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Math Help - Mechanics air resistance question

  1. #1
    C.E
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    Mechanics air resistance question

    Hi, I was hoping someone could offer some guidance on the following question:
    An airplane in flight is subject to an air resistance force proportional to the square of its speed v. But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward. The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to v^2, so that the total air resistance force can be expressed by F=av^2+b/v^2, where a and b are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use a = 0.310 Ns^2/m^2 and b = 3.48105 Nm^2/s^2. In steady flight, the engine must provide a forward force that exactly balances the air resistance force.

    a. Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel.

    b.Calculate the speed for which the airplane will have the maximum endurance (that is, will remain in the air the longest time).

    I have already done part a and got the correct answer which is v=32.6 m/s (I just included it in this thread in case it helps with part b). With regards to part b I do not know how to start. Any guidance would be much appreciated. (By the way the numerical answer should be 24.7 m/s).
    Last edited by C.E; March 19th 2009 at 08:40 AM.
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  2. #2
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    max endurance translates to minimum fuel flow

    fuel flow is proportional to engine power

    instantaneous power = Fv

    P = \left(av^2 + \frac{b}{v^2}\right)v

    P = av^3 + \frac{b}{v}

    \frac{dP}{dv} = 3av^2 - \frac{b}{v^2}

    minimum power occurs when \frac{dP}{dv} = 0

    3av^2 = \frac{b}{v^2}

    v^4 = \frac{b}{3a}

    v = \sqrt[4]{\frac{b}{3a}} = 24.7 \, m/s
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by skeeter View Post

    fuel flow is proportional to engine power
    This is an unstated assumption in the original question. At best this is only approximately the case. There will always be an efficiency term that depends on speed, power, altitude, ...

    CB
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    This is an unstated assumption in the original question. At best this is only approximately the case. There will always be an efficiency term that depends on speed, power, altitude, ...

    CB
    that's why I used the term "proportional" ... after 2400 flight hours and over 300 arrested carrier landings, I think i know what I'm saying.
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