The Physics Teacher ◆ Vol. 51, April 2013 199which the airplane’s velocity vector points relative to the +x-axis). The wind speed is assumed to be negligible comparedto the airplane’s speed v.The forces acting on the airplane are the thrust FT (which,assuming for simplicity that the engines point parallel to thechord line of the airfoil,13 points at an angle q above the +x-axis), the lift FL (which points perpendicular to the velocity ofthe airplane assuming zero wind speed), the drag FD (whichpoints opposite of the velocity of the airplane assuming zerowind speed), and the weight FW (which points downward).Applying Newton’s second law to this free-body diagramresults in the equationsFT cos (θ)+ FL sin (ϕ) – FD cos (ϕ) = max (1)FT sin (θ) – FL cos (ϕ) – FD sin (ϕ) – FW = may. (2)Here, m is the mass of the airplane, and ax and ay are thex- and y-components (respectively) of the airplane’s accelera-tion. The lift force and drag force are calculated by the stan-dard equations14,15 (3)(4)A .Here, r is the density of the air, A is the planform area of thewings,16 and CL and CD are the lift and drag coefficients (re-spectively) of the airplane. The lift and drag coefficients areassumed to depend on the angle of attack α θ – ϕ in thestandard ways15,17,18CL = aα + b (5)CD = cα2 + d, (6)where a, b, c, and d are constants that depend on the charac-teristics of the wing.In typical introductory explorations of this problem, Eqs.Simulation of the Physics of FlightW. Brian Lane, Jacksonville University, Jacksonville, FLComputer simulations continue to prove to be a valu-able tool in physics education. Based on the needsof an Aviation Physics course, we developed thePHYSics of FLIght Simulator (PhysFliS), which numericallysolves Newton’s second law for an airplane in flight based onstandard aerodynamics relationships. The simulation can beused to pique students’ interest, teach a number of physicsconcepts, and teach computational investigation techniques.This paper describes the development and operation of thissimulation, illustrates an example study that can be per-formed using it, and suggests further ideas for its use.BackgroundComputer simulations, now ubiquitous in the physicslearning experience, can appeal to students of various back-grounds at many course levels, stimulate exploration, andequip students with the terminology and context of coursematerial.1 Even simulations that have deliberately been pro-grammed incorrectly can stimulate inquiry,2 and simulationsmay sometimes impact learning more than lab equipment.3Simulations offer advantages over other computational tools(such as spreadsheets) because they feature “real-time” actionand they hide mathematical details behind the scenes (impor-tant when eliciting students’ interest).The simulation described here (PhysFliS) was developedfor use in a single-semester introductory physics course forstudents pursuing a degree in aviation management and flightoperations while training to become commercial or militarypilots.4 These aviation majors comprise approximately 10%of the university’s student population. The Aviation Physicscourse was designed to fulfill these students’ laboratory sci-ence requirement and apply introductory physics principlesto aviation.These students typically need opportunities to explicitlydevelop their sense of relevance and confidence in phys-ics concepts and skills.5 PhysFliS was thus designed to showthem the application of some of the central concepts of in-troductory physics to their interests and develop their confi-dence.The topic of flight is also of interest in general introductorymechanics courses, as it represents an exciting applicationthat can generate student interest,8 as evidenced by continu-ing theoretical and experimental explorations of flight byphysics educators.6-12 PhysFliS can supplement these explora-tions with computational activities.Model development and applicationPhysFliS models the physics of a flying airplane by apply-ing Newton’s second law to the free-body diagram in Fig. 1.The angle q is the pitch of the airfoil’s chord line (the angleat which the wings are oriented relative to the +x-axis) andf is the angle of the airplane’s current trajectory (the angle at Fig. 1. Free-body diagram of an airplane in flight.
