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Courtesy of T. L. Pearson, Professor of Mathematics, Acadia University (Retired) Nova Scotia, Canada An aircraft flying under the guidance of a nondirectional beacon (a fixed radio transmitter, abbrevi- ated NDB) moves so that its longitudinal axis always points toward the beacon (see Figure 3.19). A pilot sets out toward an NDB from a point at which the wind is at right angles to the initial direc- tion of the aircraft; the wind maintains this direction. Assume that the wind speed and the speed of the aircraft through the air (its "airspeed") remain constant. (Keep in mind that the latter is differ ent from the aircraft's speed with respect to the ground.) Locate the flight in the xy-plane, placing the start of the trip at (2, 0) and the destina- tion at (0, 0). Set up the differential equation describing the aircraft's path over the ground. Hint: dy/dx (dy/dt) /(dx/dt). NDB Wind Figure 3.19 Guided aircraft Projects for Chapter 3 147 (b Make an appropriate substitution and solve this equation. [Hint: See Section 2.6.] (c) Use the fact that x-2 and y 0 at t0 to determine the appropriate value of the arbitrary constant in the solution set. (d) Solve to get y explicitly in terms of x. Write your solution in terms of a hyperbolic function. (e Let y be the ratio of windspeed to airspeed. Using a software package, graph the solutions for the cases γ = 0.1, 03, 05, and 0.7 all on the same set of axes. Interpret these graphs. (f) Discuss the (terrifying!) cases y 1 and y>1 Courtesy of T. L. Pearson, Professor of Mathematics, Acadia University (Retired Nova Scoda, Canada An aircraft flying under tho guidance of a nondirectional boacon (a fisod radio tra aled NDB) moves so that its longitudinal axis always points tow and the beacon(se Fi 319 Apkt sets out toward an NDES from a pont at which the wndbatnght angles ao the tion ofthe arran; the wind maintains this dinction. Assume that the windspoed.nd pedof he aircraf through the ir ts "aispeed) remain consant Keep in mind ht theer is ent from the iran's speed with f 10the pund) ⑩ucale-Aght inthexy-plane, placing the stat ofthe trip at (2.0) andthe- ion at (0,0). Set up the differential oquation desonibing the aincralt's path over the ground. Hint: ddx-(dy)ja) Projects for Chpter 3 147 D Make an appropriate substitution and solve this-mplie See Soton 2 rhitary constant in the solution set Let γ be the ratio of wind poed toanpod Using a solbware atag Fapháe olutions for the cacs γ-0.1.03.05, aed07 all on -sctofna hese graphs Discuss the 0orrifyingreas γ-1 and γ >1. (f)

Answer 1

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