I'm sure pirates knew the answer. Probably fighter pilots as well.
It can never escape because its turning speed helps nothing while the distance is big, so the pursuing ship can always catch up to it again.
The only reason a fighter pilot has a chance to escape a faster missile is when the missile's targeting system can only see in front of it, so when it overshoots it loses its target.
But with a faster turning speed, the chased ship can evade the pursuer forever, if the captain always turns at the perfect moment.
I wonder if with missiles it also helps because the missile would eventually run out of fuel which I'd presume a rocket would be burning through quite quickly. Maybe you could also evade a pursuing ship in this manner, although you'd have to hope they didn't have much fuel left to begin with.
That was the evasion strategy for the SR 71, just fly faster than anything else in the air
If anyone wants to, now is the time to post one of the several relevant copypastas
There were a lot of things we couldn't do in an Cessna 172, but we were some of the slowest guys on the block and loved reminding our fellow aviators of this fact. People often asked us if, because of this fact, it was fun to fly the 172. Fun would not be the first word I would use to describe flying this plane. Mundane, maybe. Even boring at times. But there was one day in our Cessna experience when we would have to say that it was pure fun to be some of the slowest guys out there, at least for a moment.
It occurred when my CFI and I were flying a training flight. We needed 40 hours in the plane to complete my training and attain PPL status. Somewhere over Colorado we had passed the 40 hour mark. We had made the turn back towards our home airport in a radius of a mile or two and the plane was performing flawlessly. My gauges were wired in the left seat and we were starting to feel pretty good about ourselves, not only because I would soon be flying as a true pilot, but because we had gained a great deal of confidence in the plane in the past ten months. Bumbling across the mountains 3,500 feet below us, I could only see the about 8 miles across the ground. I was, finally, after many humbling months of training and study, ahead of the plane.
I was beginning to feel a bit sorry for my CFI in the right seat. There he was, with nothing to do except watch me and monitor two different radios. This wasn't really good practice for him at all. He'd been doing it for years. It had been difficult for me to relinquish control of the radios, as during my this part of my flying career, I could handle it on my own. But it was part of the division of duties on this flight and I had adjusted to it. I still insisted on talking on the radio while we were on the ground, however. My CFI was so good at many things, but he couldn't match my expertise at sounding awkward on the radios, a skill that had been roughly sharpened with years of listening to LiveATC.com where the slightest radio miscue was a daily occurrence. He understood that and allowed me that luxury.
Just to get a sense of what my CFI had to contend with, I pulled the radio toggle switches and monitored the frequencies along with him. The predominant radio chatter was from Denver Center, not far below us, controlling daily traffic in our sector. While they had us on their scope (for a good while, I might add), we were in uncontrolled airspace and normally would not talk to them unless we needed to ascend into their airspace.
We listened as the shaky voice of a lone SR-71 pilot asked Center for a readout of his ground speed. Center replied:"Aspen 20, I show you at one thousand eight hundred and forty-two knots, across the ground."
Now the thing to understand about Center controllers, was that whether they were talking to a rookie pilot in a Cessna, or to Air Force One, they always spoke in the exact same, calm, deep, professional, tone that made one feel important. I referred to it as the " Houston Center voice." I have always felt that after years of seeing documentaries on this country's space program and listening to the calm and distinct voice of the Houston controllers, that all other controllers since then wanted to sound like that, and that they basically did. And it didn't matter what sector of the country we would be flying in, it always seemed like the same guy was talking. Over the years that tone of voice had become somewhat of a comforting sound to pilots everywhere. Conversely, over the years, pilots always wanted to ensure that, when transmitting, they sounded like Chuck Yeager, or at least like John Wayne. Better to die than sound bad on the radios.
Just moments after the SR-71's inquiry, an F-18 piped up on frequency, in a rather superior tone, asking for his ground speed. "Dusty 52, Center, we have you at 620 on the ground." Boy, I thought, the F-18 really must think he is dazzling his SR-71 brethren. Then out of the blue, a Twin Beech pilot out of an airport outside of Denver came up on frequency. You knew right away it was a Twin Beech driver because he sounded very cool on the radios. "Center, Beechcraft 173-Delta-Charlie ground speed check". Before Center could reply, I'm thinking to myself, hey, that Beech probably has a ground speed indicator in that multi-thousand-dollar cockpit, so why is he asking Center for a readout? Then I got it, ol' Delta-Charlie here is making sure that every military jock from Mount Whitney to the Mojave knows what true speed is. He's the slowest dude in the valley today, and he just wants everyone to know how much fun he is having in his new bug-smasher. And the reply, always with that same, calm, voice, with more distinct alliteration than emotion: "173-Delta-Charlie, Center, we have you at 90 knots on the ground."
