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Math Help - Vector Addition Question

  1. #1
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    Vector Addition Question

    An aircraft is flying at a speed of 2000km/h, wishes to reach a city 3000 km due East. Encounters wind blowing from N70W with speed of 100 km/h. Find the velocity of the plane relative to the ground. How long will it take to reach the city?
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  2. #2
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    Hello, peon123!

    A plane, flying at 2000 km/h, wishes to reach a city 3000 km due East,
    encounters wind blowing from N70W with speed 100 km/h.
    (a) Find the velocity of the plane relative to the ground.
    (b) How long will it take to reach the city?
    Code:
                   D      N
                    *     :
                      *70:
                        * :
                          oC
             2000t    *   : *  100t
                  *       :   *
              *              20*
          o - - - - - - - - - - - o
         A           3000          B

    The plane intends to fly from A to B: . AB = 3000 km.

    The wind is blowing from D to B: . \angle DCN = 70^o
    . . Hence: . \angle CBA = 20^p

    Because of the wind, the plane flies from A to C at 2000 km/hr.
    . . In t hours, it flies 2000t km: . AC = 2000t

    In the same t hours, the wind blows from C to B: . CB = 100t km.


    In \Delta ABC, Law of Cosines: . (2000t)^2 \:=\:3000^2+(100t)^2 - 2(3000)(100t)\cos20^o

    This simplifies to: . 399t^2 + 56.382t - 900 \:=\:0

    Quadratic Formula: . t \;=\;\frac{-56.382 \pm\sqrt{56.382^2 + 4(399)(900)}}{2(399)}

    . . which has the positive root: . t \:\approx\:1.433


    (b) The flight took 1.433 hours.


    (a) The plane's speed was: . \frac{3000\text{ km}}{1.433\text{ hrs}} \;\approx\;2094\text{ km/hr}

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