Old Man River... In Figure 3.2.8(a) suppose that the y-axis and the dashed verti

Question

Old Man River... In Figure 3.2.8(a) suppose that the y-axis and the dashed vertical line x 1 represent, respectively, the straight west and east beaches of a river that is 1 mile wide. The river ows northward with a velocity vr, where mi/h is a constant. A man enters the current at the point (1, 0) on the east shore and swims in a direction and rate relative to the river given by the vector vs, where the speed mi/h is a constant. The man wants to reach the west beach exactly at (0, 0) and so swims in such a manner that keeps his velocity vector vs always directed toward the point (0, 0). Use Figure 3.2.8(b) as an aid in showing that a mathematical model for the path of the swimmer in the river is
[Hint: The velocity v of the swimmer along the path or curve shown in Figure 3.2.8 is the resultant v vs vr. Resolve vs and vr into components in the x- and y-directions. If are parametric equations of the swimmer’s path, then .

Explanation / Answer

The diagrams are missing, please add the images.

Let velocity of the river vr = xi+yj ;
Velocity of the swimmer (in still water) be = ai + bj;
The net 'v' of vr and vs = vr+vs = (x+a) i + (y+b) j
this v = (0-1) i + (0-0) j (i.e. from point (1,0) to (0,0) )
v= -i
Thus x+a = -1 or x= a-1
y+b =0 or y=-b

Thus vr = xi + yj
vs = (x+1) i - y j

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