And I thought to myself, is this a ripe situation, or what? As my hand instinctively reached for the mic button, I had to remind myself that my CFI was in control of the radios. Still, I thought, it must be done - in mere minutes we'll be out of the sector and the opportunity will be lost. That Beechcraft must die, and die now. I thought about all of my training and how important it was that we developed well as a crew and knew that to jump in on the radios now would destroy the integrity of all that we had worked toward becoming. I was torn.
Somewhere, half a mile above Colorado, there was a pilot screaming inside his head. Then, I heard it. The click of the mic button from the right seat. That was the very moment that I knew my CFI and I had become a lifelong friends. Very professionally, and with no emotion, my CFI spoke: "Denver Center, Cessna 56-November-Sierra, can you give us a ground speed check?" There was no hesitation, and the replay came as if was an everyday request. "Cessna 56-November-Sierra, I show you at 76 knots, across the ground."
I think it was the six knots that I liked the best, so accurate and proud was Center to deliver that information without hesitation, and you just knew he was smiling. But the precise point at which I knew that my CFI and I were going to be really good friends for a long time was when he keyed the mic once again to say, in his most CFI-like voice: "Ah, Center, much thanks, we're showing closer to 72 on the money."
For a moment my CFI was a god. And we finally heard a little crack in the armor of the Houston Center voice, when Denver came back with, "Roger that November-Sierra, your E6B is probably more accurate than our state-of-the-art radar. You boys have a good one."
It all had lasted for just moments, but in that short, memorable stroll across the west, the Navy had been owned, all mortal airplanes on freq were forced to bow before the King of Slow, and more importantly, my CFI and I had crossed the threshold of being BFFs. A fine day's work. We never heard another transmission on that frequency all the way to our home airport.
For just one day, it truly was fun being the slowest guys out there.
As a retired Center Air Traffic Controller, thank you for that. I've got a million stories, and it's good to hear one from the other side of the mike.
Oh you haven’t seen this one? In jealous you are getting to read it for the first time today. It’s so good
Lmao I hadn't seen that reversed copypasta one yet. It's beautiful
Lol is this not a story about you? Is it a wall of text meme I've never heard of?
Something, something, "Tower, can we get a speed check", something, something, kings of the skies that day, something, something...
🛩️: 💨?
🗼: 🐌.
🚁: 💨?
🗼: 🐢.
⚓: 💨?
🗼: 🐆.
✈️: 💨?
🗼: 🚄+🏎️.
✈️: 🚀.
🗼: ☑️.
🛩️,🚁,⚓:🦗
Funny copypastas aside, I think that's missing the point of this question and my addition to it. In this scenario as the OP described, the thing being pursued is slower than the pursuer, but faster at turning. Among the various answers they got one saying that the superior turning ability would not help with completely escaping because even if the pursued could temporarily increase the gap between themselves and their pursuers by well timed turnings, the gap would always be shrunk again by the pursuer's superior speed. They noted however that though this meant they'd never escape, they could theoretically never be caught either because they could keep just evading with well timed turns. They also made an analogy to missiles and aircraft saying that in that instance, aircraft turning away from a missile that's faster than the aircraft can be an effective evasion strategy because the missile can 'lose' the target if the aircraft gets behind the missile making it unable to detect the aircraft with its forward facing sensors.
What I'm saying is that another reason that trick might work with aircraft and missiles is just the fact that the missile can only chase for so long before it has no fuel left, even if it is faster than the aircraft its chasing. I figure the same goes for the hypothetical with the ships. If the pursued ship can continue to shake off the pursuer with well timed turns for long enough, it could maybe escape just through attrition, as long as they had more fuel than their pursuers, who are still faster but obviously unable to continue the chase indefinitely.
If there were a series of obstacles to avoid the more agile ship could escape the faster one.
This sounds right to me. Thank you
In a situation where wind isn't a factor, and there are no obstacles, this is true.
