table 1: Table of values that shows the coefficient of drag for each formation of cupcakes. As shown, the
coefficient of drag increases as cupcake holders are added to the formation. This means that as
cupcakes are added, the resistance the formation has to air resistance increases. Formations from one
to seven also show a linear relationship to the coefficient of drag 2, shown in graph 2.
Graph 2: Coefficient of Drag 2 over the number of cupcake holders in a formation
Graph 2: Final concluding graph that shows the relationship between the number of cupcake holders in
a similar formation and their coefficients of drag. The y axis is the coefficient of drag, and it is squared to
allow a linear relationship. This shows that the relationship between the coefficient of drag and multiple
formations of cupcake holders is a square function. However, this is only applicable to a certain extent,
since the formation of 13 cupcakes does not fall in the linear relationship.

6. Sample Calculations:
i. Average Terminal Velocity
(Trial 1 velocity + Trial 2 velocity + Trial 3 velocity)/3
(1.75+1.73+1.69)/3
1.72 m/s
ii. Uncertainty of Average Terminal Velocity
(Highest velocity trial-Lowest velocity trial)/2
(1.75-1.69)/2
0.03 m/s
iii. Cross Sectional Surface Area
(π*(Diameter/2)2)/1000
(π*42)/1000
(50.3 cm2)/1000
0.0053 m2
iv. Uncertainty of Cross Sectional Surface Area
(π((diameter + 0.1)/2)2 – π((diameter-.01)/2)2)/2
(π(8.1/2)2-π(7.9 /2)2)/2
(51.53-49.02)/2
1.255
± 1 cm2
v. Coefficient of Drag
(2gm) / (Aρ(Vt2)
(2*9.8*0.00051) / (0.005*1.19*1.722)
0.56
vi. Uncertainty of Coefficient of Drag
((2*g*highest m) / (lowest A*ρ*lowest Vt2)- (2*g*lowest m) / (highest A*ρ*highest Vt2)) / 2
((2*9.8*0.00052) / (0.0049*1.19*1.692) – (2*9.8*0.0005) / (0.0051*1.19*1.752))/2
(0.611988 – 0.527269) / 2
± 0.04

7. Conclusion and Evaluation:
The research conducted is aimed to find the relationship between the amount of cupcakes in a
formation while free falling and the change in coefficient of drag. By manipulating the data, it is found
that the relationship is a square function, but supported only by values when the coefficient of drag is
less than 1. This means that as cupcakes are added, the coefficient of drag decreases in an exponential
fashion. The data that was attained supports this conclusion with confidence, since the uncertainty on
the derived coefficient of drag is only around 7%.However, the last formation, with 13 cupcakes, does
not support this conclusion. As a result, it is not included in the final graph, and shows that the
applicability of this relationship only lasts for a small range of cupcake holders. The coefficient of drag
attained for the cupcake formation of seven cupcakes was 0.99. This figure, when squared, is also equal
to 0.99. This shows that the range of applicability of this research ends at a formation of seven cupcakes,
following the procedure carried out. When the coefficient of drag exceeds 1, the square function does
not apply, as shown with the formation of 13 cupcakes, where the coefficient of drag is 1.06. When the
coefficient of drag exceeds 1, the square does not decrease the coefficient of drag, but rather increases
it, and it does not lie on the linear fit of Graph 2 anymore. While this conclusion holds true for this case,
different types of cupcake holders and formations must be investigated to further this concept.
There were many weaknesses in this investigation. As the cupcakes were handled and dropped
over and over again, the structure deteriorated, and it was ultimately not controlled. Taping the cupcake
holders together was also a problem, as it added additional weight to the formation, and changed the
shape of the cupcake holder. This affected the mass to surface area ratio, and skewed the data, because
it was assumed that mass was proportional to cross sectional surface area. In addition, every cupcake
used was different. This weakness was not controlled, and cannot be controlled very well unless the
cupcakes are made of a different material that will allow the structure not to deteriorate while being
handled and attached together. Special cupcakes must be used where external sources are not used to
attach them together. The mass to cross sectional surface must stay constant throughout further
research for data to be reliable. ρ, or the density of the air, was a weakness in this investigation. A value
of 1.19 was used, but in reality, the temperature of the room in which the investigation was carried out
fluctuated, since it took nearly an hour to compile all the data. Along with air temperature was the issue
of wind or turbulence within the room. Slower moving cupcake formations had prolonged exposure to
the air in the room, which could have had an impact. Controlling this in future research may include
dropping the formations down a tube, where the temperature and wind or turbulence is controlled.
Further research may be carried out, varying the types of cupcake holders used, and the ways the
formations are put together. This investigation only tested one type of cupcake holder, and in
symmetrical formations. These variables can be changed, and observed to test whether they agree with
the conclusion from this investigation.