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An Analysis of Non-
Uniformly Accelerated
Motion for a Quasi-rigid
Object with Varying Mass
By: Mr. Mac
AP Physics C
AP Physics C, G4
9 November 2012
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Abstract
Stacks of coffee filters of varying mass were dropped repeatedly under a motion sensor.
Distance versus time graphs were produced on data-logging software which showed acceleration
decreased to zero over time. Further analysis showed that the terminal velocity, , of each
stack increased with the mass. It was hypothesized that via Newton’s 2nd
Law, acting under ideal
conditions for drag, the relationship between and mass would display a linear relationship,
with a slope equal to (
). The hypothesis was generally supported with a linear slope yielding
a correlation coefficient of .9427. However, the last 6 data points did not fall on the line of best-
fit within uncertainty, and thus reveal significant systematic error. These systematic errors
involved two main sources: 1) the determination of higher terminal velocities was unreliable, and
2) Assumed constants, such as the drag coefficient, varying due to limitations in the procedure.
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Researchable Question and Hypothesis
I. Research Question:
How will the mass of coffee filters influence their terminal velocity, given that the drag
coefficient, air density, and the cross-sectional surface area of the coffee filters are held constant?
II. Hypothesis:
The coffee filter’s terminal velocity squared (VT2) will be directly proportional to its mass, and
the graph will be linear (since VT = √
, thus VT
2 = (
)m, where (
) is a constant). This is
due to the definition of terminal velocity. It is reached as the acceleration approaches zero and
the weight and drag forces are balanced. The drag force set equal to weight yields the
aforementioned relationship twixt terminal velocity and mass. As a result, the graph will be a
line with mass as the independent variable and VT2
as the dependent variable.
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Procedure with Materials
Materials:
1. 10 [No.8] coffee filters, preferably unused and intact
2. Pasco CI-7859 Motion Detector (x1)
3. Mass Recorder/ Digital Weighing Machine
4. Computer, sensors, cabling, and data-logging software, specifically the programs Excel
2012 and Logger Pro.
5. Optional: Timer, Chair for elevation, offed air conditioning to prevent bias from coffee
filters moving.
6. Meter sticks for calibrating distance measurements of sensor.
7. There would be a picture here of setup.
Procedure:
1. Start by placing the motion detector on the ceiling. Have it stick firmly in place, parallel
to the ground with the bean pointed perpendicular to the ground.
2. Initialize the experiment by powering up the sensor and keeping 10 filters together, one
filter resting in another.
3. Before dropping, record mass on a Digital Weighing Machine (Electronic Balance). Note
down the mass seen, with a plus/minus.
4. Using the chair, one person will stand on it and carefully hold the ten stacked filters
approximately six inches away from the beam with bottom parallel to the ground and the
beam reflecting off the center of the top filter. Check this distance using a ruler.
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5. The filters together must open towards the ceiling, the center being lower in height than
the edges; however, the CENTER must be at least six inches away from the motion
detector. The initial position will have no effect on the results, since the terminal velocity
is reached later.
6. Have Logger Pro start recording the value (started by someone else as another is still
holding the filters) and drop the filters approximately five seconds after the program has
started recording values.
7. Stop recording the data around two seconds after the filters have reached the ground and
save the data on Logger Pro.
8. Ensure the graph should be a smooth curve with a slope that shows initial acceleration
decreasing to zero. If there are data that are not smooth trends (like a weirdly reflected
sound wave), redo the trial.
9. With each mass, perform 3 trials, averaging the results.
10. Perform these steps again with only nine filters, followed by eight, and so on until the
trial with only one filter has been attempted.
11. On the raw data graphs of distance versus time, select at least 6 data points corresponding
to the coffee filter moving at a constant velocity (straight linear line). If more can be
selected then great. Do not choose a data point where the graph curves or the graph
shows a discontinuity due to hitting the ground.
12. When selecting these data points, ensure that the r-value is at least 0.99.
13. Record uncertainties from graph.
14. Clean up and print out graphs of raw data.
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Raw Data:
Here would be a nice picture of your notebook with a data table, uncertainties, and observations
recorded. This should be at least a page in your notebook.
