Results 1 to 2 of 2

Math Help - River mechanics question

  1. #1
    0-)
    0-) is offline
    Newbie
    Joined
    Jan 2008
    Posts
    21

    River mechanics question

    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.
    Last edited by 0-); January 20th 2008 at 12:54 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by 0-) View Post
    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.
    The swimmer's speed in the j direction is constant (v sin θ). So at time t, when he has travelled a distance y in that direction, y will be equal to (v sin θ)t. For that value of y, the corresponding value of u will be a(v sin θ)t(W (v sin θ)t).

    To get the total distance travelled downstream, we have to integrate the velocity u + v cos θ as t goes from 0 to W/v.sin θ:

    . . . . . .Distance = \int_0^{W/v\sin\theta} \!\!\!\!\!\!\bigl(av\sin\theta.t(W - v\sin\theta.t) + v\cos\theta\bigr)dt.

    Integrate that and you should get the right answer.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Mechanics question
    Posted in the Math Topics Forum
    Replies: 4
    Last Post: March 13th 2011, 04:59 PM
  2. Mechanics Question
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: March 8th 2010, 12:33 PM
  3. Mechanics Question
    Posted in the Advanced Applied Math Forum
    Replies: 5
    Last Post: March 13th 2009, 09:49 AM
  4. Mechanics question
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 4th 2008, 09:47 AM
  5. River mechanics problem
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: January 22nd 2008, 05:15 PM

Search Tags


/mathhelpforum @mathhelpforum