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Math Help - Extremely difficult question on relative velocity?

  1. #1
    Super Member fardeen_gen's Avatar
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    Extremely difficult question on relative velocity?

    A man is walking in the north-east direction and wind appears to blow from north. If the man doubles his speed, wind appears at angle arccot 2 east of north. Find the actual direction of the wind.
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  2. #2
    MHF Contributor

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    The wind comes from the west. You can check that it works; anyway, here is how I found this out:

    Let us denote by \vec{w}={w_E\choose w_N} and \vec{v}={\alpha \choose \alpha} the respective speeds of the wind and the man, in an orthonormal basis (\vec{e}_E,\vec{e}_N) (E for east, N for north). Since the man goes NE, both components of v are equal to some positive \alpha.

    We know that the relative speed, i.e. the vector \vec{w}-\vec{v}={w_E-\alpha\choose w_N-\alpha}, is along the N direction, hence the E component is zero: w_E=\alpha.

    Then we are told that the vector \vec{w}-2\vec{v}={w_E-2\alpha\choose w_N-2\alpha} is (positively) colinear with -{1\choose 2}. This is the content of the statement about "arccot 2" (draw a right triangle with perpendicular sides of length 1 and 2; the least angle measures {\mathop{arccot}}(2)). As a consequence, \frac{w_N-2\alpha}{w_E-2\alpha}=2. Replace \alpha by w_E (as seen above), to get 2-\frac{w_N}{w_E}=2, hence w_N=0.

    Finally, \vec{w}={\alpha\choose 0} (the same \alpha as in \vec{v}), and \alpha>0, which means that the wind comes from the west.

    I hope my explanations are clear.
    Laurent.
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