Processed Data:
Examined Values
trial mass(kg)
mass+-
(kg)
mass+-
% Vt(m/s)
Vt+-
(m/s) Vt+-% R Vt2
Vt2+-
(%)
Vt2+-
(m/s)
1 0.0011 0.00005 4.5455 1.22 0.01 0.82 1.000 1.4884 1.64 0.0244
2 0.0023 0.00005 2.1739 1.62 0.02 1.23 1.000 2.6244 2.47 0.0648
3 0.0034 0.00005 1.4706 2.03 0.01 0.49 1.000 4.1209 0.99 0.0406
4 0.0047 0.00005 1.0638 2.33 0.03 1.29 1.000 5.4289 2.58 0.1398
5 0.0059 0.00005 0.8475 2.43 0.03 1.23 1.000 5.9049 2.47 0.1458
6 0.0071 0.00005 0.7042 2.42 0.02 0.83 1.000 5.8564 1.65 0.0968
7 0.0083 0.00005 0.6024 2.84 0.03 1.06 1.000 8.0656 2.11 0.1704
8 0.0095 0.00005 0.5263 2.69 0.05 1.86 0.999 7.2361 3.72 0.269
9 0.0108 0.00005 0.4630 3.1 0.04 1.29 1.000 9.61 2.58 0.248
10 0.0118 0.00005 0.4237 3.03 0.04 1.32 0.999 9.1809 2.64 0.2424
Velocity vs. Time
Time v1 (m/s) Time v2 (m/s) Time v3 (m/s)
0.8 2.875 1.02 2.428571 1.12 2.875
0.82 2.875 1.04 2.428571 1.14 2.875
0.84 2.875 1.06 2.428571 1.16 2.875
0.86 2.875 1.08 2.428571 1.18 2.875
0.88 2.875 1.1 2.428571 1.2 2.875
0.9 2.875 1.12 2.428571 1.22 2.875
0.92 2.875 1.14 2.428571 1.24 2.875
0.94 2.875 1.16 2.428571 1.26 2.875
0.96 2.875
1.28 2.875
Time v4 (m/s) Time v5 (m/s) Time v6 (m/s)
0.8 2.13 1.6 2.8 3.55 2.56
0.85 2.13 1.65 2.8 3.6 2.56
0.9 2.13 1.7 2.8 3.65 2.56
0.95 2.13 1.75 2.8 3.7 2.56
1.8 2.8 3.75 2.56
3.8 2.56
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Time v7 (m/s) Time v8 (m/s) Time v9 (m/s) Time v10(m/s)
2 2.1 2.3 1.53 3.5 2.6 4.3 1.28
2.1 2.1 2.4 1.53 3.6 2.6 4.4 1.28
2.2 2.1 2.5 1.53 3.7 2.6 4.5 1.28
2.6 1.53
4.6 1.28
4.7 1.28
4.8 1.28
Sample Calcs
would be hand-
written here!
Calculations Mass uncertainty
(%) = mass
uncertainty
(kg)/mass (kg)*100
Ex: 4.5455% =
0.00005/0.0011*100
Terminal velocity
uncertainty (%) =
terminal velocity
uncertainty
(m/s)/terminal
velocity (m/s)*100
Ex: 0.82% =
0.01/1.22*100
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Terminal velocity
squared uncertainty
(%) = 2*Terminal
velocity uncertainty
(%)
Ex: 1.64% =
2*0.82%
Terminal velocity
squared uncertainty
(m/s) = Terminal
velocity squared
uncertainty (%)*
Terminal velocity
squared (m/s)^2/100
Ex: 0.0244 =
1.64*1.4884/100
(m/s)^2
v9
(m/s) Time
v10
(m/s)
2.6 4.3 1.28
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2.6 4.4 1.28
4.5 1.28
4.6 1.28
4.7 1.28
4.8 1.28
Processed Data
y = 714.56x + 1.3142 R² = 0.9427
0
2
4
6
8
10
12
0 0.005 0.01 0.015
Tem
inal
ve
loci
ty s
qu
are
d (
m^2
/s^2
)
Mass (kg)
Graph A: Terminal velocity squared vs mass Trials 1-10
Linear (vt2)
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Analysis
The line of best fit for all trials (Graph A) was given as y = 714.56x + 1.3142 with an R-squared of
0.9427. (Sample calculations are shown in the processed data section.) While this generally supports the
hypothesis, the last 6 data points do not fall on the line of best fit. Graph B shows that the first four data
points 1 to 4 filters) supports the data beautifully with an R-squared of 0.9955. Graph C shows the last 6
trials by themselves. With a much weaker line of best fit, these data could be said to have too much
y = 1117.5x + 0.2028 R² = 0.9955
0
1
2
3
4
5
6
0 0.001 0.002 0.003 0.004 0.005
Vt^
2 (
m^2
/s^2
)
Mass (kg)
Graph B: Terminal Velocity vs. Mass, 1-4 coffee filters
Series1
Linear (Series1)
y = 645.7x + 1.8955 R² = 0.8142
0
2
4
6
8
10
12
0 0.005 0.01 0.015
Vt^
2 (
m^2
/s^2
)
Mass (kg)
Graph C: Terminal Velocity vs. Mass, 5-10 coffee filters
Series1
Linear (Series1)
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systematic error to support the hypothesis. Another thing to point out is the y-intercept of each graph.