But since we're talking boats, I'm assuming wind speed/direction is a factor, so the ship that can adjust their orientation and sails to maximally take advantage of the wind could have an advantage.
When did the ships from OP's math question turn into sailboats?
Wind affects all boats, but I may have made an assumption when they asked about it in the context of pirates.
If the faster ship is smart I can't see a way for them to fail to catch the nimble ship eventually.
I think if the faster ship is "dumb" and always just drives straight at the slower ship, then the slow ship might be able to keep itself moving perpendicular to the fast ship which will be unable to spend it's entire speed advantage catching up since it can't keep itself pointed at slow ship.
There's a distance component here, since the closer the ships are to each other the more significant difference in turning speed becomes. (At long distance, one unit of turn might equal ten units of distance, but at close distance one unit of turn might only equal one unit of distance. In math terms, as the radius gets smaller, so does arc length)
I don't care to do the math at the moment, but at some distance between them, the distance the slow ship can travel in a time unit will be greater than the arc length the fast ship can turn at that distance in that time. If the slow ships turning rate is high enough to keep it perpendicular at that distance, then it should be able to avoid capture, but never escape. (My gut says turning radius equal to or smaller than the distance between the ships, but I could be wrong).
If the ship can't turn fast enough to make a circle, then it'll grow further away than the magic distance and the fast ship will be able to point at them and get closer. If the faster ship is smart, it can just increase the distance and make a more favorable approach by pointing where the slow ship will be.
Don't they end up in two concentric circles with the slower ship on the inside and the faster ship on the outside?
I think so, but I can't prove it without actually using a pencil and paper.
I don't know the answer (my gut says no, fwiw), but your question made me remember this numberphile video about a cat and mouse that I think is pretty interesting and roughly relates to your problem.
I'm a little sad nobody with the relevant mathematics background has jumped in. These puzzles are considered; a simple version is the lion-hunting-man where both have the same speed and infinite turning speed (eg, this paper, where the arena they play in varies).
Very cool. I love that it exists, I love humanity that some of us are capable of understanding, or even generating such things, but wow. Some people are a lot smarter than me.
If it's any consolation, you are almost certainly within ~3 years of understanding the solution and a dozen variants. It's not a super deep area. Probably doesn't really require calculus (you need continuous as in 'the lion doesn't teleport; that's cheating', but I think not much more).
We must always consider capability. If it requires 3 quarts of understanding and I've only got a 1/2 gallon container, it's not all getting in there.
We aren't all the same. I am content that there are people way smarter than me. I've met some. They're usually cool.
Fortunately containers can get bigger =)
While we aren't all the same, there's a difference between things that require holding 8 complicated things in mind at once, and things that require a little language learning and the intelligence to solve a crossword. This is closer to the latter - like doing a crossword in Spanish. You need to know a bunch of little things, but learning them is basically all tedium and not brilliant insights. (Taking these puzzles, creating a dozen new variants, and solving all of those probably does require managing a lot of complexity. But to understand the work of others, is not so bad)
Agreed. I think "Flowers for Algernon" hit me hard. And I had an experience in college where I accidentally took a philosophy class called "Critical Theory Since Plato" when I first realized that I'm not very smart, just a little above average, and some people live life on a while different level than I do.
And don't get me started on "Electromechanical Wave Theory," a book I bought from Goodwill. I wanted to learn more about that, but I think it was written by aliens.
My experience in eve online says that the faster ship can always maneuver close enough to do damage. Maybe you over shoot it, but if you know the direction of the more agile ship you can course correct more frequently to make an approach.
Probably depends also on the range of your weapons. The faster ship might not be able to ram the more nimble ship but could get close enough to launch a missile.
Given a bit of lead time, I believe the slower ship may avoid capture if there are any sort of obstacles like an island or rocks.
However, if we're talking perfectly flat ocean with no time limit, no obstacles and pefect decision making: I'm guessing the faster ship eventually catches up, as turning from far away doesn't slow your persuer much.
It depends on what you mean by "escape", and what you view as the alternative.
I suspect that the pursuer could never converge on the same instantaneous point, given sufficient initial distance (and orientation). At a certain distance, the prey could enter a stable orbit around the pursuer. I don't have a mathematical proof but I strongly suspect this to be the case,and I can envision the structure of a proof.
