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Math Help - vector problems

  1. #1
    Member Mr Rayon's Avatar
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    vector problems

    An aeroplane can travel at a speed of 120 km/h there's no wind. Today there's a wind velocity of 3i + 2j km/h. Find:

    a) the speed of the wind
    b) the poisition vector if the aeroplane is to end up due north of its starting point after 1 hour
    c) the actual bearing required.
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  2. #2
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    Quote Originally Posted by Mr Rayon View Post
    An aeroplane can travel at a speed of 120 km/h there's no wind. Today there's a wind velocity of 3i + 2j km/h. Find:

    a) the speed of the wind
    b) the poisition vector if the aeroplane is to end up due north of its starting point after 1 hour
    c) the actual bearing required.
    a) |v_w| = \sqrt{3^2 + 2^2}


    a bit easier to do part (c) first ...

    to travel due north relative to the ground, the airplane will have to steer west of north because of the wind blowing to the northeast.

    let \theta = air velocity vector, \vec{v_a} , angle relative to east.

    \vec{v_a} = 120(\cos{\theta} \vec{i} + \sin{\theta}\vec{j})

    wind velocity vector ...

    \vec{v_w} = 3\vec{i} + 2\vec{j}

    ground velocity vector ...

    \vec{v_g} = |v_g|\vec{j}


    now, since \vec{v_a} + \vec{v_w} = \vec{v_g} ...

    in the x-direction, 120\cos{\theta} + 3 = 0

    \cos{\theta} = -\frac{1}{40}

    \theta \approx 91.4^\circ relative to east

    the airplane needs to steer 1.4 ^\circ west of north.

    in the y-direction ...

    120\sin{\theta} + 2 = |v_g| \approx 122 km/hr

    and for part (b) ... the position vector of the airplane after 1 hr is

    \vec{r_g} = 122\vec{j} km
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