Math people: would a ship that is 1/4 slower but can complete a 180° turn 1/4 faster escape its pursuer?

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.

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).

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.

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.

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

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.

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.

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 :)