Could the prey infinitely extend the gap between themselves and the pursuer? No. I don't have the tooling to actually present such a proof, but of that one I am confident.
I think if you introduced concepts of obstacles and a "radius of escape" (where if the gap meets a threshold the predator is permanently foiled), then there are almost certainly scenarios where the prey could escape.
We actually see this scenario play out in nature all the time
WOW, really nice explanation.
Thank you!
Although, I'm realizing that for completeness, there probably are mathematical constraints around the relationship between the required absolute values of turning speeds and movement speeds. They're kinda egde-casey for any practically imagined scenarios, but would come into play for a rigorous proof.
Concerning the proof, I'd consider that at any given point where both objects haven't converged yet, there has to be a next point that can be reached by the ship with the higher maneuverability but not by the faster ship. It's probably calculus from that point on and I'm not really good at that. If there's always such a possibility, the slower ship can always outmaneuver the other one. And seems to me like vectors in a polar coordinate system would be made for this.
Set vector1 equal to vector2 plus an arbitrary distance. See if there's a solution for phi2 < phi1.
Yeah, I've got a similar thought.
There do exist scenarios where there would be a solution... For example if the base turning speed is 0, or the distance between them is already sufficiently small (and their relative orientations are aligned).
Sure. And I mean the "sufficiently small" distance is exactly the question. I mean it's not really an interesting question to ask if they're still 12 nautical miles apart... The initial distance isn't really of concern. It just has to work for any given initial state. And the next question is, are we talking about entering a ship or using cannons? Then it's either can the distance become 0 or can it get less than something.
I agree they aren't practically interesting egde cases, because in order to hit them, they're no longer meaningful for realistically describing pirate ships. At very high turning speeds, it also ceases to matter that one is 3/4 than that the other either. But at that point, we're talking about pirate ships spinning like ballerinas across the seven seas.
Feel free to extend that problem to fighter jets or ballerinas playing tag 😆
However, I'm pretty sure it's already solved. Doesn't seem difficult to prove and has had applications for centuries already. And I've played the Robots Game when I was 12 or something...
Any amount of turn in between 0 and 180 i presume, and the game is "tag" correct? Do they have to be caught from the back, or would a head to head collision count? Does the trailing ship have perfect response time? I don't see how turning around 180 degrees would help evade capture. In summary, no, I think you'll always be tagged by the faster ship.
Too much thinking is beyond me. That's why I'm asking smart people.
Even assuming open 2D sea and a pursuer turn radius of, let's say, 20 ship lengths, I'm sure that depends on boundary conditions like how far it has to go to escape and the position/speed/orientation the ships start with.
Obviously, if we start with a pursuer right off the stern, there's no escaping.
For real ships, there also will be a time limit, because someone will run out of supplies first. The way I imagine doing this would basically be to find a pattern where no matter what the pursuer does they can't board, but it doesn't matter if you run out of hard tack while the people chasing are still well-fed. Cannons would change the logic here too.
Guess:
The slower ship would never be in a position to get caught without making a mistake, but it's turning advantage rapidly reduces with with distance, so it would never be able to get away in the long run.
Confounding factors:
Ship lengths. As the ships get longer relatively their turning circle, the easier the slow ship becomes to catch. The length of the ship makes it harder to dodge.
Conditions of capture and escape. Are we talking about being side by side, close enough to through grappling hooks, or just a point touching? Or cannon range? Is escaping getting out of line of sight? RADAR range?
Islands. This favours the more manoeuvrable ship as the fast ship cannot take a straight line course. It also increases the fast ships cost for falling for bluffs.
Acceleration and speed cost to turn. Overall, introducing these makes the engagement longer. That said, it would also affect the fast ship more if islands are introduced.
Outside objectives. Ships can't be at sea forever, and likely both ships have places to be. Is a vital shipment of vaccines on the slow ship? Does the fast ship know this? Etc
Ill defined. Slower/faster than what? The pirate ship? Whats the initial distance between the ships? I'll assume a sufficiently large distance so the ships all get away :)
Slower or faster than the pursuing ship, presumably.
It can never escape because its turning speed helps nothing while the distance is big, so the pursuing ship can always catch up to it again.
The only reason a fighter pilot has a chance to escape a faster missile is when the missile's targeting system can only see in front of it, so when it overshoots it loses its target.
