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Thread: Where should a pilot start descent

  1. #1
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    Oct 2008
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    Where should a pilot start descent

    Here is a diagram of the problem.

    http://www.stewartcalculus.com/data/...p_0205_stu.pdf

    Where Should a Pilot Start Descent?
    An approach path for an aircraft landing is shown in the figure and satisfies the following conditions:
    i. The cruising altitude is h when descent starts at a horizontal
    distance l from touchdown at the origin.
    ii. The pilot must maintain a constant horizontal speed v
    throughout descent.
    iii. The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity).

    1. Find a cubic polynomial P(x) = ax^3 + bx^2 + cx + d that satisfies condition (i) by imposing suitable conditions on P(x) and P'(x) at the start of descent and touchdown.
    2. Use conditions (ii) and (iii) to show that .
    (6hv^2)/l^2?k




    i completed the first part, and i got p(x) = -2hx^3/l^3 + 3hx^2/l^2, but im a bit confused about the second part. I know how to get the 6h/l^2 but i dont know how to place the v^2 (or how it even got there).

    to get the 6h/l^2, i simply took the third derivative of the polynomial p''(x) = 6ax + 2b, and i subbed in my a and b values from question one to get

    p''(x) = 6(-2h/l^3)x + 2(3h/l^2)

    i then subbed in x=0, since that is when the maximum accelreation will occur, and this value cannot be greater than a value k,

    p''(0) = 6h/l^2

    i dont know how i would include the v^2,

    thanks,

  2. #2
    Elite Member
    Join Date
    Jun 2007
    Posts
    12,289

    Re: Where should a pilot start descent

    Quote Originally Posted by evox
    Here is a diagram of the problem.

    http://www.stewartcalculus.com/data/...p_0205_stu.pdf

    Where Should a Pilot Start Descent?
    An approach path for an aircraft landing is shown in the figure and satisfies the following conditions:
    i. The cruising altitude is h when descent starts at a horizontal
    distance l from touchdown at the origin.
    ii. The pilot must maintain a constant horizontal speed v
    throughout descent.
    iii. The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity).

    1. Find a cubic polynomial P(x) = ax^3 + bx^2 + cx + d that satisfies condition (i) by imposing suitable conditions on P(x) and P'(x) at the start of descent and touchdown.
    2. Use conditions (ii) and (iii) to show that .
    (6hv^2)/l^2?k




    i completed the first part, and i got p(x) = -2hx^3/l^3 + 3hx^2/l^2, but im a bit confused about the second part. I know how to get the 6h/l^2 but i dont know how to place the v^2 (or how it even got there).

    to get the 6h/l^2, i simply took the third derivative of the polynomial p''(x) = 6ax + 2b, and i subbed in my a and b values from question one to get

    p''(x) = 6(-2h/l^3)x + 2(3h/l^2)

    i then subbed in x=0, since that is when the maximum accelreation will occur, and this value cannot be greater than a value k,

    p''(0) = 6h/l^2

    i dont know how i would include the v^2,

    thanks,
    This is a classic problem

    Read the discussions at:

    http://www.freemathhelp.com/forum/viewt ... =3&t=26449

    http://answers.yahoo.com/question/index ... 057AAEvCWc

    http://groups.google.com.sg/group/alt.m ... 2005-11%3F

    http://mathforum.org/kb/thread.jspa?thr ... tstart=134

    http://slumz.boxden.com/showthread.php?t=865008

    However, do not be tempted to copy the answers - your instructor might be looking at these reolies too.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
    New Member
    Join Date
    Oct 2008
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    2

    Re: Where should a pilot start descent

    Does my answer to question 1 look correct? it seems it is correct because if i sub in x=l, i get h and thats satisfies condition i

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