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Math Help - A woman bails out of an airplane...

  1. #1
    MHF Contributor alexmahone's Avatar
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    A woman bails out of an airplane...

    A woman bails out of an airplane at an altitude of 10,000 ft, falls freely for 20 s, then opens her parachute. How long will it take her to reach the ground? Assume linear air resistance \rho v\ ft/s^2, taking \rho=0.15 without the parachute and \rho=1.5 with the parachute. (Suggestion: First determine her height above the ground and velocity when the parachute opens.)

    Without the parachute:

    \frac{dv}{dt}=32-0.15v

    \int \frac{dv}{32-0.15v}=\int dt

    \frac{-1}{0.15}ln\ (32-0.15v)=t+C_1

    v(0)=0

    C_1=\frac{-1}{0.15}ln\ 32

    \frac{-1}{0.15}ln\ (32-0.15v)=t-\frac{1}{0.15}ln\ 32

    When t=20,

    \frac{-1}{0.15}ln\ (32-0.15v_{20})=20-\frac{1}{0.15}ln\ 32

    -ln\ (32-0.15v_{20})=3-ln\ 32=-0.466

    32-0.15v_{20}=1.593

    v_{20}=202.712

    -ln\ (32-0.15v)=0.15t-ln\ 32

    ln\ \left(\frac{32}{32-0.15v}\right)=0.15t

    \left(\frac{32}{32-0.15v}\right)=e^{0.15t}

    32-0.15v=32e^{-0.15t}

    0.15\frac{dx}{dt}=32(1-e^{-0.15t})

    \int \frac{3}{640}dx=\int (1-e^{-0.15t})dt

    \frac{3x}{640}=t+\frac{e^{-0.15t}}{0.15}+C_2

    x(0)=0

    C_2+\frac{1}{0.15}=0

    C_2=-\frac{1}{0.15}

    \frac{3x}{640}=t+\frac{e^{-0.15t}}{0.15}-\frac{1}{0.15}

    When t=20,

    \frac{3x_{20}}{640}=20+\frac{e^{-3}}{0.15}-\frac{1}{0.15}

    x_{20}=2915.253

    With the parachute:

    \frac{dv}{dt}=32-1.5v

    \frac{-1}{1.5}ln\ |32-1.5v|=t+C_3

    ln\ |32-1.5v|=-1.5(t+C_3)

    32-1.5v=\pm e^{-1.5C_3}e^{-1.5t}=Be^{-1.5t} [where B=\pm e^{-1.5C_3}].

    v(20)=202.712

    -272.068=B*e^{-30}

    B=-2.907*10^{15}

    32-1.5v=-2.907*10^{15}e^{-1.5t}

    32-1.5\frac{dx}{dt}=-2.907*10^{15}e^{-1.5t}

    1.5\frac{dx}{dt}=32+2.907*10^{15}e^{-1.5t}

    \int 1.5dx=\int (32+2.907*10^{15}e^{-1.5t})dt

    1.5x=32t-1.938*10^{15}e^{-1.5t}+C_4

    x(20)=2915.253

    4372.88=640-1.938*10^{15}e^{-30}+C_4

    C_4=3914.231

    1.5x=32t+1.938*10^{15}e^{-1.5t}+3914.231

    When x=10,000,

    15000=32t+1.938*10^{15}e^{-1.5t}+3914.231

    32t+1.938*10^{15}e^{-1.5t}=11085.769

    t\approx 346\ s=5\ min\ 46\ s
    Last edited by alexmahone; September 21st 2011 at 05:29 AM. Reason: Solved it!!
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  2. #2
    Grand Panjandrum
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    Re: A woman bails out of an airplane...

    It's a pity that you have had this set with an air resistance model appropriate for low Reynolds number motion in a situation where the Reynolds number is high.

    CB
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