Chapter 3.2, Problem 28E

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 flows northward with a velocity v_r where |v_r| = v_r 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 v_s, where the speed |v_s| = v_s 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 v_s 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

$\frac{dy}{dx}=\frac{v_{s}y-v_r\sqrt{x^2+y^2}}{v_{s}x}$.

[Hint: The velocity v of the swimmer along the path or curve shown in Figure 3.2.8 is the resultant v = v_s + v_r. Resolve v_s and v_r into components in the x- and y-directions. If x = x(t), y = y(t) are parametric equations of the swimmer’s path, then v = (dx/dt, dy/dt).]