But with a faster turning speed, the chased ship can evade the pursuer forever, if the captain always turns at the perfect moment.
I wonder if with missiles it also helps because the missile would eventually run out of fuel which I'd presume a rocket would be burning through quite quickly. Maybe you could also evade a pursuing ship in this manner, although you'd have to hope they didn't have much fuel left to begin with.
That was the evasion strategy for the SR 71, just fly faster than anything else in the air
If anyone wants to, now is the time to post one of the several relevant copypastas
There were a lot of things we couldn't do in an Cessna 172, but we were some of the slowest guys on the block and loved reminding our fellow aviators of this fact. People often asked us if, because of this fact, it was fun to fly the 172. Fun would not be the first word I would use to describe flying this plane. Mundane, maybe. Even boring at times. But there was one day in our Cessna experience when we would have to say that it was pure fun to be some of the slowest guys out there, at least for a moment.
It occurred when my CFI and I were flying a training flight. We needed 40 hours in the plane to complete my training and attain PPL status. Somewhere over Colorado we had passed the 40 hour mark. We had made the turn back towards our home airport in a radius of a mile or two and the plane was performing flawlessly. My gauges were wired in the left seat and we were starting to feel pretty good about ourselves, not only because I would soon be flying as a true pilot, but because we had gained a great deal of confidence in the plane in the past ten months. Bumbling across the mountains 3,500 feet below us, I could only see the about 8 miles across the ground. I was, finally, after many humbling months of training and study, ahead of the plane.
I was beginning to feel a bit sorry for my CFI in the right seat. There he was, with nothing to do except watch me and monitor two different radios. This wasn't really good practice for him at all. He'd been doing it for years. It had been difficult for me to relinquish control of the radios, as during my this part of my flying career, I could handle it on my own. But it was part of the division of duties on this flight and I had adjusted to it. I still insisted on talking on the radio while we were on the ground, however. My CFI was so good at many things, but he couldn't match my expertise at sounding awkward on the radios, a skill that had been roughly sharpened with years of listening to LiveATC.com where the slightest radio miscue was a daily occurrence. He understood that and allowed me that luxury.
Just to get a sense of what my CFI had to contend with, I pulled the radio toggle switches and monitored the frequencies along with him. The predominant radio chatter was from Denver Center, not far below us, controlling daily traffic in our sector. While they had us on their scope (for a good while, I might add), we were in uncontrolled airspace and normally would not talk to them unless we needed to ascend into their airspace.
We listened as the shaky voice of a lone SR-71 pilot asked Center for a readout of his ground speed. Center replied:"Aspen 20, I show you at one thousand eight hundred and forty-two knots, across the ground."
Now the thing to understand about Center controllers, was that whether they were talking to a rookie pilot in a Cessna, or to Air Force One, they always spoke in the exact same, calm, deep, professional, tone that made one feel important. I referred to it as the " Houston Center voice." I have always felt that after years of seeing documentaries on this country's space program and listening to the calm and distinct voice of the Houston controllers, that all other controllers since then wanted to sound like that, and that they basically did. And it didn't matter what sector of the country we would be flying in, it always seemed like the same guy was talking. Over the years that tone of voice had become somewhat of a comforting sound to pilots everywhere. Conversely, over the years, pilots always wanted to ensure that, when transmitting, they sounded like Chuck Yeager, or at least like John Wayne. Better to die than sound bad on the radios.
Just moments after the SR-71's inquiry, an F-18 piped up on frequency, in a rather superior tone, asking for his ground speed. "Dusty 52, Center, we have you at 620 on the ground." Boy, I thought, the F-18 really must think he is dazzling his SR-71 brethren. Then out of the blue, a Twin Beech pilot out of an airport outside of Denver came up on frequency. You knew right away it was a Twin Beech driver because he sounded very cool on the radios. "Center, Beechcraft 173-Delta-Charlie ground speed check". Before Center could reply, I'm thinking to myself, hey, that Beech probably has a ground speed indicator in that multi-thousand-dollar cockpit, so why is he asking Center for a readout? Then I got it, ol' Delta-Charlie here is making sure that every military jock from Mount Whitney to the Mojave knows what true speed is. He's the slowest dude in the valley today, and he just wants everyone to know how much fun he is having in his new bug-smasher. And the reply, always with that same, calm, voice, with more distinct alliteration than emotion: "173-Delta-Charlie, Center, we have you at 90 knots on the ground."
