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Math Help - Tough Calculus Problem...Can't figure out.

  1. #1
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    Tough Calculus Problem...Can't figure out.

    (Miracle on the Hudson) Airline pilot Sully Sullenberger is flying his Airbus A320 when a flock of geese fly into one of his engines. He needs to make an emergency landing and sees a point on the Hudson River, l meters away. At time t=0, his altitude is h meters and his flight trajectory is horizontal. In order to successfully land, he must.

    1) Maintain a horizontal speed v throughout the flight.
    2) Keep his vertical acceleration less than k at the time of landing so that his passengers don't feel sick.
    3) Keep his flight trajectory exactly horizontal initially and when he lands.

    a) Assuming that his flight trajectory is a cubic polynomial of the form y=ax^3+bx^2+cx+d, deduce the relations a = -2h/l^3, b = 3h/l^2, c = d = 0.

    b) If the position of his plane is (x(t),y(t), show that x(t) = l-vt, y''(t) is less than k.

    c) By differentiating the flight trajectory y = ax^3+bx^2+cx+d with respect to t (not x) and using your calculations from above, deduce that Captain Sullenberger must fly with horizontal speed v satisfying 6hv^2/l^2 is less than k.

    Thanks for the help!
    Last edited by seilhsub; October 23rd 2009 at 09:27 AM.
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  2. #2
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    We begin with the assumption that (x(t),y(t)) will travel left as t increases. This gives us

    (x(0),y(0))=(l,h).

    a) In order to deduce that c=d=0, we first note that (0,0) must lie on the trajectory, from which it follows that d=0. We then remember that when the plane lands, \frac{dy}{dt}=0, allowing us to solve a quadratic equation to prove that c=0.

    b) Knowing that Captain Sullenberger flies left at speed v, we may deduce that x(t)=l-vt. Here is a hint to prove that y''(t)<k: we may use the fact that a < 0.

    c) Here, we may use the Chain Rule, \frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}, together with the fact that \frac{dx}{dt}=-v.

    One final note: this problem is made tricky by the fact that y can be considered either a function of x or a function of t. The notation y(0) is therefore ambiguous, as it could refer to the value of y when x=0 (which is 0) or when t=0 (which is h).
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