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CM: Calculus of Variations Airplane problem

  1. Jan 13, 2009 #1
    1. The problem statement, all variables and given/known data
    "An aircraft whose airspeed is vo has to tfly 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 = Vy(x-hat) [the wind shear is in the x direction...]. 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.

    Part a): find the plane's grounds peed in terms of vo, V, and phi (the angle by which teh plane head sto the north of east), and the plane's position

    b): write down the time of flight as an integral of the form Integral[ f(x), x, {0, D}] (for those of you familiar with mathematica I used their wacky notation... basically its just an integral of f(x)dx from 0-> D). Show that if we assume that y' and phi both remain small then the integrand f takes the approximate form: f = (1+ 1/2 * y'^2 ) / (1+ ky) where k = V/v0

    the remaining two parts involve finding y(x) though I need help on this part first.

    2. Relevant equations

    The euler-lagrange equation

    3. The attempt at a solution

    This is a problem in calculus of variations, so basically what we're doing here is optimizing the Time integral.

    for part a), i said that V-plane = Vx *(x-hat) + Vy*(y-hat) (we don't need to worry about the altitude here). Vx = vo * cos( phi ) + Vy where Vy is the wind shear

    Vy = vo * sin(phi)

    for part b) here's what I don't understand here... t = distance/velocity, , so shouldn't T = Integral[ ds/v-wind] from 0 -> D, where ds = (dx^2 + dy^2)^1/2? I don't understand how i'm supposed to find the shortest time using only dx, or f(x) * dx... Can someone please explain this to me? The problem tells what what f(x) should be given the approximation but I'm unsure of how to set it up... the problem set is due wednesday morning.

    I feel like I can adequately use the Euler - lagrange equation once i figure this part out and set it up.


    Last edited: Jan 13, 2009
  2. jcsd
  3. Jan 13, 2009 #2
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