An Investigation of a Model for Air Resistance

This experiment was done to determine the exponential value of the velocity of an object falling in the presence of air resistance. To measure this coffee filters were dropped from a standard height and their velocity was measured by a motion sensor. The value that was collected was 1.6922 which is closest to an exponent of 2. The value was about 5 standard deviations of .076 away from 2. Thus it was concluded that terminal velocity was proportional to v2. The error collected was

Ryan MillerAP Physics

6 December 2010

1. Introduction and Background

This experiment is done in order

to find out whether an object’s terminal

velocity is proportional to v or v2. Also

it is to investigate how air resistance

affects an object and its terminal

velocity. To do this one person drops a

coffee filter from a set height while

another person presses the collect button

on the data collect program (such as

Vernier LabPro) which is connected to

the motion sensor below the filter. This

allows the group to measure the terminal

velocity of the coffee filter as it floats

down to the ground. The equation for

the force of gravity observed on the

coffee filter is:

F = mg

However, the force of the gravity was

being counteracted by the force of the air

resistance and thus the total force

equation or Fnet equation is:

Mg – FDrag = ma

with FDrag being the force caused by the

air resistance (see derivation 1 for a

better equational analysis).

After dropping a total of eight

coffee filters for a total of five trials (at

least) of each number of filters, one has

enough data to make a scatter plot (see

graph 1) of the natural log of the data

and a best fit line. The slope of this line

is then the exponent of v that is

proportional to terminal velocity.

2. Procedure

The main equipment needed for

this experiment is a computer (with

Vernier Logger Pro and Vernier LabPro

software and Microsoft Excel for data

collection / graph making), coffee filters

(8 for testing), and a Vernier Motion

Probe (for measuring velocity).

Then a height needs to be set

from which the coffee filters are dropped

(preferably two meters for accuracy).

Each set of 5 (at least, more if wanted)

trials is then accompanied by an addition

of one more coffee filter. To collect data

we used a Vernier software program

known as LabPro with a motion sensor,

but any data collection connected to a

motion sensor would work.

After the eight coffee filters are

successfully dropped, the data collected

from the drops is then put into excel to

create a natural log graph (see graph 1).

Also the standard deviations and other

statistical data analysis should be found

using excel to help supplement data and

explain one’s findings. Finding the

slope of the best fit line of said natural

log graph indicates which exponent of v

is proportional to terminal velocity.

3. Results

The slope of my best fit line was

1.6922. Thus, the terminal velocity is

more closely proportional to v2.

Although the value that I measured was

about 5 standard deviations away from

the accepted value of 2, the value was

much further from 1 due to the fact that

it was about 10 standard deviations away

from 1. Yet the standard deviations

within my own data were very small,

meaning that my data was not very far

apart within each trial.

4. Error Analysis

There were a total of forty trials

executed with five trials per number of

coffee filter.

The value I determined was

1.6922 which is technically 5 standard

deviations away from the accepted value

of 2. Despite being that many standard

deviations away from the accepted

value, the data that I collected was very

closely correlated and was within not

many standard deviations of itself.

Possible sources of error for this

experiment are numerous. One includes

how each coffee filter was dropped, the

slightest change in shape or crinkle

could change the way the coffee filter

fell and throw the data off. Also any

technical errors within the measuring

device itself could throw the data off

enough to drastically move our value

away from 2. As always, every

measurement taken has a slight error in

it due to rounding by computers and

because of any shortcuts or error taken

or made in graph making and best fit

lines by Microsoft Excel (see derivation

1 for equation and graph 1 for graph).

To top it off, doing the experiment in a

closed environment, such as a school (on

the third level no less) may cause a

change in air density which can change

the speed at which the filter falls, as well

as how much resistance there is.

Graph 1:

Natural Log of Coffee Filters and Air Resistance Graph

y = 1.6922x + 0.8177

-0.5

0

0.5

1

1.5

2

2.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Natural Log of Terminal Velocity

Nat

ura

l L

og

of

Co

ffee

Fil

ters

Derivation 1:

(R = resistive force opposite to motion)= ma (where ma = 0 because velocity is constant)

(y=mx+b is used to as a comparison for the final equation to show that the slope is equal to the exponent of v)