Asklemmy

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.

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

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.

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.

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 :)

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).

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.

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.

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.