In a naval exercise a ship S₁ is pursued by a submarine S₂ as shown in Figure 5.3.8. Ship S₁ departs point (0, 0) at t = 0 and proceeds along a straightline course (the y-axis) at a constant speed v₁. The submarine S₂ keeps ship S₁ in visual contact, indicated by the straight dashed line L in the figure, while traveling at a constant speed v₂ along a curve C. Assume that ship S₂ starts at the point (a, 0), a > 0, at t = 0 and that L is tangent to C.

(a) Determine a mathematical model that describes the curve C.

(b) Find an explicit solution of the differential equation. For convenience define r = v₁/v₂.

(c) Determine whether the paths of S₁ and S₂ will ever intersect by considering the cases r > 1, r < 1, and r = 1.

Where s is arc length measured along C.

Figure 5.3.8.