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Math Help - relative velocity

  1. #1
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    relative velocity

    A canal of width 2a has parallel straight banks and the water flows due north . The points A and B are on opposite banks and B is due east of A , with the point O as the midpoint of AB . THe x-axis and y-axis are taken in the east and north directions respectively with O as the origin . The speed of the current in the canal , v_c=v_o(1-\frac{x^2}{a^2}) , where v_o is the speed of the current in the middle of the canal and x the distance eastwards from the middle of the canal . A swimmer swims from A towards the east at speed v_r , relative to the current in the canal . Taking y to denote the distance northwards travelled by the swimmer , show that

     <br />
\frac{dy}{dx}=\frac{v_o}{v_r}(1-\frac{x^2}{a^2})<br />
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  2. #2
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    Hello hooke
    Quote Originally Posted by hooke View Post
    A canal of width 2a has parallel straight banks and the water flows due north . The points A and B are on opposite banks and B is due east of A , with the point O as the midpoint of AB . THe x-axis and y-axis are taken in the east and north directions respectively with O as the origin . The speed of the current in the canal , v_c=v_o(1-\frac{x^2}{a^2}) , where v_o is the speed of the current in the middle of the canal and x the distance eastwards from the middle of the canal . A swimmer swims from A towards the east at speed v_r , relative to the current in the canal . Taking y to denote the distance northwards travelled by the swimmer , show that

     <br />
\frac{dy}{dx}=\frac{v_o}{v_r}(1-\frac{x^2}{a^2})<br />
    Draw a triangle of velocities such that the velocity of the swimmer relative to the ground = velocity of swimmer relative to the water plus the velocity of the water relative the ground.

    This is a right-angled triangle, in which the velocity of the swimmer relative to the ground has a component v_r eastward and v_c northward. So, if (x,y) is his position relative to the ground at time t, we get:
    \frac{dx}{dt}= v_r

     \Rightarrow \frac{dt}{dx}=\frac{1}{v_r}
    and
    \frac{dy}{dt}=v_c=v_o\left(1-\frac{x^2}{a^2}\right)

    \Rightarrow \frac{dy}{dx}=\frac{dy}{dt}\times\frac{dt}{dx}
    =\frac{v_o}{v_r}\left(1-\frac{x^2}{a^2}\right)
    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello hookeDraw a triangle of velocities such that the velocity of the swimmer relative to the ground = velocity of swimmer relative to the water plus the velocity of the water relative the ground.

    This is a right-angled triangle, in which the velocity of the swimmer relative to the ground has a component v_r eastward and v_c northward. So, if (x,y) is his position relative to the ground at time t, we get:
    \frac{dx}{dt}= v_r

     \Rightarrow \frac{dt}{dx}=\frac{1}{v_r}
    and
    \frac{dy}{dt}=v_c=v_o\left(1-\frac{x^2}{a^2}\right)

    \Rightarrow \frac{dy}{dx}=\frac{dy}{dt}\times\frac{dt}{dx}
    =\frac{v_o}{v_r}\left(1-\frac{x^2}{a^2}\right)
    Grandad
    Thanks a lot Grandad , i would like some clarification on some terms used here ,

    velocity of swimmer relative to water means his velocity is 'affected' by the velocity of water , so in this case would be the hypothenus of the triangle ?

    velocity of swimmer relative to the ground - 'nett' velocity ?
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  4. #4
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    Relative velocity

    Hello hooke
    Quote Originally Posted by hooke View Post
    Thanks a lot Grandad , i would like some clarification on some terms used here ,

    velocity of swimmer relative to water means his velocity is 'affected' by the velocity of water
    Any velocity is made up of (a) magnitude; and (b) direction. For the velocity of the swimmer relative to the water:

    • (a) the magnitude is his speed in still water; in this question this is v_r.


    • (b) the direction is the direction in which he is pointing (which is not necessarily the direction in which he's moving, because the current will take him 'off-course'). In this question he's pointing due East.

    To get a clearer picture of what this means, imagine that you're in a helicopter flying due North (the direction of the current) directly over the river, moving at exactly the same speed as the water. So the water appears still to you. As you look down, you'll see the swimmer as though he were swimming through still water. That's his velocity relative to the water (and, indeed, relative to you at this moment).
    so in this case would be the hypothenus of the triangle ?
    No, this is not the hypotenuse of the triangle. The hypotenuse is what you have referred to as his 'nett' velocity - the velocity of the swimmer relative to the ground.

    Study the diagram I have attached, and you should see what I mean. If you're still not sure, imagine that the movements of the swimmer and the water take place separately. First the swimmer has a turn, and he swims a certain distance eastwards; then it's the water go, and it carries the swimmer a little way northwards. Well, just allow those two movements to happen at the same time, and you're there.

    Grandad
    Attached Thumbnails Attached Thumbnails relative velocity-untitled.jpg  
    Thanks from takho95
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    Re: relative velocity

    This is another part of the question.. If the width of the canal is 12m, the speed of the current in the middle of the canal is 10 meter per second and the speed of the swimmer is 2 meter per second relative to the current in the canal. Find the distance of the swimmer from O when he is at the middle of the canal and his distance from B when he reaches the east bank of the canal. Please help.. Thanks
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