Let me try...(I assume you meant the 'best' strategy I gave, as my assertion)QuoteOriginally posted by: sevvostTo me it seems logical that the swimmer should maneuver himself into a right position relative to the runner and then make a dash for the edge. At least, that is how the escape strategy for speed ratio = 4 works.Let me take it from there. I think you will agree that the 'right position' is the one that minimizes the distance of the swimmer(S) from the edge and maximizes the distance between S and chaser(C). And that is achieved by maintaining collinearity of S,center and C. And as shown below, it can be maintained only up to a certain radius(r) above which the (angular) velocity stays at the maximum velocity vs and radial component of velocity vanishes.From that point, S has the option to choose any path that will give him an advantage over C in reaching a point on the edge and simultaneously C continues closing down the angle constantly from one side (doing so otherwise will be like retracing its path while S is moving further towards the edge). It can be either a radial path (which I used below) or any curved/straight non-radial path. If it is non-radial, it makes sense only if the relative angular velocity of S with respect to C is positive (in other words, their angular seperation widens). But, this cannot happen or else it will nullify the radial velocity by virtue of the maximum velocity specified for S (this infact is the very reason for discontinuing the original collinear motion tactic) QuoteOriginally posted by: TheTheorist.....and since vs is max possible speed of S, vs_r = (vs^2 - (vc/R*r)^2)^0.5 ; r=instataneous distance of S from centervs_r >= 0 ----> r <= R/y ; y=(vc/vs)For maximum success S will follow the collinear strategy till r=R/y after which he will head straight to the edge which is at a distance (R-R/y)..... This is not a rigorous proof, i agree... but, intuitive.

QuoteOriginally posted by: TheTheoristThis is not a rigorous proof, i agree... but, intuitive.This is not a proof. And no, I don't agree with any of your erroneous assumptions. Again, your answer is wrong - in case you care about a trifle like that.QuoteOriginally posted by: sevvostThis question does indeed have a definite answer - given earlier in this thread. The precise statement: if speed ratio x < A (the root of the equation cited in the earlier post), there exists an escape strategy for the swimmer. If x > A, there exists a catching strategy for the runner.QuoteOriginally posted by: sevvost.

Sevvost, an outline of how you obtained that solution and why is it different from the strategy that I (or most others, for that matter) stated might make things clearer for everyone, right?

Well, I could in fact post a complete solution here. The only snag is that it would take me a few hours, at the very least, and I happen to have a few other things to entertain myself with at the moment. Even a precise description of the optimal strategy is a non-trivial thing. And, btw, that is exactly why I think that this particular problem might not be really suitable for a forum like this. I myself would never post a problem like this here. Quite frankly, I just feel that posting a solution should not really be my duty, but rather that of the person who had originated this thread.Having said that, I will see what I can do over the weekend or whenever I have some free time.

Nice. That will be interesting to compare with my vector mechanics solution. I am very curious to see where my mistake lies, if indeed there is one.

QuoteOriginally posted by: TheTheoristNice. That will be interesting to compare with my vector mechanics solution. I am very curious to see where my mistake lies, if indeed there is one.I am not sure what "mistake" you are talking about. I think before discussing mistakes it makes sense to formulate an exact result you are trying to prove (if any) and then come up with an actual proof.

sevvost - quoting a formula without any information about how it is derived does not really help those of us on this thread who are scratching our heads over this problem. TheTheorist's strategy is the one I intuitively grasp as being optimal, although I am happy to admit that I cannot prove it.If your proof is really that complex, then perhaps you can at least point us in the right direction i.e. what concepts are you using? Maybe we can then figure it out for ourselves.

Fair enough. As I said, I will see what I can do. Meanwhile - well, do a little more head-scratching (don't hurt yourself, though.)Here is an idea. The strategy you mention has the swimmer at some moment head radially for the edge. Wouldn't it be nice if we could have him instead head for the edge tangentially and at the same time somehow make sure that the chaser would run after him along the longer arc?

I have posted my argument here several weeks ago. Since there are a lot of discussions now, and I find that my last post could not explain fully what I thought. It would be much better for me to explain more in details. Please kindly comment my solution.I definitely agree that writing down an answer without putting any argument does not help anything. It is much better to post the argument along with the solution such that everyone can examine the logic of the argument and the validity of the answer. However, I do hope the discussion here in this forum is limited to a gentleman discussion.The key point is the intelligence of the runner and the swimmer as well as the free flows of information. The runner may not know the maximum speed of the swimmer. Will the swimmer let the runner know his maximum speed? I do not think so. The runner may not know the max speed of the swimmer. Thus, it opens the door for the intelligence of these two players in determining the answer of this question. If the runner knows the swimmer’s max speed, it is a simple maths problem. However, it may not be the case.OK! Let us begin with the point that the swimmer reaches the circle of radius r and the swimmer is 180 degrees out of phase of the runner. (Please see my last post for definition of the symbol.) As I said before, the swimmer employs the ziz-zag pattern strategy. When the swimmer swims to the right at an angle almost near 90 degrees to the line AB, the runner will of course run in the direction to the right. If suddenly the swimmer changes his direction and swims to the left at an angle almost near 90 degrees to the line AB, the runner must determine whether he should change the direction as well. If the runner knows the swimmer’s max speed, he can of course easily determine. However, if he does not know, he will ask, “Well, is this his max speed? If I continue in this direction, I must run nearly 270 degrees (Actually less than 270 degrees). What happens if I continue in this direction, and the swimmer suddenly swims very fast, I may not be able to catch him as it is a longer course. If I change direction, I need to run only 90 degrees. (Actually more than 90 degrees)”The runner may change his direction. If it happens that the runner has followed the swimmer’s ziz-zag pattern for several times, the swimmer has already been much nearer to the edge of the pool. When the runner discovers the swimmer’s trick, and if the swimmer also discovers that his trick has been known at the same time, the swimmer just simply swims toward the edge of the pool in the direction of minimum distance and at the max speed. The max speed required for the runner to catch the swimmer depends on how soon he knows the trick.What happens if the runner does not change his course at the first time that the swimmer changes his direction. That is the runner not buying the ziz-zag game. The runner will run nearly 270 degrees depending on the angle of the direction of the swimmer to AB. Let us keep the problem simple. We assume that the swimmer swims at 90 degrees to AB to the first order of approximation. The distance that the swimmer must swim to reach the edge is where x = v/s.The distance that the runner must run to catch the swimmer is Equating the time for the swimmer to travel and the runner to travel, , we get .Solving this equation, we get x= 4.6.I should point out that I do not mean that x= 4.6 is the answer. Even though in the case that the runner knows the swimmer's max speed, x may not equal to 4.6.I have not proved that x = 4.6 is the speed required for all the cases. However, as I point out that earlier, the runner may not know the swimmer's max speed and x may bemuch larger than 4.6 depending how smart the runner is.Hope I can explain more about my solution.

