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Title: THE CHASE Post by pcbouhid on Dec 2nd, 2005, 8:24am Ship A is chasing ship B and making 30 knots to 15 for the pursued, and neither is equipped with radar. Ship B enters a cloud bank which makes its further visibility from ship A impossible. Captain A correctly assume that Captain B will take advantage of the fog to immediately change his course and will mantain his new direction unchanged at full speed. Based on this assumption, what plan should Captain A follow to insure that he will intercept ship B? |
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Title: Re: THE CHASE Post by Joe Fendel on Dec 2nd, 2005, 12:52pm I may get really seasick :P, but I'll try this anyway. [hide] We're not given the shape of the cloud, but let's assume the "worst case" scenario, that the cloud is spreading in every direction faster than 15 knots, so that Captain B has full 360-degree freedom with his angle choice. The set of possible locations of B can be described by a ripple moving outward from the point when B entered the cloud. If (theta, x) are the polar coordinates associated with B's location with respect to this point, then we have x = 15t and theta = thetaB, a constant. Thus A should continue his chase until his boat meets this ripple, at time t0. At this point, A begins searching around the ripple, by following a route given by polar coordinates (thetaA(t), 15t). What is left is to find the function thetaA(t). My calculus is a bit rusty, but I think the speed of A's boat can be shown to be equal to 15*sqrt(1 + (t * thetaA'(t))^2), which we know is 30. Thus thetaA'(t) = sqrt(3) / t, so thetaA(t) = sqrt(3)*ln(t) - C, where C = sqrt(3)*ln(t0). Since the natural log function increases unboundedly, A will eventually traverse the whole ripple and catch B. [/hide] |
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