And I thought to myself, is this a ripe situation, or what? As my hand instinctively reached for the mic button, I had to remind myself that my CFI was in control of the radios. Still, I thought, it must be done - in mere minutes we'll be out of the sector and the opportunity will be lost. That Beechcraft must die, and die now. I thought about all of my training and how important it was that we developed well as a crew and knew that to jump in on the radios now would destroy the integrity of all that we had worked toward becoming. I was torn.
Somewhere, half a mile above Colorado, there was a pilot screaming inside his head. Then, I heard it. The click of the mic button from the right seat. That was the very moment that I knew my CFI and I had become a lifelong friends. Very professionally, and with no emotion, my CFI spoke: "Denver Center, Cessna 56-November-Sierra, can you give us a ground speed check?" There was no hesitation, and the replay came as if was an everyday request. "Cessna 56-November-Sierra, I show you at 76 knots, across the ground."
I think it was the six knots that I liked the best, so accurate and proud was Center to deliver that information without hesitation, and you just knew he was smiling. But the precise point at which I knew that my CFI and I were going to be really good friends for a long time was when he keyed the mic once again to say, in his most CFI-like voice: "Ah, Center, much thanks, we're showing closer to 72 on the money."
For a moment my CFI was a god. And we finally heard a little crack in the armor of the Houston Center voice, when Denver came back with, "Roger that November-Sierra, your E6B is probably more accurate than our state-of-the-art radar. You boys have a good one."
It all had lasted for just moments, but in that short, memorable stroll across the west, the Navy had been owned, all mortal airplanes on freq were forced to bow before the King of Slow, and more importantly, my CFI and I had crossed the threshold of being BFFs. A fine day's work. We never heard another transmission on that frequency all the way to our home airport.
For just one day, it truly was fun being the slowest guys out there.
As a retired Center Air Traffic Controller, thank you for that. I've got a million stories, and it's good to hear one from the other side of the mike.
Oh you haven’t seen this one? In jealous you are getting to read it for the first time today. It’s so good
Lmao I hadn't seen that reversed copypasta one yet. It's beautiful
Lol is this not a story about you? Is it a wall of text meme I've never heard of?
The original story is from the SR-71 pilot's perspective https://www.thesr71blackbird.com/Aircraft/Stories/sr-71-blackbird-speed-check-story
Well it made perfect sense to me, and was funny and rang true. Real pilot humor there.
lol yea. It’s a reversed copy pasta. The original story is from the SR71’s perspective. It’s pet great both ways.
Something, something, "Tower, can we get a speed check", something, something, kings of the skies that day, something, something...
🛩️: 💨?
🗼: 🐌.
🚁: 💨?
🗼: 🐢.
⚓: 💨?
🗼: 🐆.
✈️: 💨?
🗼: 🚄+🏎️.
✈️: 🚀.
🗼: ☑️.
🛩️,🚁,⚓:🦗
Funny copypastas aside, I think that's missing the point of this question and my addition to it. In this scenario as the OP described, the thing being pursued is slower than the pursuer, but faster at turning. Among the various answers they got one saying that the superior turning ability would not help with completely escaping because even if the pursued could temporarily increase the gap between themselves and their pursuers by well timed turnings, the gap would always be shrunk again by the pursuer's superior speed. They noted however that though this meant they'd never escape, they could theoretically never be caught either because they could keep just evading with well timed turns. They also made an analogy to missiles and aircraft saying that in that instance, aircraft turning away from a missile that's faster than the aircraft can be an effective evasion strategy because the missile can 'lose' the target if the aircraft gets behind the missile making it unable to detect the aircraft with its forward facing sensors.
What I'm saying is that another reason that trick might work with aircraft and missiles is just the fact that the missile can only chase for so long before it has no fuel left, even if it is faster than the aircraft its chasing. I figure the same goes for the hypothetical with the ships. If the pursued ship can continue to shake off the pursuer with well timed turns for long enough, it could maybe escape just through attrition, as long as they had more fuel than their pursuers, who are still faster but obviously unable to continue the chase indefinitely.
If there were a series of obstacles to avoid the more agile ship could escape the faster one.
