The Physics Teacher ◆ Vol. 51, April 2013 199

which the airplane’s velocity vector points relative to the +x-axis). The wind speed is assumed to be negligible compared to 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 the chord line of the airfoil,13 points at an angle q above the +x-axis), the lift FL (which points perpendicular to the velocity of the airplane assuming zero wind speed), the drag FD (which points opposite of the velocity of the airplane assuming zero wind speed), and the weight FW (which points downward).

Applying Newton’s second law to this free-body diagram results in the equations

FT 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 the x- 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 the wings,16 and CL and CD are the lift and drag coefficients (re-spectively) of the airplane. The lift and drag coefficients are assumed to depend on the angle of attack α θ – ϕ in the standard ways15,17,18

CL = 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, FL

Computer simulations continue to prove to be a valu-able tool in physics education. Based on the needs of an Aviation Physics course, we developed the

PHYSics of FLIght Simulator (PhysFliS), which numerically solves Newton’s second law for an airplane in flight based on standard aerodynamics relationships. The simulation can be used to pique students’ interest, teach a number of physics concepts, and teach computational investigation techniques. This paper describes the development and operation of this simulation, illustrates an example study that can be per-formed using it, and suggests further ideas for its use.

BackgroundComputer simulations, now ubiquitous in the physics

learning experience, can appeal to students of various back-grounds at many course levels, stimulate exploration, and equip students with the terminology and context of course material.1 Even simulations that have deliberately been pro-grammed incorrectly can stimulate inquiry,2 and simulations may sometimes impact learning more than lab equipment.3 Simulations offer advantages over other computational tools (such as spreadsheets) because they feature “real-time” action and they hide mathematical details behind the scenes (impor-tant when eliciting students’ interest).

The simulation described here (PhysFliS) was developed for use in a single-semester introductory physics course for students pursuing a degree in aviation management and flight operations while training to become commercial or military pilots.4 These aviation majors comprise approximately 10% of the university’s student population. The Aviation Physics course was designed to fulfill these students’ laboratory sci-ence requirement and apply introductory physics principles to aviation.

These students typically need opportunities to explicitly develop their sense of relevance and confidence in phys-ics concepts and skills.5 PhysFliS was thus designed to show them 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 introductory mechanics courses, as it represents an exciting application that can generate student interest,8 as evidenced by continu-ing theoretical and experimental explorations of flight by physics 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 angle at which the wings are oriented relative to the +x-axis) and f 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 2013

simulation by clicking the “Play” button in the main window. PhysFliS then applies the standard Euler’s method to calculate the x- and y-components vx and vy (respectively) of the velocity and the values of x and y recursively:

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 of Eqs. (1) and (2) divided by m, with ϕ = arctan (vy/vx) and FL and 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 traced out in real time in the plotting frame.

Example case – Cruising flightThe array of options in PhysFliS allows the user to explore

many flying scenarios. Here, we closely examine the results of a cruising flight scenario as an example. Cruising flight is characterized by constant horizontal velocity with a level chord line. Mathematically, these conditions mean zero val-ues for ax, ay, q, and ϕ. Equations (1) and (2) become

FT = FD (7)FL = FW. (8)

Inserting Eqs. (3) and (4) yields the thrust and speed required for cruising flight:

(9)

(10)

To explore a simplified cruising flight scenario,19 consider constant CL = 0.4, constant CD = 0.03, g = 9.8 m/s2, A = 15 m2,

(1) and (2) are solved for ax, ay = 0 to quantitatively examine equilibrium scenarios such as cruising flight.14,15 By solving these 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 using Euler’s method to model the flight of an airplane. The angle of attack determines CL and CD, which along with v and r determine FL and FD, which along with θ and ϕ determine the acceleration components, which determine v and ϕ (and therefore α ), and the recursive process repeats. At any time, the user may change FT and θ, much as a pilot would during flight.

