An aircraft whose airspeed is v₀ has to fly from town O (at the origin) to town P, which is a distance D due east. There is a steady gentle wind shear, such that v_wind = V y X, where x and y are measured east and north respectively. Find the path, y = y(x), which the plane should follow to minimize its flight time, as follows: (a) Find the plane's ground speed in terms of v₀,V,Ï• (the angle by which the plane heads to the north of east), and the plane's position. (b) Write down the time of flight as an integral of the form

Show that if we assume that y' and Ï• both remain small (as is certainly reasonable if the wind speed is not too large), then the integrand f takes the approximate form (times an uninteresting constant) where(c) Write down the Euler—Lagrange equation that determines the best path. To solve it, make the intelligent guess that which clearly passes through the two towns. Show that it satisfies the Euler-Lagrange equation, provideHow far north does this path take the plane, if D = 2000 miles, v₀= 500 mph, and the wind shear is V = 0.5 mph/mi? How much time does the plane save by following this path? [You'll probably want to use a computer to do this integral.]