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        riddles >> hard >> THE CHASE
        
(Message started by: pcbouhid on Dec 2^nd, 2005, 8:24am)

Title: THE CHASE
Post by pcbouhid on Dec 2^nd, 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?

Title: Re: THE CHASE
Post by Joe Fendel on Dec 2^nd, 2005, 12:52pm

I may get really seasick :P, but I'll try this anyway.

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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 = theta_B, 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 (theta_A(t), 15t).

What is left is to find the function theta_A(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 * theta_A'(t))^2), which we know is 30. Thus theta_A'(t) = sqrt(3) / t, so theta_A(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.
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