200 The Physics Teacher ◆ Vol. 51, April 2013simulation by clicking the “Play” button in the mainwindow. PhysFliS then applies the standard Euler’smethod to calculate the x- and y-components vx and vy(respectively) of the velocity and the values of x and yrecursively:vx (t + Dt) = vx (t) + ax (t + Dt)Dtvy (t + Dt) = vy (t) + ay (t + Dt)Dtx(t + Dt) = x(t) + vx (t + Dt)Dty(t + Dt) = y(t) + vy (t + Dt)Dt.Here, Dt is a small increment of time (0.05 s). The accel-eration components are evaluated using the left-hand sides ofEqs. (1) and (2) divided by m, with ϕ = arctan (vy/vx) and FLand FD evaluated using Eqs. (3) through (6).As PhysFliS runs, the current values of v, ϕ, the accelera-tion magnitude, FL, and FD are displayed in the main window,enabling the user to track how the flight’s physical character-istics change. The airplane’s trajectory (y versus x) is tracedout in real time in the plotting frame.Example case – Cruising flightThe array of options in PhysFliS allows the user to exploremany flying scenarios. Here, we closely examine the resultsof a cruising flight scenario as an example. Cruising flightis characterized by constant horizontal velocity with a levelchord line. Mathematically, these conditions mean zero val-ues for ax, ay, q, and ϕ. Equations (1) and (2) becomeFT = FD (7)FL = FW. (8)Inserting Eqs. (3) and (4) yields the thrust and speedrequired for cruising flight: (9) (10)To explore a simplified cruising flight scenario,19 considerconstant CL = 0.4, constant CD = 0.03, g = 9.8 m/s2, A =15m2,(1) and (2) are solved for ax, ay = 0 to quantitatively examineequilibrium scenarios such as cruising flight.14,15 By solvingthese equations numerically for general (ax, ay ≠ 0) behavior,PhysFliS can show how the airplane approaches these equilib-rium scenarios.PhysFliS evaluates Eqs. (1) through (6) recursively usingEuler’s method to model the flight of an airplane. The angleof attack determines CL and CD, which along with v and rdetermine FL and FD, which along with θ and ϕ determinethe acceleration components, which determine v and ϕ (andtherefore α ), and the recursive process repeats. At any time,the user may change FT and θ, much as a pilot would duringflight.Before beginning the simulation, the user specifies thevalues of a, b, c, d, g, FW, A, r and the initial values of altitude,v, and f in the Initial Conditions window (Fig. 2). The user isfree to consistently employ any system of units (although an-gles must be in degrees). Entering 0 for r and 9.8 (or 32) for gwill cause PhysFliS to vary the air density with the altitude bylinearly interpolating between standard air density values15in units of kg/m3 (or slugs/ft3). Entering a non-zero value forr imposes a uniform air density.Before and during the simulation, the user can change FTand q by adjusting the sliders in the main PhysFliS window(Fig. 3). The range of values for FT scales with the weight ofthe airplane (since required thrust generally increases withairplane weight) and θ can be varied between -180° and 180°(although extreme values of θ cause the calculations to be-come unrealistic). This slider control system enables the userto maintain “perfect” control of the airplane’s pitch duringflight, therefore ignoring the effects of external torques onthe airplane (or assuming that the pilot is able to adjust to thedesired pitch rapidly).After setting the initial conditions, the user begins theFig. 2. PhysFliS Initial Conditions window. The constants a, b, c, and d thatdetermine the behavior of the lift and drag coefficients are entered as theyappear in Eqs. (5) and (6). Except for angles (which must be measured indegrees), any consistent system of units can be used.Fig. 3. Main PhysFliS simulation window. This exampletrajectory is the result of the airplane’s initial speedbeing below the cruising flight value but the thrust andwing pitch being set to their cruising flight values.Fig. 4. This simulation differs from that in Fig. 3 inthat the lift coefficient is not constant. The plane stillapproaches cruising flight as a stable equilibrium point.
