An aircraft whose airspeed is v, 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 th

An aircraft whose airspeed is v, 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 vindVy k, 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 vos V, d (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 o fdx. Show that if we assume that y and d both remain small (as is certainly reasonable if the wind speed is not too large), then the integrand f takes the approximate form f = (1+ y)/1+ ky) (times an uninteresting constant) where k == V/v,. (C) Write down the Euler-Lagrange equation that determines the best path. To solve it, make the intelligent guess that y(x) = x (D - x), which clearly passes through the two towns. Show that it satisfies the Euler- Lagrange equation, provided . = (V4 + 2k2D2- 2)/(kD). How far north does this path take the plane, if D = 2000 miles, v, = 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.]