The current speed vr of a straight river such as that in Problem 28 is usually not a con...

The current speed v_r of a straight river such as that in Problem 28 is usually not a constant. Rather, an approximation to the current speed (measured in miles per hour) could be a function such as whose values are small at the shores (in this case, v_r(0) =0 and v_r(1) =0) and largest in the middle of the river. Solve the DE in Problem 30 subject to y(1) =0, where v_s = 2 mi/h and v_r(x) is as given. When the swimmer makes it across the river, how far will he have to walk along the beach to reach the point (0, 0)?

(reference problem 28 )

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 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 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

[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 are parametric equations of the swimmer’s path, then .]

(reference of problem 30)

Old Man River Keeps Moving . . . Suppose the man in Problem 28 again enters the current at (1, 0) but this time decides to swim so that his velocity vector v_s is always directed toward the west beach. Assume that the speed mi/h is a constant. Show that a mathematical model for the path of the swimmer in the river is now

(reference problem 28)

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 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 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

[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 are parametric equations of the swimmer’s path, then .]