An aircraft whose airspeed is epsilon 0 has to fly from town O (at the origin) t

Question

An aircraft whose airspeed is epsilon 0 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 wind = V y , where x and y are measured due east and north respectively. Find the path, y = y(x), which the plane should follow to minimize its flight time, as follows: Find the plane's ground speed in terms of epsilon 0, V, phi (the angle by which the plane heads to the north of east, assumed to be small), and the plane's position. Write down the time of flight as an integral of the form fdx. Show that if we assume that y' and phi both remain small (as it certainly reasonable if the wind speed is not too large), then the integrand f takes the approximate form f = 1 + (1/2)y'2 / 1 + ky (times an uninteresting constant) where k = V/ epsilon 0. Write down the Euler-Lagrange equation that determines the best path. To solve it, make the intelligent guess that y(x) = lambda x(D - x), which clearly passes through the two towns. Show that it satisfies the Euler- Lagrange equation, provided How far north does this path take the plane, if D = 2000 miles, epsilon 0 = 500 mph, and the wind shear is V = 0.5 mph/mi? How much time does the plane save by following this path? (You may need to use MATHEMATICA or some other means to do this integral).

Explanation / Answer

take help 125km/h = 125 x 1000/3600 = 34.72m/s Here are 2 methods. _____________________ Since it starts from rest, average speed during take-off = (0+34.72)/2 = 17.36m/s Time = distance/average speed = 277/17.36 = 15.96s Acceleration = change in speed/time = 34.72/15.96 = 2.18m/s² _____________________ The other method is to use the standard formula v² = u² +2as (or vf² = vi² + 2as if you use those symbols) v² = u² +2as 34.72² = 0² + 2a x 277 a = 1205/(2x277) = 2.18m/s² EDIT. The points made in the other answers are correct. The question should state that the plane starts from rest and that you are to find the minimum average acceleration (or that acceleration is constant).

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