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A boat moving because of the wind

  1. Dec 18, 2012 #1
    1. The problem statement, all variables and given/known data
    A boat has a sail that has a surface S towards the wind.
    The wind blows at constant velocity u.

    Also given the density of the air, p.

    Find the velocity of the boat as a function of time.
    What will be the velocity of the boat after a long enough time? What exactly is this "long enough" time?


    2. Relevant equations



    3. The attempt at a solution

    I figured I could write the forces equation, since I know that the force that the wind asserts on an object is proportional to its surface and to the velocity^2.

    k is some constant,
    So F = k*S*(u-v)^2 = ma

    But then I get a diff. equation with v, v dot, and v^2, which I dont know how to solve.
    Is there an easier way to solve this?
     
    Last edited: Dec 18, 2012
  2. jcsd
  3. Dec 18, 2012 #2

    haruspex

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    Please post the equation you get. If you get what I get, you should find that it is easily turned into a quite straightforward integral.
     
  4. Dec 18, 2012 #3

    SteamKing

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    Is the force from the sail the only force acting on the boat? Think about what happens after the boat begins to move.
     
  5. Dec 18, 2012 #4
    This is the equation I get:
    k*S*(u-v)^2 = ma
    k*S*(u^2-2uv+v^2)= ma

    How does that become a nice integral?
     
  6. Dec 19, 2012 #5

    haruspex

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    u is constant, a = dv/dt:
    k*S*dt = m(u-v)-2dv

    As Steamking points out, in the real world there is also a drag related to v. But typically that is taken to be a quadratic in v also, so the equation doesn't change much.
     
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