Lets say you have 2 ships that travel at the exact same speed on a infinite sea. Ship A is trying to get away from ship B but (ship A) must change direction every so often. Ship B can instantly react to the change of direction of ship A. Direction changes are instant and involve no loss of speed. Will ship B ever catch ship A, or will it be a limit that B can get very very close but never catch it?
If they travel the same speed, how can B catch A? Do they both have instantaneous direction changes? How far apart were they when they started?
I think the point is that if ship A is travelling directly away from ship B, then any change of direction turns it in a course that is not directly away from B, and puts it on a heading toward a point that B can reach more directly (A is travelling two sides of a triangle, B is travelling only one)
I think the question is based on the fact that ship B can “cut the corner” on ship A. For example, if A is 10 miles ahead and makes a 90 degree right hand turn and goes straight for 10 miles before turning back to its original course, A will have traveled 20 miles total. B will only have to travel 14.2 miles to get to the point where A changed back to the original course, so it will have closed the distance between them by 6 miles.
With only 4 miles between them, A can’t do another 10 miles straight, 10 miles up or B will catch it. Even if it does less than 10 miles so that it isn’t caught right away, hatever A does, some of its speed is wasted in side-to-side motion that B can take a shorter path on.
I think that B will eventually catch A but I can’t think of a way to prove it.
Here is a diagram.
Boat at point B has already travelled A->B at the same speed as boat A
Boat B now turns toward point C
Boat at point A could continue to point B, then turn to point C, but it has the option of turning directly toward C as it sees the other boat turn, and by doing so, it travels a shorter distance, gaining on the other boat a little.
The question is: can it ever actually catch up by doing this.
I think the answer is: for certain angles of turn, certainly yes (consider the extreme case in which the pursued boat turns and heads back - directly toward or nearly toward its pursuer).
For smaller angles of turn, and assuming infinitely small boats, the pursuer won’t ever catch up, because the distance gained by cutting the corner is a percentage portion of the distance travelled by the pursued boat.
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I don’t think the triangle inequality is quite strong enough if ship A is changing direction continuously. But if there’s a limit on how often the ships are allowed to change direction, then yes, B will be able to catch A eventually. A’s best course of action is to not change direction at all.
I think B can still catch A if A’s direction does change continuously, but the math needed to show that is considerably more advanced.
Oops. I keep doing that.
Take the graphic of the function y = cos x starting at (0,1). Ship A starts at (0, 1) with speed V. At the same moment ship B starts at (0, 0) with a constant speed V and direction constantly changing so that it constantly points at A. B will describe a sinusoidal track of smaller amplitude than A so it is obvious that B will catch up with A.
I see it now. I think I was limiting the scenario too much to the triangle, imagining that there was some restriction where no further changes would be made until both ships had eventually passed through point C (even then, I’m not sure I was grasping it properly).
It is in essence, Xeno’s paradox, at sea, isn’t it?
It’s not completely clear what this means. One possible meaning is the the rules of the game require A to make occasional course changes. But A can meet this requirement by making a course change and then immediately changing back to the course that leads directly away from B. Since course changes involve no loss of distance, under this scheme B never gains on A.
No. Any course change will slow down the speed made good.
It’s a matter of intelligence and prediction.
If ship B knows what ship A will do, it can pick a closer path to get there. I believe, though can’t prove, that if ship A was moving randomly (like a random walk) then ship B, manned by a mathematician, would still have the advantage. (It could maintain an optimum trajectory, make good decisions, and ultimately wait until luck nets it its bird.) But if ship A’s captain understood ship B’s math and its assumptions, it could trick ship B. (Ship B may then figure out ship A’s tricks, and the cat-mouse game continues. Ultimately luck will be a large component, but cunning as well. If ship B is predictable enough or ship A hyper-intelligent enough, then all the luck won’t help ship B catch ship A after a million years.)
Not according to this from the OP: “Direction changes are instant and involve no loss of speed.”
No loss of speed, but a loss of direct distance traveled from ship B.
No loss of distance if the course changes are “instant”, as specified in the OP.
It can fairly be argued that in the real world there’s no such thing - as is the case with many of these hypotheticals.
Ship A can only hope to maintain distance by traveling directly away from B. Anything else results in B getting closer. If A is forced to change course then B will catch up.
If ship A can also instantly change directions, as ship B can, then ship B will never catch up.
Let’s say that A chooses the point at which B is initially and decides to drive in a perfect circle around that point.
B can’t drive to the edge of the circle and wait. If it did that, A would simply choose a new, larger circle the instant it was further away from B.
Ultimately, any time B is facing any direction other than directly towards A, it is losing ground on A. Since direction change is instantaneous, this isn’t a problem.
In end result, B will simply trace a spiral out to meet A. If A changes it’s direction to head the other way on the circle, B will merely reverse the direction of the spiral, resulting to no actual change. The precise amount of time this will take can most likely be calculated. Indeed, assuming that A starts off heading directly away from B and turns at a regular angle constantly (following a curve), you can most likely calculate the exact time that they will collide.