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Homework Help: A difficult boat and river problem

  1. Jul 24, 2010 #1
    1. The problem statement, all variables and given/known data
    A boat is rowed with constant speed u starting from a point A on
    the bank of a river of width d , which flows with a constant speed
    nu . The boatman always points the boat at a point O on the other
    side of the bank opposite to A. Find the equation of the path
    r = f (theta ) for the boat.

    2. Relevant equations
    dx/dt = v

    3. The attempt at a solution
    I was able to find dr/dt in terms of theta. here is my expression dr/dt= u(1-nsin(theta)). The problem is when I integrate it i have to express dt in terms of d(theta) which I cannot find.
     

    Attached Files:

  2. jcsd
  3. Jul 25, 2010 #2
    I don't understand the problem much. Is it the boat's velocity relative to the ground, or its velocity relative to the water, which always points towards the water? I guess it's the latter, because if n>1 then there is no chance that the boatman can guide the boat towards O.

    Let's assume so. I think your equation is a little bit incorrect (just a little bit, most is fine). First, you should write down the formula of velocity in polar coordinates, i.e. write down [tex]\vec{v}[/tex] in term of [tex]r[/tex] , [tex]\theta[/tex] and the unit vectors [tex]\vec{e}_r[/tex] and [tex]\vec{e}_{\theta}[/tex]. Then use the condition that [tex]\vec{v}[/tex] is the sum of 2 velocity: u in the direction towards O and nu in the direction to the left. You should arrive at these 2 equations:

    [tex]nusin\theta-u=\frac{dr}{dt}[/tex]

    [tex]nucos\theta=r\frac{d\theta}{dt}[/tex]

    From here, you can solve for [tex]r(\theta)[/tex].
     

    Attached Files:

  4. Jul 25, 2010 #3
    can u explain me how u arrive at the second equation. I dont understand it
     
  5. Jul 25, 2010 #4
    Can you do step 1, i.e. writing down the formula of velocity in polar coordinates?
     
  6. Jul 25, 2010 #5
    oh ok I got it. I didnt think of v=r(omega). Thx...
     
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