This sounds right to me. Thank you
In a situation where wind isn't a factor, and there are no obstacles, this is true.
But since we're talking boats, I'm assuming wind speed/direction is a factor, so the ship that can adjust their orientation and sails to maximally take advantage of the wind could have an advantage.
When did the ships from OP's math question turn into sailboats?
Wind affects all boats, but I may have made an assumption when they asked about it in the context of pirates.
If the faster ship is smart I can't see a way for them to fail to catch the nimble ship eventually.
I think if the faster ship is "dumb" and always just drives straight at the slower ship, then the slow ship might be able to keep itself moving perpendicular to the fast ship which will be unable to spend it's entire speed advantage catching up since it can't keep itself pointed at slow ship.
There's a distance component here, since the closer the ships are to each other the more significant difference in turning speed becomes. (At long distance, one unit of turn might equal ten units of distance, but at close distance one unit of turn might only equal one unit of distance. In math terms, as the radius gets smaller, so does arc length)
I don't care to do the math at the moment, but at some distance between them, the distance the slow ship can travel in a time unit will be greater than the arc length the fast ship can turn at that distance in that time.
If the slow ships turning rate is high enough to keep it perpendicular at that distance, then it should be able to avoid capture, but never escape. (My gut says turning radius equal to or smaller than the distance between the ships, but I could be wrong).
If the ship can't turn fast enough to make a circle, then it'll grow further away than the magic distance and the fast ship will be able to point at them and get closer. If the faster ship is smart, it can just increase the distance and make a more favorable approach by pointing where the slow ship will be.
Don't they end up in two concentric circles with the slower ship on the inside and the faster ship on the outside?
I think so, but I can't prove it without actually using a pencil and paper.
I don't know the answer (my gut says no, fwiw), but your question made me remember this numberphile video about a cat and mouse that I think is pretty interesting and roughly relates to your problem.
Good video! I really enjoyed that!
Here is an alternative Piped link(s):
this numberphile video about a cat and mouse
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I'm a little sad nobody with the relevant mathematics background has jumped in. These puzzles are considered; a simple version is the lion-hunting-man where both have the same speed and infinite turning speed (eg, this paper, where the arena they play in varies).
Very cool. I love that it exists, I love humanity that some of us are capable of understanding, or even generating such things, but wow. Some people are a lot smarter than me.
If it's any consolation, you are almost certainly within ~3 years of understanding the solution and a dozen variants. It's not a super deep area. Probably doesn't really require calculus (you need continuous as in 'the lion doesn't teleport; that's cheating', but I think not much more).
We must always consider capability. If it requires 3 quarts of understanding and I've only got a 1/2 gallon container, it's not all getting in there.
We aren't all the same. I am content that there are people way smarter than me. I've met some. They're usually cool.
Fortunately containers can get bigger =)
While we aren't all the same, there's a difference between things that require holding 8 complicated things in mind at once, and things that require a little language learning and the intelligence to solve a crossword. This is closer to the latter - like doing a crossword in Spanish. You need to know a bunch of little things, but learning them is basically all tedium and not brilliant insights. (Taking these puzzles, creating a dozen new variants, and solving all of those probably does require managing a lot of complexity. But to understand the work of others, is not so bad)
Agreed. I think "Flowers for Algernon" hit me hard. And I had an experience in college where I accidentally took a philosophy class called "Critical Theory Since Plato" when I first realized that I'm not very smart, just a little above average, and some people live life on a while different level than I do.
And don't get me started on "Electromechanical Wave Theory," a book I bought from Goodwill. I wanted to learn more about that, but I think it was written by aliens.
My experience in eve online says that the faster ship can always maneuver close enough to do damage. Maybe you over shoot it, but if you know the direction of the more agile ship you can course correct more frequently to make an approach.
Probably depends also on the range of your weapons. The faster ship might not be able to ram the more nimble ship but could get close enough to launch a missile.
Given a bit of lead time, I believe the slower ship may avoid capture if there are any sort of obstacles like an island or rocks.
However, if we're talking perfectly flat ocean with no time limit, no obstacles and pefect decision making: I'm guessing the faster ship eventually catches up, as turning from far away doesn't slow your persuer much.
It depends on what you mean by "escape", and what you view as the alternative.
