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Math Help - Vectors In Component Form

  1. #1
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    Exclamation Vectors In Component Form

    Hey Everyone,
    If Possible need some help with the following Question. I have attempted (1) and (2) and need some help on (3). Thanks. Any Contribution will be appreciated.


    A plane can fly at 350 km/h in still air. The pilot wishes to fly from airport A to airport B where AB= (2000i + 5000j) km



    1)
    1) There is a wind blowing with velocity ( 30i – 20j) km/h. Find the velocity vector, in the form ai + bj, which the pilot should set so that the plane flies directly from A to B.

    My Working Out:


    AB = p + w

    λ(2000i +5000j) = (ai+bj) + (30i-20j)

    2000iλ + 5000jλ = (a+30)i + (b-20)j

    a+30 = 2000λ ... (1)
    b-20 = 5000λ ... (2)

    (2)/(1) =

    b-20/a+30 = 2.5

    b-20 + 2.5 a + 75

    b = 2.5 a + 95

    __

    a2 + b2 = 3502
    a2 + (2.5a + 95 ) 2 = 3502
    a2 + 6.25a2 + 475a + 9025 = 122500
    7.25 a2 + 475a – 113475 = 0

    Using the quadratic equation

    a = 96.56 and a = - 162.083
    b = 2.5 (96.56 ) + 95 = 336.4
    b = 2.5 (- 162.083) + 95 = -310.2075

    Therefore p = -310i + 336j km/h



    2)
    How Long will this flight take?




    Velocity = Distance / Time



    Time = Distance / Velocity




    Time = Square Root of [(-310)^2 + (336)^2]
    =457 km



    Time = 457 / 350 = 1.306




    Time = 1 hour and 18 Minutes





    3)
    a)How long will it take for the plane to fly from A to B in still air?


    Thankyou.
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  2. #2
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    Quote Originally Posted by Pandp77 View Post
    ...

    A plane can fly at 350 km/h in still air. The pilot wishes to fly from airport A to airport B where AB= (2000i + 5000j) km
    ....

    3)
    a)How long will it take for the plane to fly from A to B in still air?
    1. Calculate the length of \overrightarrow{AB}:

    |\overrightarrow{AB}| = \sqrt{2000^2+50002}^=\sqrt{29,000,000}\approx 5,385.2\ km

    2. time = \frac{distance}{speed}

    t = \frac{5385.2\ km}{350\ \frac{km}{h}} \approx 15.3863 \ h \approx 15h; 23min;11 s
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  3. #3
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    Exclamation Hmmmmmmmmmmmmm

    Thanks for that earmoth.

    But could anyone please see if my working out for question 1 is correct? and explain to me how to do number 3.

    Please do help me as this is due today.


    Thankyou.
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  4. #4
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    Re: Vectors In Component Form

    Do you still want help? :P
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  5. #5
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    Re: Vectors In Component Form

    Hello, Pandp77!

    \text{A plane can fly at 350 km/h in still air. \;The pilot wishes to fly}
    \text{ from airport }A\text{ to airport }B\text{, where }\overrightarrow{AB}\,=\, (2000i + 5000j)

    \text{(1) There is  a wind blowing with velocity }(30i  20j)\text{ km/hr.}
    . . . \text{Find the velocity vector, in the form }ai + bj\text{, which the pilot}
    . . . \text{should set so that the plane flies directly from }A\text{ to }B.
    Code:
                  C
                  *
                 *  *
                *     * B
               *    *
              *   *
             *  *
            * *
           **
          *
          A
    \text{We have: }\:\overrightarrow{AB} \:=\:\langle 2000,5000\rangle,\;\;\overrightarrow{CB} \:=\:\langle 30,\text{-}20\rangle

    \text{And: }\:\overrightarrow{AC} + \overrightarrow{CB} \:=\:\overrightarrow{AB}

    . \overrightarrow{AC} + \langle 30,\text{-}20\rangle \:=\:\langle2000,5000\rangle

    . . . . . . . . \overrightarrow{AC} \:=\:\langle 1970,5020\rangle

    . . . . . . . . \overrightarrow{AC} \:=\:1970i + 5020j




    \text{(2) How long will this flight take?}

    \frac{\text{Distance}}{\text{Speed}} \:=\:\frac{\sqrt{1970^2+5020^2}}{350} \:=\:\frac{\sqrt{29,\!081,\!300}}{350}

    . . . . . . . =\:\frac{5392.70804}{350} \:=\:15.40773726

    \text{Time} \;\approx\;15\text{ hours, }24.5\text{ minutes}




    \text{(3) How long will it take for the plane to fly from }A\text{ to }B\text{ in still air?}

    \left|\overrightarrow{AB}\right| \:=\:\sqrt{2000^2+5000^2} \:=\:\sqrt{29,\!000}

    \frac{\text{Distance}}{\text{Speed}} \:=\:\sqrt{29,\!000}}{350} \:=\:15.38618516

    \text{Time}\;\approx\;15\text{ hours, }23.2\text{ minutes}

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