writing reflection
Resistance Is Not Futile: Air Resistance in an Algebra-Based Course Ian Lovatt and Bill Innes Citation: 43 , 544 (2005); doi: 10.1119/1.2120387 View online: http://dx.doi.org/10.1119/1.2120387 View Table of Contents: http://aapt.scitation.org/toc/pte/43/8 Published by the American Association of Physics Teachers distance for an object dropped from rest is beyond the scope of an algebra-based course, one can at least list some distances at which the speed of the object is 95% (for example) of its terminal speed. Brancazio 4 has done this but has not presented the equations from which the values are calculated. Such a list might open a discussion of why one doesn’t calculate the distance fallen when the object has actually reached its terminal speed. (The object only asymptotically approaches terminal speed.) We have found another surprising reason for in - troducing air resistance. What would happen if you treated these distances and terminal speeds as data and I n this paper we show that an object’s terminal speed due to air resistance depends not on any of the object’s details, but only on the distance at which an object reaches a particular fraction of its ter - minal speed. We show this graphically and algebraical - ly. Although a mathematical treatment of air resistance is beyond the scope of an algebra-based, introductory physics course, some of the concepts involved are im - portant for (at least) three reasons. First, the equations used for uniform acceleration only approximately (and perhaps badly!) describe projectiles students know (a home-run baseball, for example). With the equation for terminal speed, students can estimate the speeds at which the simple kinematic equations no longer pro - duce “reasonable” approximations. Second, we can construct the equation for terminal speed using dimensional analysis (in the spirit of Ref.
1, for example). This emphasizes the importance of units. Third, with the equation for terminal speed (1) v mg C AT= 2 ρ , we have a source of scaling questions and questions about graphs (in the spirit of Arnold Arons). 2 Here are two examples: If you double the mass of a falling object (without changing anything else) 3 by what factor does the terminal speed get multiplied? In the three graphs of Fig. 1, the square of the terminal speed is plotted against the mass of the falling object.
Imagine three different densities of air ( A, B, and C); in which of the situations shown is the air densest? Although deriving the equations for the speed and Resistance Is Not Futile:
Air Resistance in an Algebra-Based Course Ian Lovatt and Bill Innes, Mount Royal College, Calgary, Alberta, Canada square of terminal speed objec t's mass A B C Fig. 1. Three sets of hypothetical measurements of an object’s terminal speed in air. Rank the sets in order of the air’s density. 544 DOI: 10.1119/1.2120387 THE PHYSICS TEACHER ◆ Vol. 43, November 2005 THE PHYSICS TEACHER ◆ Vol. 43, November 2005 545 plotted a graph ( v2T versus distance)? The “data” for many different objects fall on the same straight line (see Fig. 2)! This allows a student who can calculate the terminal speed of any object to also calculate the object’s “95% distance,” further illuminating the lim - its of the simple kinematic equations. (The student might use the graph or the regression equation, whose slope in this case is 8.42 m .s-2.) This feature of air resistance is yet another source of scaling questions. For instance, my cell phone’s termi - nal speed is 50 m/s; your cell phone’s terminal speed is 55 m/s (10% greater than mine). When I drop my cell phone from a great height, it reaches 95% of its ter - minal speed after falling 100 m; how far will your cell phone fall before it reaches 95% of its terminal speed? Why is v2T versus “95% distance” a straight line?
Newton’s second law for an object (mass m) falling through air (density ρ) is (2) m dv dt m m g bv m g b v = − − = ′− ( ) . ρ 2 2 (3) ′ = − g m m g 1 ρ . In Eq. (2), mρg represents the buoyant force; if the object is significantly denser than air one can ignore mρ [although see Ref. 5 and Eq. (7) below].
Experimental evidence for a resistive force propor - tional to the square of the object’s speed has been presented in many pedagogical articles (for instance, Refs. 3 and 6–10). At equilibrium (4) v mg b T= ′.
Reference 3 experimentally shows that terminal speed is proportional to the square root of the object’s mass. Reference 9 experimentally shows that the resistive force is proportional to the object’s cross-sectional area so that terminal speed is inversely proportional to the square root of the object’s cross- sectional area. If the only relevant property of the air is its density (as opposed to its viscosity, perhaps) the only dimensionally correct equation for terminal speed is Eq. (1), v mg C AT= ′ 2 12 ρ .
