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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.