A Study of Falling Bodies and Aid Resistance

A Mini-Module by Paul Kertay, Joe Sosine,

Fred Dittman and Michalinos Zembylas

Problem Statement

What are the effects of mass, surface area, air resistance and aerodynamic shape on the motion of falling objects?

Objectives

1. To compare the rate at which different objects fall.

2. To determine whether some objects reach a constant (terminal) velocity as they fall.

3. To explore the characteristics of those objects that fall at different rates and determine the role that: (a) weight, (b) surface area, (c) air density and (d) aerodynamic shape have on their rate.

For a reference about air resistance check the following web sites:

<http://www.ksc.nasa.gov/facts/faq04.html>

<http://www.usatoday.com/wdencalc.htm>

Mathematical Extension

To find a mathematical model to describe the relationship between the weight of an object and its terminal velocity.

Equipment [Each group of 4-5 students]

TI-82, 83 or 85, 86, 89, 92 graphic calculators

CBR or CBL with motion detector and motion program

10 basket style coffee filters

ping pong ball

tennis ball

Procedure

PART A: Getting started with CBR--In three steps [Please see CBR Pages 1, 2, and 3]

** Notes: **#1.

#2. The CBR needs to be clamped down. Perhaps a good way to do this and take care of #1 would be to make a bird "perch" out of PVC piping and an umbrella stand (PVC piping setup would be less than $20 and could be used for a water pressure module)

#3. Be careful with prompts. CBRs have different prompts than CBLs.

**PART B:** (Optional) __Worksheet__ "Match the Graph" to become familiar with CBR.[Insert worksheet]

PART C: Behavior of different objects falling through the atmosphere

1) Hold a tennis ball, a ping pong ball, and a stack of 5 coffee filters the same distance above the floor. What force(s) act on these objects causing them to fall? Predict whether all objects will strike the floor simultaneously. If not, indicate which one (s) will hit later and why. Drop them simultaneously. Is your prediction correct?

2) Repeat the drop, observe the objects: Do any seem to reach a constant velocity? If so which one(s)? According to Newton’s law of acceleration what does this say about the forces acting on the object(s)?

PART D: The study of the role weight has on behavior of falling bodies.

1) Make a data table to record number of filters and terminal velocity. (Optional: You can use Excel or any other spreadsheet program to record the data)

2) __One group of students__: Get a stack of 9 paper coffee filters, be careful not to alter their shape.

3) Connect the CBR unit to your TI-82 calculator using the calculator-to-CBR cable.

4) Transfer the RANGER program from CBR to your calculator.

5) Secure the CBR unit to a stand so that it is at least 2.5 meters above the floor with the sonic sensor facing down.

6) Select the RANGER program and run it.

7) Hold your stack of 9 filters, concave side up, 0.5m below the detector. Hold in one hand parallel to the detector. DO NOT MOVE THE FILTERS FROM THIS LOCATION UNTIL YOU START GATHERING DATA.

8) When prompted enter the collection time (no less that 0.02 seconds).

9) Follow calculator prompt to hit enter to start collecting data, and release filters keeping __both__ hands and body out of probes path.

10) When data collection is complete, display plot option 1: Distance-Time. D=f(t)

11) Examine plot to determine if there is a region of constant slope (this would indicate a period of constant velocity)

12) Use trace to determine the time period during which this occurs

13) To determine the velocity during this period use plot option 2: Velocity-TIme. V=f(t)

14) See both position and velocity plots simultaneously

15) Use the Horizontal line option in Draw determine the terminal velocity in this region and record in data table.

16) Remove 3 filters from the inside of the stack (this is to insure that the frontal surface of the filter stack does not change)

17) Repeat steps 6-15 until you have done 5 different trials (with 9:6:3:2:1 filters). The other group should try the arrangement 1:2:3:6:9 filters. Question: What are the differences in these two experiment setups ("fluffed" and "unfluffed" coffee filters)

18) Make a scatter plot of velocity as a function of weight (number of filters)

19) Perform regression analysis to determine the relationship between these variables. A linear relationship is consistent with laminar flow of air over the filters, a power relationship is consistent with turbulent flow.

PART C: Effect on aerodynamic shape and terminal velocity.

1) Change the shape of the filters by cutting holes (decreasing surface area) and determine the effect on aerodynamic shape and terminal velocity.

2) Research the meaning of laminar and turbulent flow.

3) Relate laminar and turbulent flow to automotive and aerodynamic design.

4) Determining the aerodynamic properties of the filters.

A property of bodies moving through air is known as the drag coefficient. It is a factor which determines the amount of friction the object experiences as it falls through the air. Since an object must have no net forces acting on it for it to move at constant velocity, the force of gravity (F_{g}) must be counteracted by air friction (F _{air f}) when it reaches terminal velocity (F_{g}-F _{air f}=0). As a result F_{g}=F _{air f}. Since F_{g}=(m)(a) which is weight, then the weight of the filters equals the F _{air f}.

a) Determine the mass of 9 coffee filters and calculate the mean value. This should be a close approximation of those used previously.

b) Using weight=(m)(a), determine the cumulative weight of 9, 6, 3, 2, 1 filters.

c) The friction experienced by a body depends on a number of factors and is described as follows

F _{air f}=(CArv^{2} )/2,

where C is the drag coefficient, A is the surface area of the object, r is the density of air, v is the velocity of the falling object (terminal velocity).

d) Determine a method to estimate the surface area of the filter.

e) Using weight=(m)(a)=(CArv^{2} )/2, solve for the drag coefficient for each trail. Determine the average value for the drag coefficient.

f) Compare to values reported for automobiles to see if they are reasonable.

**Calculus extension**: Determine the time it takes to reach terminal velocity, and how this relates to weight. Compare this to the theoretical value from F_{g}-F _{air f}=ma, using the a=dv/dt.

Reference

Bower, B., & Heil J. __A study of falling bodies__.