I suspect that the pursuer could never converge on the same instantaneous point, given sufficient initial distance (and orientation). At a certain distance, the prey could enter a stable orbit around the pursuer. I don't have a mathematical proof but I strongly suspect this to be the case,and I can envision the structure of a proof.
Could the prey infinitely extend the gap between themselves and the pursuer? No. I don't have the tooling to actually present such a proof, but of that one I am confident.
I think if you introduced concepts of obstacles and a "radius of escape" (where if the gap meets a threshold the predator is permanently foiled), then there are almost certainly scenarios where the prey could escape.
We actually see this scenario play out in nature all the time
WOW, really nice explanation.
Thank you!
Although, I'm realizing that for completeness, there probably are mathematical constraints around the relationship between the required absolute values of turning speeds and movement speeds. They're kinda egde-casey for any practically imagined scenarios, but would come into play for a rigorous proof.
Concerning the proof, I'd consider that at any given point where both objects haven't converged yet, there has to be a next point that can be reached by the ship with the higher maneuverability but not by the faster ship. It's probably calculus from that point on and I'm not really good at that. If there's always such a possibility, the slower ship can always outmaneuver the other one. And seems to me like vectors in a polar coordinate system would be made for this.
Set vector1 equal to vector2 plus an arbitrary distance. See if there's a solution for phi2 < phi1.
Yeah, I've got a similar thought.
There do exist scenarios where there would be a solution... For example if the base turning speed is 0, or the distance between them is already sufficiently small (and their relative orientations are aligned).
Sure. And I mean the "sufficiently small" distance is exactly the question. I mean it's not really an interesting question to ask if they're still 12 nautical miles apart... The initial distance isn't really of concern. It just has to work for any given initial state. And the next question is, are we talking about entering a ship or using cannons? Then it's either can the distance become 0 or can it get less than something.
I agree they aren't practically interesting egde cases, because in order to hit them, they're no longer meaningful for realistically describing pirate ships. At very high turning speeds, it also ceases to matter that one is 3/4 than that the other either. But at that point, we're talking about pirate ships spinning like ballerinas across the seven seas.
Feel free to extend that problem to fighter jets or ballerinas playing tag 😆
However, I'm pretty sure it's already solved. Doesn't seem difficult to prove and has had applications for centuries already. And I've played the Robots Game when I was 12 or something...
Any amount of turn in between 0 and 180 i presume, and the game is "tag" correct? Do they have to be caught from the back, or would a head to head collision count? Does the trailing ship have perfect response time? I don't see how turning around 180 degrees would help evade capture. In summary, no, I think you'll always be tagged by the faster ship.
Too much thinking is beyond me. That's why I'm asking smart people.
Even assuming open 2D sea and a pursuer turn radius of, let's say, 20 ship lengths, I'm sure that depends on boundary conditions like how far it has to go to escape and the position/speed/orientation the ships start with.
Obviously, if we start with a pursuer right off the stern, there's no escaping.
For real ships, there also will be a time limit, because someone will run out of supplies first. The way I imagine doing this would basically be to find a pattern where no matter what the pursuer does they can't board, but it doesn't matter if you run out of hard tack while the people chasing are still well-fed. Cannons would change the logic here too.
Guess:
The slower ship would never be in a position to get caught without making a mistake, but it's turning advantage rapidly reduces with with distance, so it would never be able to get away in the long run.
Confounding factors: Ship lengths. As the ships get longer relatively their turning circle, the easier the slow ship becomes to catch. The length of the ship makes it harder to dodge.
Conditions of capture and escape. Are we talking about being side by side, close enough to through grappling hooks, or just a point touching? Or cannon range? Is escaping getting out of line of sight? RADAR range?
Islands. This favours the more manoeuvrable ship as the fast ship cannot take a straight line course. It also increases the fast ships cost for falling for bluffs.
Acceleration and speed cost to turn. Overall, introducing these makes the engagement longer. That said, it would also affect the fast ship more if islands are introduced.
Outside objectives. Ships can't be at sea forever, and likely both ships have places to be. Is a vital shipment of vaccines on the slow ship? Does the fast ship know this? Etc
Ill defined. Slower/faster than what? The pirate ship? Whats the initial distance between the ships? I'll assume a sufficiently large distance so the ships all get away :)
Slower or faster than the pursuing ship, presumably.