The constant C usually falls between 0 and 1 (although see Ref. 11). If the fluid’s viscosity is important, then the associated drag force is propor - tional to v, not to v2.
The solutions to Eq. (2) (speed and distance as functions of elapsed time) are well-known. 7,12 Since we want to express the terminal speed as a function of the “95% distance,” a more direct solution is appro - priate. Substitute (5) dv dt v dv dx d v dx = = ( ) 1 2 2 into Eq. (2) and integrate. The solution is (6) x m b bv mg = − ′ − 2 1 2 1 ln . Substitute mg b v ′= T2[Eq.(5)] and f = v vT ; the result is (7) x v g f v g m m f = ′ −( ) = − −( ) − − T T 2 2 1 2 2 1 2 1 2 1 1 ln ln ρ (8) v g f x T2 2 1 2 1 ≈ −( ) − ln . 0 10000 20000 0 500 1000 1500 2000 2500 95% distance (m) square of terminal speed (m/s) 2 Fig. 2. Calculated values of a falling object’s terminal speed in air as a function of the distance at which the object reaches 95% of its terminal speed. 546 THE PHYSICS TEACHER ◆ Vol. 43, November 2005 Equation (8) is approximately true when the object is significantly denser than the fluid through which it falls. A graph of v2T versus x is a straight line whose slope depends only on the local free-fall acceleration and the fraction ( f ) chosen and not on any property of the object! [Compare the slope expected for such a graph with the slope of Fig. 2 (8.42 m .s-2).] Acknowledgments We thank Alfredo Louro at the University of Calgary for carefully reading the manuscript and for making valuable suggestions; and Paul van der Pol, also at the University of Calgary, for translating our dia - grams into jpeg. We also thank the anonymous refer - ee for comments that made our paper more precise. References 1. Thomas A. McMahon and John Tyler Bonner, On Size and Life (Scientific American Books, W.H. Freeman, 1983).
2. Arnold Arons, A Guide to Introductory Physics Teaching (Wiley, New York, 1990).
3. Frank Weichman and Bruno Larochelle, “Air resis - tance,” Phys. Teach. 25, 505–507 (Nov. 1987).
4. Peter Brancazio, Sport Science (Simon and Schuster, New York, 1984).
5. David Keeports, “Drag and the two-bullet problem,” Phys. Teach. 35, 188–191(March 1997). 6. Carl Angell and Trond Ekern, “Measuring friction on falling muffin cups,” Phys. Teach . 37, 181–182 (March 1999).
7. Norman F. Derby, Robert G. Fuller, and Phil W. Gronseth, “The ubiquitous coffee filter,” Phys. Teach. 35, 168–171 (March 1997).
8. Peter Timmerman and Jacobus P. van der Weele, “On the rise and fall of a ball with linear or quadratic drag,” Am. J. Phys. 67, 538–546 (June 1999).
9. Ken Takahashi and D. Thompson, “Measuring air re - sistance in a computerized laboratory,” Am. J. Phys . 67 , 709–711 (Aug. 1999).
10. M.E. Brandan, “Measurement of the terminal velocity in air of a ping-pong ball using a time-to-amplitude converter in the millisecond range,” Am. J. Phys . 52, 890–893 (Oct. 1984).
11. Vasilis Pagonis, David Guerra, Sean Chauduri, Brian Hornbecker, and Nathan Smith, “Effects of air resis - tance,” Phys. Teach . 35, 364–368 (Sept. 1997).
12. Clifford E. Swartz, Used Math, 2nd ed. (American Association of Physics Teachers, College Park, MD, 1993).
PACS codes: 46.01A, 46.02B Ian Lovatt and Bill Innes teach mathematics, astronomy, and physics at Mount Royal College. Mount Royal College, 4825 Mount Royal Gate S.W., Calgary, Alberta, Canada T3E 6 K6 Used Math by Clifford E. Swartz is not a math text. It is a physics teacher’s tutorial on all the math for the first two years of university physics.
Instead of rigorous proofs, there are plausibility explanations and applied examples. The book emphasizes approximation methods. Topics are:
error analysis, units and dimensions, the simple functions of applied math, statistics, analytical geometry, series, common differential equations, and much more. (264 pp.) by Clifford E. Swartz Order online at www.aapt.org/store or call 301-209-3333 members: $27 • nonmembers: $34
