Chapter 5.3, Problem 18E

Pursuit curve In another naval exercise a destroyer S₁ pursues a submerged submarine S₂. Suppose that S₁ at (9, 0) on the x-axis detects S₂ at (0, 0) and that S₂ simultaneously detects S₁. The captain of the destroyer S₁ assumes that the submarine will take immediate evasive action and conjectures that its likely new course is the straight line indicated in Figure 5.3.10. When S₁ 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.

1. (a) Explain why the captain waits until S₁ reaches (3, 0) before ordering a course change to C.
2. (b) Using polar coordinates, find an equation r = f (θ) for the curve C.
3. (c) Let T denote the time, measured from the initial detection, at which the destroyer intercepts the submarine. Find an upper bound for T.