Graph A should theoretically pass through the origin, but it clearly does not. When you see that Graph C
has a lower slope than Graph B, it is clear that the last 6 data points “skewed” the data toward a + y-
intercept. Graph B, with the most weak trials eliminated, shows a y-intercept much closer to zero. Graph
C shows the opposite. It is to say that
⁄ = slope. The last 6 trials, were the filter
stack trials with the most mass. These most likely did not have enough time to speed up to terminal
velocity in the 2.6 meter fall. On the Raw Data graphs (the Data Studio ones I gave you!) you can even
see this trend. For the first 2 trials it could be said that the line was not linear and only had 5-6 points to
use for a determination of terminal velocity. The last trial with 1 filter had over 15 data points as the most
clearly linear. Also, to elaborate, Graph C shows that the trials vary widely and seem much less linear
than in Graph B. Why would these vary so much more than one filter? Although less filters caused more
wobbling during the falls, the data was clearly linear on the distance vs. time graphs and showed no
anomalies. The only other confounding variables here are the assumed constants. Due to the stacked
filters, the surface Area for more filters may have slightly exceeded 1 filter, thus decreasing terminal
velocity (ie-less slope, since mass had very little error). The g-value should not have changed as well as
the air density (no elevation change and temperature change.) So, for the last 6 points to yield a lower
slope than expected, the drag coefficient would have had to increase. Perhaps the less porous nature of
many stacked filters had an effect on this. I have no evidence for this, but it could be argued that there
could be less air drag for air passing through the porous single filter than AROUND the multiple in the
larger stacks. I do not understand fluid dynamics, but this seems reasonable. (The slope can also be
determined by the fact that (
) = (
) = 717.7, with an assumed drag coefficient of
0.8 according to internet resources, and an elevation of 6000 ft. and temp. at 20c). With this
number to work with, the slope of the first 4 trials EXCEEDS this calculation. If the cross-
sectional surfaced are did not change significantly, perhaps the only explanation is that, for the
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first 4 filters, which are more porous, the drag coefficient is less. This would lower the
denominator of the fraction for the theoretical slope. The interesting fact is that, while the last 6
filters had a more erratic nature on their graph, the average slope was almost exactly the
expected slope based on a known drag coefficient! It must be that the terminal velocity
determination had major systematic errors. There is a slight chance that using a balance from
Harbor Freight that costs 12 bucks might actually not be correct. Next time I would use a good
balance.
Finally, errors lasted throughout this lab, giving misinterpreted data. Another way to see
this could be claiming that the door was open during one of the tests and closed for another, a
sort of atmospheric pressure that was minute, but still affected the values. The judgment of
where the filter reaches terminal velocity is also subjective, thus skewing the correlation between
terminal velocity squared and mass. One last source of error to keep in mind is how accurately
Logger Pro recorded the data or how the data was read. Were the filters dropped in the exact
same way from the exact same altitude? Was it calibrated? This leads to methods for dropping
the filter. The filter cannot be dropped perfectly parallel to the floor. One side is let go before the
other and then it straightens out. The higher mass stacks were more stable With these thoughts in
mind, the calculations can be understood.
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Conclusion
This lab generally supports the hypothesis that the terminal velocity squared of an object is linearly
correlated with its mass. In other words, the terminal velocity of an object is directly proportional to its
mass, but the graph is will contain a different slope nearing the force of gravity because it is the square
root. There are significant limitations discussed in the analysis. The Coffee Filter Lab Also proved that
(
) is constant, more so for lighter filter stacks, from our given data points and correlation
coefficients. The fact that the expected slope was closest to the poor line of best fit for the
lasrgest 6 masses revealed major systematic errors.
The graphed showed greater reliability for the first five data points, or the relatively light filter
drops. After the first five points, the graph starts to become unpredictable and non-linear, a
source of bias that still requires greater analysis. Referring to the graphs, error bars have been
added to not overlap with the line of best fit. The irregularity is perhaps caused by systematic
error. The experiment began with a person dropping all the coffee filters and taking one out after
five repeated trials and replicating the same steps with one less filter in the stack. In the
beginning, the person dropping who was to drop the filter was inconsistent because he needed to
find a method to drop it precisely. Repeating the trials would help to solve for a precise and
accurate average. Although all coffee filters reached terminal velocity a half second after the
drop, the lighter coffee filters reached terminal velocity in less time (0.4s) than the heavier filters
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as expected, because their terminal velocity is less. It is easy to claim that they had less time to
accelerate, and should travel less distance. It is also important to note, however, that the lighter
coffee filters stayed at terminal velocity for less than 0.2s, while heavier filters floated for at least
0.3s.
This experiment could be improved when attempted in the future by using varied equipment,
techniques and methods. For example, more precise release mechanisms, such as cutting a string
to drop the coffee filters every time will prevent initial systematic errors. Also to make the set up
easier, the motion sensor could be place on the ground to collect data, avoiding the use of chairs
to drop the filters. Overall, this Coffee Filter Lab helped identify the force of gravity, prove that
(
) is indeed a constant, and help clarify that the value of the Terminal Velocity squared has a
direct relationship or a linear relationship to its mass.