Pursuit Curve In another naval exercise a destroyer S1

Chapter 5, Problem 18E

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Pursuit Curve In another naval exercise a destroyer \(S_{1}\) pursues a submerged submarine \(S_{2}\). Suppose that \(S_{1}\) at (9, 0) on the x-axis detects \(S_{2}\) at (0, 0) and that \(S_{2}\) simultaneously detects \(S_{1}\). The captain of the destroyer \(S_{1}\) assumes that the submarine will take immediate evasive action and conjectures that its likely new course is the straight line indicated in Figure 5.3.9. When \(S_{1}\) is at (3, 0), it changes from its straight-line course toward the origin to a pursuit curve C. Assume that the speed of the destroyer is, at all times, a constant 30 mi/h and that the submarine’s speed is a constant 15 mi/h.

(a) Explain why the captain waits until \(S_{1}\) reaches (3, 0) before ordering a course change to C.

(b) Using polar coordinates, find an equation \(r=f(\theta)\) for the curve C.

(c) Let T denote the time, measured from the initial detection, at which the destroyer intercepts the submarine. Find an upper bound for T.

Text Transcription:

S_1

S_2

r=ftheta