1148 words - 5 pages

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

5

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.

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