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.

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.

Wind affects all boats, but I may have made an assumption when they asked about it in the context of pirates.

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.

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.