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Math Help - River mechanics problem

  1. #1
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    River mechanics problem

    I'm having trouble with this question and I appreciate anyone's help.

    A man swims across a straight river of uniform width W, starting from a point O on one bank of the river. The velocity of the river at a distance y from the bank is
    u(y)=ay(W − y), where a is a positive constant. The man travels at a constant speed v relative to the current and steers a course set at a
    constant angle θ (0 < θ < pi) to the downstream direction.

    (a) Show that the velocity of the man relative to a Cartesian coordinate system with origin at O, i pointing in the downstream direction, and j pointing across the river is given by (u + v cos θ)i + (v sin θ)j. I've done this part.

    (b) At what time does the man reach the other bank?
    I get t = W/v.sinθ - I think that's correct.

    (c) Show that when the man has reached the other bank, the downstream distance he has travelled is equal to aW^3/6v.sinθ + W.cot θ.

    This is the part i'm having problems with. I thought about replacing t from part (b) into the integrated i component from the velocity in part (a). I'm not sure what to do with the u(y) though.
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  2. #2
    Senior Member JaneBennet's Avatar
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    When the swimmer has swum for time t, he is distance (v\sin{\theta})t away from the shore. The downstream current speed at that spot is a(v\sin{\theta})t[W-(v\sin{\theta})t]=aW(v\sin{\theta})t-a(v^2\sin^2{\theta})t^2. So the total horizontal component of the manís velocity is aW(v\sin{\theta})t-a(v^2\sin^2{\theta})t^2+v\cos{\theta}.

    Hence, the total horizontal displacement of the man is

    \color{white}.\quad. \int_0^{\frac{W}{v\sin{\theta}}}{[aW(v\sin{\theta})t-a(v^2\sin^2{\theta})t^2+v\cos{\theta}]}\,\mathrm{d}t
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