QuoteOriginally posted by: MsccubeWhat happens if the runner does not change his course at the first time that the swimmer changes his direction. That is the runner not buying the ziz-zag game. The runner will run nearly 270 degrees depending on the angle of the direction of the swimmer to AB. Let us keep the problem simple. We assume that the swimmer swims at 90 degrees to AB to the first order of approximation. The distance that the swimmer must swim to reach the edge is where x = v/s.The distance that the runner must run to catch the swimmer is Equating the time for the swimmer to travel and the runner to travel, , we get .Solving this equation, we get x= 4.6.I should point out that I do not mean that x= 4.6 is the answer. Even though in the case that the runner knows the swimmer's max speed, x may not equal to 4.6.I have not proved that x = 4.6 is the speed required for all the cases. However, as I point out that earlier, the runner may not know the swimmer's max speed and x may bemuch larger than 4.6 depending how smart the runner is.Ah, yes. I see what you mean. To solve for an optimal path for S from the 'right position' is pretty difficult. I doubt if it is even solvable. It is like the chicken and egg problem. I would say that C should know S's max speed. Without that information, whatever C chooses to do is on blind faith and random and S can escape anytime. Even with S's speed, the better strategy for C will be to pick a direction and run as long as S is outside 'that' circle. Else, as you said, S will zig-zag around a mean radius vector (with a non-zero angle) and confuse and fix C at a point and escape.The strategy to stick with a 270deg run is such a worst-case "pick and run" strategy. While S may have started in a tangential direction (which, if continued, would have made C catch up with S with the above 4.6 ratio), S can change his direction of swim midway (i.e. choose a point where to make a change and also a new angle to change to). When this happens, C may or may not be able to catch up depending on the extra time needed to cover the extra arc length compared to the difference of S's new distance to the edge and the old distance to the edge. In a different case, S may have started out radially first and then change to a path of tangent to 'that' (critical) circle. And any case inbetween.This reduces to the question: From any point(A) within a circle of radius R (and outside radius R/4.6), draw a straight line to the boundary(B) such that it does not intesect the inner circle of radius R/4.6. Now, from 'A' draw another line to the boundary (C) with the same property. Is {(arcBC)/4.6} <= (AC-AB)? I am not sure. If it is true, then 4.6 is indeed the solution.

Hi The Theorist,An interesting fact: supposing that the runner and the swimmer know each other very well and there is no secert between them,the runner knows the max speed of the swimmer and the final landing point of swimmer whatever the course the swimmer takes.It means that it does not matter how the swimmer swims, the runner knows what the swimmer is thinking and his intended finallanding point such that the runner can choose the shorest path to reach that point. It also assumes that the swimmer knowsthat the runner has known everything.If it is the case, the answer should be x = 1+pi since the swimmer must take the shortest path once reaching the circle of radius r.(That is swimming in the radial direction toward the edge.) The runner will run only 180 degrees.Msccube [Edit] : I am sorry that there is a typo. I mean x = 1 + pi , not x = 4.3 . My apology.

I fail to grasp what all this mind reading has to do with this problem. Here is my understanding of the problem. I think it would be reasonable to assume that at all times each one is aware of the location of the other one. Based on this, each one needs to come up with the optimal strategy. Our problem is: find the value A such that the swimmer can always escape (no matter how the chaser moves) if the speed ratio is < A and the chaser can always catch him (no matter how the swimmer moves) if the ratio > A. Or rather prove that A is the root of the equation cited at least 3 times in this thread.

I wonder if a chapter in this paper could pass for a proof. See section 5.4, pages 29-33.

Lol. How many academic papers have a chapter called 'The Homicidal chauffeur'?Who said mathematicians have no sense of humour?

I think that I should summarize my answer to this problem.Case 1: If the runner and swimmer know each other well. They know each other's max speed and the runnerknow the actual landing point of the swimmer. The answer is x = 1+pi.Case 2. If the runner and swimmer keep their max speeds as the top secert, and the runner does not know the actual landingpoint of the swimmer, there is no answer to this problem. It depends on the relative intelligence and skills.My arguments have been posted in this thread.PS. I do not have the time to go through all the maths given in this Paper,It seems to me that this paper has not considered the ziz-zag strategy. It just claims that the best strategy is for the lady to run in this direction .......