Pursuit Curve In a naval exercise a ship S1 is pursued by

Chapter 5, Problem 17E

(choose chapter or problem)

Pursuit Curve In a naval exercise a ship \(S_{1}\) is pursued by a submarine \(S_{2}\) as shown in Figure 5.3.8. Ship \(S_{1}\) departs point (0, 0) at t  0 and proceeds along a straightline course (the y-axis) at a constant speed \(v_{1}\). The submarine \(S_{2}\) keeps ship \(S_{1}\) in visual contact, indicated by the straight dashed line L in the figure while traveling at a constant speed \(v_{2}\) along a curve C. Assume that ship \(S_{2}\) starts at the point \((a, 0), a>0, \text { at } t=0\) 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 \(r=v_{1} / v_{2}\).

(c) Determine whether the paths of (S_{1}\) and  \(S_{2}\) will ever intersect by considering the cases r>1, r<2, and r = 1. [Hint: \(\frac{d t}{d x}=\frac{d t}{d s} \frac{d s}{d x}\), where s is arc length measured along C.]

Text Transcription:

S_1

S_2

v_1

v_2

(a, 0),a>0,att=0

r=v_1v_2

fracdtdx=fracdtdsfracdsdx