1
Mass effects on the Terminal Velocity of a Coffee Filter Falling in Air
February 30
th
, 2012
Written By: Ima Cool Performed with: Notso Lame
Question: How does increasing the mass of a coffee filter falling through air affect the
terminal velocity it reaches?
Design:
The mass of the falling coffee filter, independent variable, will be varied by nesting one
to four filters instead the initial filter. The terminal velocity, dependent variable, will be
read from a velocity-time graph generated from the motion sensor. The shape and
surface area of the filter will be controlled. In addition, all trials will take place from the
same height and within the same medium, air.
Materials: 5 coffee filters
electronic balance (±0.1 g)
computer with Data Studio & motion sensor
thermometer (±0.5
o
C)
30 cm ruler (±0.05cm)
meter stick (±0.5cm)
retort stand
Figure 1: Set-up of Materials used for data collection
motion sensor
coffee
filter
2.0m
lab bench
2
Procedure:
1) Materials were set-up according to Figure 1. Within Data Studio, the motion sensor
was connected and a graphical display of velocity vs time was set-up.
3) Air temperature in the room was measured and recorded with the thermometer. The
air temperature was monitored throughout the data collection for any variations. Using
the electronic balance the mass of each coffee filter was measured. The diameter across
the opening of each coffee filter was measured using the 30 cm ruler.
4) Using the 30cm ruler, a single coffee filter was held 20.0 cm from the motion sensor.
The velocity-time graph was generated and the terminal velocity was recorded. This
process was repeated to generate a total of 3 trials.
5) Step 4 was repeated for 2, 3, 4 and 5 filters, nested inside one another (to maintain
surface area and shape of the filter) to generate the data set in Table 1.
Note: Any trial in which the filter `floated` out of the path of the beam was discarded.
Observations:
Table 1: Several trials of the terminal velocity reached by varying masses of coffee filter
as they fall through air
Trial
Number of
Coffee Filters
Mass of Coffee
Filter
(±0.1g)
Terminal
Velocity
(±0.05m/s)
1
1 1.1
1.34
2 1.30
3 1.36
4
2 2.2
1.70
5 1.90
6 1.87
7
3 3.3
2.11
8 2.05
9 2.08
10
4 4.4
2.59
11 2.64
12 2.55
13
5 5.5
2.75
14 2.56
15 2.89
* The coffee filters were dropped from 2 meters above the floor, the air temperature of
the room stayed at 22.2±0.5
o
C, the surface area of the filter increased slightly over the
trials (diameter from 14.1±0.5 cm to 14.9±0.5 cm) and the mass of each coffee filter was
1.1±0.1g
3
Analysis:
Sample Calculations
To determine the average velocity for the 1.1 g filter:
Ave terminal velocity =
=
= 1.33m/s
Table 2: Average Terminal Velocity of Falling Coffee Filters in Air versus The Mass of the Coffee Filter
Mass of Coffee
Filters
(x10
-3
g)
Average
Terminal
Velocity
(m/s [down])
1.1 1.33
2.2 1.82
3.3 2.08
4.4 2.59
5.5 2.73
Figure 2: Terminal Velocity vs Mass of Coffee Filter
The data was fit to a linear trend as well as a power trend. Since the R
2
value of
the power trend is closer to one, it will be used to find the relationship between the two
variables. From the equation, vT = 1.2657m
0.4548
, it is noted that the exponent of 0.4548
can be approximated as 0.5 which then identifies the relationship between the two
variable as the square root function. Hence the terminal velocity of the coffee filter
varies with the square root of its mass. The mass data was modified according to the
vT = 0.3245m + 1.039
R2 = 0.9742
vT = 1.2657m
0.4548
R2 = 0.9881
0
0.5
1
1.5
2
2.5
3
0 0.001 0.002 0.003 0.004 0.005 0.006
T
e
rm
in
a
l
V
e
lo
c
it
y
(
m
/s
[
d
o
w
n
])
Mass of Coffee Filter (kg)
4
sample calculation below and a graph of terminal velocity versus square root of mass was
generated.
Sample Calculation
To determine the square root of the mass of the 1.1 x 10
3
kg filter:
=
= 0.033 √kg
Table 3: Terminal Velocity versus the square root of the mass
Mass of Coffee
Filters
(x 10
-3
kg)
Square root of the
Mass of the
Coffee Filter
(√kg)
Average
Terminal
Velocity
(m/s [down])
1.1 0.033 1.33
2.2 0.047 1.82
3.3 0.057 2.08
4.4 0.066 2.59
5.5 0.074 2.73
vT= 35.147√m + 0.1557
R2 = 0.984
0
0.5
1
1.5
2
2.5
3
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
T
e
rm
in
a
l
V
e
lo
c
it
y
(
m
/s
[
d
o
w
n
])
√ Mass of Coffee Filters (√kg)
Figure 3: Terminal Velocity vs √Mass for Falling Coffee
Filters
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Conclusion & Evaluation:
From Figure 2, it is evident that as the mass of the coffee filter increases, the
terminal velocity for the filter increases. Since the R
2
value of the square root line of best
fit was closer to 1, Figure 3 was constructed to show this relationship that the terminal
velocity of the filter varies with the square root of the mass of the filter. Since the data
generated a linear line, with an R
2
= 0.984, very close to 1, this confirms that the
prediction of VT α √m is valid. In addition, it was determined that the equation between
the 2 variables was vT= 35.147√m + 0.1557.
One source of error is attributed to the surface area of the coffee filter. It was
noted that as more coffee filters were nested inside each other, the diameter across the
opening increased. This would cause the measured terminal velocities to be lowered. It
is suggested that this factor could be reduced by starting with 5 nested filters out of the
box and then remove one at a time so that there is no stretch when adding more filters.
A second source of error occurred in the reading of the terminal velocities. A
judgement had to be made where the velocity-time line levelled off. In some of the trials
the lines were erratic. This could cause the terminal velocities to be either larger or
smaller. A solution would be to have the motion sensor set at a higher sampling rate.
In addition, the trials for the 4 and 5 filters, there wasn't enough time for the filters
to completely reach terminal velocity. This may have caused the terminal velocity
measurements to be higher. A solution to this problem would be to use larger coffee
filters with larger surface areas to lower the terminal velocities or to have the filters fall
through a larger distance allowing them to reach their terminal velocity.