Before beginning the simulation, the user specifies the values 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 is free to consistently employ any system of units (although an-gles must be in degrees). Entering 0 for r and 9.8 (or 32) for g will cause PhysFliS to vary the air density with the altitude by linearly interpolating between standard air density values15 in units of kg/m3 (or slugs/ft3). Entering a non-zero value for r imposes a uniform air density.

Before and during the simulation, the user can change FT and q by adjusting the sliders in the main PhysFliS window (Fig. 3). The range of values for FT scales with the weight of the airplane (since required thrust generally increases with airplane 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 user to maintain “perfect” control of the airplane’s pitch during flight, therefore ignoring the effects of external torques on the airplane (or assuming that the pilot is able to adjust to the desired pitch rapidly).

After setting the initial conditions, the user begins the

Fig. 2. PhysFliS Initial Conditions window. The constants a, b, c, and d that determine the behavior of the lift and drag coefficients are entered as they appear in Eqs. (5) and (6). Except for angles (which must be measured in degrees), any consistent system of units can be used.

Fig. 3. Main PhysFliS simulation window. This example trajectory is the result of the airplane’s initial speed being below the cruising flight value but the thrust and wing pitch being set to their cruising flight values.

Fig. 4. This simulation differs from that in Fig. 3 in that the lift coefficient is not constant. The plane still approaches cruising flight as a stable equilibrium point.

The Physics Teacher ◆ Vol. 51, April 2013 201

References1. Carl E. Wieman, Katherine K. Perkins, and Wendy K. Adams,

“Oersted Medal Lecture 2007: Interactive simulations for teaching 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, Maria Belloni, and Francisco Esquembre, “Teaching physics (and some 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, “When learning about the real world is better done virtually: A study of 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: Simple examples 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 lift with the Wright airfoils,” Phys. Teach. 49, 502–504 (Nov. 2011).

12. Rod Cross, “Measuring the effects of lift and drag on projectile motion,” Phys. Teach. 50, 80–82 (Feb. 2012).

13. This assumption is consistent with Spathopoulos (Ref. 6) and can 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/Ejs/.

wlane@ju.eduAuthor info?

FW = 6000 N, and constant ρ = 1.22 kg/m3. Equations (9) and (10) tell us that maintaining cruising flight will require a thrust 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 if it does not have sufficient initial speed to maintain cruising flight. We can predict that the plane will not have sufficient lift 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 same results in the trajectory depicted in Fig. 3. Our prediction was correct, but the behavior changes throughout the flight: During the predicted initial descent, the drag is less than the thrust, and the speed increases. Once the speed increases past the cruising flight value, the lift exceeds the weight, and the plane ascends and the drag exceeds the thrust, causing a decrease in speed. The cycle repeats, with the extremum of each oscillation closer to the cruising flight value, much like a damped spring or pendulum system oscillating around equi-librium.

Even if we remove the unphysical condition of a constant lift coefficient, we can find that the plane approaches cruising flight as a stable equilibrium state. Figure 4 shows the results of running PhysFliS with the same conditions as in Fig. 3, but with a from Eq. (5) set to 0.10 degrees-1. Again, the plane ap-proaches its cruising flight equilibrium value. The good news for pilots and passengers, therefore, is that the plane does want to stay in the air!

PossibilitiesThe previous example illustrates how PhysFliS can be used

to teach the concept of equilibrium states and how to evaluate their stability. This principle is just one of many physics les-sons that students can explore in a real-world scenario using PhysFliS. Instructors could also use PhysFliS to… • Demonstrate the importance of keeping track of units in

a real-world example. • Conduct computational experiments. For example,

students could expand on the above example of cruising flight with insufficient initial speed to explore how the distance 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 at y = 0) and discuss the relevant physics principles.

Conclusion, invitation, acknowledgments

This paper describes the development and possible uses of PhysFliS. Instructors and students may download PhysFliS free of charge at bit.ly/iEqtQA. PhysFliS was developed using Easy Java Simulations.20