I’m sure pirates knew the answer. Probably fighter pilots as well.
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.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.
You can be smarter than you are now. Effort will reward you. Take a peek at the growth mindset. I think Hubberman did a podcast on it, and his content is usually of good tier.
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.
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!
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.
Depends on constraints.
How far is their separation distance?
How fast is the turning, and how fast are the ships? If it takes a year to turn around, and a day to “escape,” then yes.
What kind of ship? Under-sea, sea, air, space? And what size?
What does the field look like? Open, hazards, obstacles, maze?
Conditions of the medium? Stormy, Choppy, Calm?
How near is “caught” and how far away is “escape?” Is “caught” within range of long distance weaponry? Is escape a destination or a condition?
It depends, but let’s assume a simple scenario of a sea ship in calm open ocean, with a turning radius of 5 ship lengths, a ship length of 40 meters, and a speed of 37.04 km/h.
- Weaponry included and coming broadside is the bad end for the slower ship: the slower ship doesn’t stand a chance.
- Boarding required: It’ll be a long game of cat and mouse, but the cat will win. However, if they were of equal speed it’d likely be an infinite game.
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.
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 :)
