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Math Help - Velocity Vector Problem

  1. #1
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    Velocity Vector Problem

    There is a plane that just took off and is climbing northwest through still air at an airspeed of 200 km/hr, and rising at a ate of 300 m/min. Resolve its velocity vector into components.


    First, I converted 200 m/min to 18 km/hr so all values have the same units. Then I said the velocity of the plane would =


    (-) 200 cos135degrees i + 200 sin135degrees j + 18 k

    = - 141.42 i + 141.42 j + 18 k

    However, the correct answer is - 140.8 i + 140.8 j + 18 k

    Can someone please explain where I made my mistake? Thanks very much!
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  2. #2
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    There is a plane that just took off and is climbing northwest through still air at an airspeed of 200 km/hr, and rising at a rate of 300 m/min. Resolve its velocity vector into components.
    300 m/min = 18 km/hr

    angle of climb relative to the x-y plane ... \theta = \arcsin\left(\frac{18}{200}\right) \approx 5.16^{\circ}

    speed along the x-y plane is 200\cos{\theta}

    x component = -(200\cos{\theta})\cos(45^{\circ}) = -140.8 \, km/hr

    y component = (200\cos{\theta})\sin(45^{\circ}) = 140.8 \, km/hr
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  3. #3
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    Hello, Frostking!

    There is a plane that just took off and is climbing northwest through still air
    at an airspeed of 200 km/hr, and rising at a rate of 300 m/min.
    Resolve its velocity vector into components.

    The answer is: . - 140.8 i + 140.8 j + 18 k

    Look at the side view of the plane climbing.
    Code:
        B *
          |   *
          |       *   200
       18 |           *
          |               *
          |                   *
        C * - - - - - - - - - - - * A
                      x

    The hypotenuse AB = 200,\;BC = 18.

    Hence: .  x\:=\:\sqrt{200^2-18^2} \:=\:\sqrt{39,\!676} \quad\Rightarrow\quad x \:\approx\: 199.19


    Now look down at the ground.
    Code:
                    |
        C * - - - - + E
          : *       |
          :   * x   |
          :     *45|
          :   45 * |
      - - + - - - - * - - -
          D         |A
                    |

    We see that: . AD \:=\:AE \:=\:\frac{x}{\sqrt{2}} \:\approx\:140.8

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