The Physics Teacher ◆ Vol. 51, April 2013 201References1. Carl E. Wieman, Katherine K. Perkins, and Wendy K. Adams,“Oersted Medal Lecture 2007: Interactive simulations forteaching physics: What works, what doesn’t, and why,” Am. J.Phys. 76 (4-5), 393–399 (April 2008).2. Anne J. Cox, William F. Junkin III, Wolfgang Christian, MariaBelloni, and Francisco Esquembre, “Teaching physics (andsome computation) using intentionally incorrect simulations,”Phys. Teach. 49, 273–276 (May 2011).3. N. D. Finkelstein, W. K. Adams, C. J. Keller, P.B. Kohl, K. K.Perkins, N. S. Podolefsky, S. Reid, and R. LeMaster, “Whenlearning about the real world is better done virtually: A studyof substituting computer simulations for laboratory equip-ment,” Phys. Rev. ST - PER 1, 010103 (2005).4. Further information can be found in the university catalog:www.ju.edu/cc1112/Pages/Aviation-Mgmnt-Flight-Operations.aspx.5. J. M. Keller, “Development and use of the ARCS model of in-structional design,” J. Inst. Dev. 10 (3), 2–10 (1987).6. Vassilis Spathopoulos, “Flight physics for beginners: Simpleexamples of applying Newton’s laws,” Phys. Teach. 49, 373–376(Sept. 2011).7. Michael Liebl, “Investigating flight with a toy helicopter,” Phys.Teach. 48, 458–460 (Oct. 2010).8. John C. Strong, “Downwash and lift force in helicopter flight,”letter to the editor, Phys. Teach. 49, 132 (March 2011).9. James J. Carr, “Toy helicopters and room fans,” letter to the edi-tor, Phys. Teach. 49, L2 (July 2011).10. Michael Liebl, “Liebl’s response,” letter to the editor, Phys.Teach. 49, L2–L3 (July 2011).11. Richard M. Heavers and Arianne Soleymanloo, “Measuring liftwith the Wright airfoils,” Phys. Teach. 49, 502–504 (Nov. 2011).12. Rod Cross, “Measuring the effects of lift and drag on projectilemotion,” Phys. Teach. 50, 80–82 (Feb. 2012).13. This assumption is consistent with Spathopoulos (Ref. 6) andcan be relaxed by using the “thrust offset” feature on PhysFliS.14. N. Dreska and L. Weisenthal, Physics for Aviation (Jeppesen,1992).15. J. D. Anderson, Introduction to Flight (McGraw-Hill, New York,1999).16. “Size Effects on Lift,” retrieved May 30, 2012, from NASA:www.grc.nasa.gov/WWW/k-12/airplane/size.html.17. “Modern Lift Equation,” retrieved May 30, 2012, from NASA:wright.nasa.gov/airplane/lifteq.html.18. “The Drag Coefficient,” retrieved May 30, 2012, from NASA:www.grc.nasa.gov/WWW/K-12/airplane/dragco.html.19. These values are reasonable for a Cessna 150, and the air den-sity is only slightly lower than that for sea level.20. “Easy Java Simulations,” retrieved May 30, 2012, from fem.um.es/Ejsemail@example.comAuthor info?FW = 6000 N, and constant ρ = 1.22 kg/m3. Equations (9)and (10) tell us that maintaining cruising flight will require athrust of 450 N and a speed of approximately 40.49 m/s.If we enter these initial conditions (with ϕ, q = 0) and click“Play,” we see a horizontal plane trajectory, as expected.A question that arises is how the airplane will behave ifit does not have sufficient initial speed to maintain cruisingflight. We can predict that the plane will not have sufficientlift to balance its weight, and the plane will accelerate down-ward. Restarting PhysFliS with a lower value of initial speed(20.0 m/s) but keeping other initial conditions the sameresults in the trajectory depicted in Fig. 3. Our predictionwas correct, but the behavior changes throughout the flight:During the predicted initial descent, the drag is less than thethrust, and the speed increases. Once the speed increasespast the cruising flight value, the lift exceeds the weight, andthe plane ascends and the drag exceeds the thrust, causing adecrease in speed. The cycle repeats, with the extremum ofeach oscillation closer to the cruising flight value, much like adamped spring or pendulum system oscillating around equi-librium.Even if we remove the unphysical condition of a constantlift coefficient, we can find that the plane approaches cruisingflight as a stable equilibrium state. Figure 4 shows the resultsof running PhysFliS with the same conditions as in Fig. 3, butwith a from Eq. (5) set to 0.10 degrees-1. Again, the plane ap-proaches its cruising flight equilibrium value. The good newsfor pilots and passengers, therefore, is that the plane does wantto stay in the air!PossibilitiesThe previous example illustrates how PhysFliS can be usedto teach the concept of equilibrium states and how to evaluatetheir stability. This principle is just one of many physics les-sons that students can explore in a real-world scenario usingPhysFliS. Instructors could also use PhysFliS to…• Demonstrate the importance of keeping track of units ina real-world example.• Conduct computational experiments. For example,students could expand on the above example of cruisingflight with insufficient initial speed to explore how thedistance between the trajectory’s extrema is determined.• Explore other special flying scenarios. For example,students can explore the behavior of an airplane in a zero-power glide or try to “land” the plane safely (with ϕ = 0 aty = 0) and discuss the relevant physics principles.Conclusion, invitation, acknowledgmentsThis paper describes the development and possible uses ofPhysFliS. Instructors and students may download PhysFliSfree of charge at bit.ly/iEqtQA. PhysFliS was developedusing Easy Java Simulations.20