The Importance Of Physical Models: How Not To Shoot Yourself In The Foot Or Anywhere Else

We take shortcuts all the time with our physical models. We rarely consider that wire has any resistance, for example, or that batteries have a source impedance. That’s fine up until the point that it isn’t. Take the case of the Navy’s Grumman F11F Tiger aircraft. The supersonic aircraft was impressive, although it suffered from some fatal flaws. But it also has the distinction of being the first plane ever to shoot itself down.

So here’s the simple math. A plane traveling Mach 1 is moving about 1,200 km/h — the exact number depends on a few things like your altitude and the humidity. Let’s say about 333 m/s. Bullets from a 20 mm gun, on the other hand, move at more than 1000 m/second. So when the bullet leaves the plane it would take the plane over three seconds to catch up with it, by which time it has moved ever further away, right?

Who Shot You Down?

No. In 1956, Tom Attridge took off from Long Island to do a weapons test over the Atlantic. He climbed to 20,000 feet, started a Mach 1 dive, and fired his cannons which ran out of ammo at about 13,000 feet.

Around the 7,000-foot mark, something hit his windshield — presumably a bird. The plane started losing power and the plane crashed leaving a 300-foot flaming path through a wooded area near the airstrip. Attridge survived but had a broken leg and broken vertebra.

But it wasn’t a bird that hit the naval aviator’s plane. It was his own bullets. The problem is, the bullet did leave the gun at a high rate of speed. However, they immediately encountered air resistance causing them to slow down. By the time the bullets slowed to 643 m/s, the plane was going at 1,400 m/s. Three bullets hit the plane, one through the nose cone, one through the windshield, and one hit the starboard engine intake. All this took a mere 11 seconds. You can see the whole story in the video, below.

The plane proved to be less reliable compared to other contemporary fighter planes. It had other undesirable characteristics, but it was used as a training aircraft and the Blue Angels used them until 1968.

Lesson Learned?

You would think this was ample evidence that something was wrong, but no. It was dismissed as a “fluke.” Of course, in 1973, an F-14 Tomcat also shot itself down with a dummy missile. A Dutch F-16 also shot itself down in 2019 in a very similar incident. You have to wonder if there aren’t other examples that went unreported.

However, there is a lesson here. Common sense isn’t always engineering sense. Think a two-micron deviation isn’t important? Think again. Or maybe you want to check for air leaks using a candle in a nuclear power plant? Engineering history is full of stories where any reasonable person would think something was fine, when, in retrospect, it was anything but.

There’s an old Russian saying “trust but verify.” That is a good adage for us, too. Trust your instincts, but verify with solid mathematical models that take everything into account.

73 thoughts on “The Importance Of Physical Models: How Not To Shoot Yourself In The Foot Or Anywhere Else

    1. Does say in a dive from high altitude and speed, and what the right mach number would be is altitude dependent.

      Doesn’t seem unreasonable to me when even the piston driven prop aircraft can get up to around the speed of sound in a dive that something already very much faster when flying level can in a dive then get way above the normal ‘airspeed’ of the bird. I don’t know how fast one of these jets can manage to get in a dive without damaging itself or becoming uncontrollable. But then that is what test pilots are for, and after shooting yourself down precise data logging would be rather impressive, so the numbers are quite possibly in doubt, the event itself isn’t… I can’t find in a quick search any great detail on the the events with cited facts other than that it definitely shot itself down.

      1. The exact Mach number is irrelevant – 1,400m/s would be roughly one and a half times as fast as the airbreathing speed record holder (the SR-71), and well on its way to X-15 territory. It’s clearly a typo.

        The author probably meant 1,400 km/hr, which is on the high end of an F-11’s capability.

        1. I’d not be surprised if you are correct its meant to be km/h (was wondering about mentioning it as that was my first thought as well, but all my searching could find no particular information on this incident beyond a dive with afterburners) however all the ‘speed records’ listed are straight and level flight – in a dive at full afterburners you have nearly 1 whole extra g of acceleration available over the engine power alone, so ending up faster than the straight and level flight record holder’s speed isn’t entirely implausible – Plus the SR-71 is held back only because the wingtips start to take harm, reportedly it can go faster if you don’t mind expensive damage). So getting that fast wouldn’t be entirely implausible without real info on the aircraft in use.

          1. All that is true, but there’s a 0% chance an F11F (or any comparable jet, or almost any air-breathing craft) would have done 5,000KM/h in a dive and survived to tell the tale without that feat alone being a huge story in aviation lore.

          2. Just the compression heating alone would destroy an F-11 in very short order at those speeds. There’s also no way those air intakes would still be taking in air at those speeds (shockwave from the intake lip would stall the intake) so the engine would definitely flame out long before it got to those speeds

          3. @NattyKathy So just how fast do these things go in a dive? The only real datapoints for modernish high speed jets seem to be climb rate – when its impressive enough to warrant quoting and the flat and level top speed.

            @ThisGuy A flame out seems very likely to me as well, along with quite likely mangling the wing tips but you are not talking so fast that heating would rapidly destroy stuff it seems to me. Lots of thermal mass and it would only be able to stay at whatever its top speed in a dive is for a very short period, doesn’t have enough altitude to loose and as soon as it pulls out its fighting gravity rather than gaining from it, so the drag will slow it down.

            Still don’t think that speed is very likely, adding a single paltry g to the engines output I wouldn’t think is near enough at those speeds with how drag scales to make that much difference to the straight and level top speed.

      2. A common error. The speed of sound in air depends on temperature, not density. It is the speed of the air molecules – or the RMS speed. Kinetic thermodynamics statistical mechanics and all that.

        I used to hear this a lot from WWII fighter pilots who had lost of lore about “propeller compression” and explanations for loosing flight controls in a too aggressive dive. They usually claimed that got control authority back as the air density increased. In fact, the air temperature was going from -50 to above freezing. The difference isn’t huge but could be 50m/s or 180km/hr (110mph).

        1. Its the real world, and in the real world all these things are correlated rather directly! Altitude is what you actually might just know and measure on an aircraft, AND therefor the sane measure to use to define such things…

          1. Nope. The place where temperature increase with altitude goes positive – the troposphere/stratosphere boundary – varies from 23,00 feet to 66,000 feet depending on location. Use temperature.

    1. “at a high rate of speed” does not parse.

      Why is that so? Translated into my language, it makes sense. Maybe English just isn’t flexible enough? It has weird comma rules, also. Punctuation is much better in, um, German? :)

      1. “rate” is a ratio, indicating the change of a quantity with respect to something else. Speed itself is a rate: meters per second. If you say something like “rate of speed” it says that the speed is changing with respect to something, but that something isn’t stated here.

        If it’s “rate of speed change”, it could logically be assumed to be “rate of speed change with respect to time”, more simply “acceleration”. But the phrase “rate of speed” by itself is nonsensical. But much of this silly language is nonsensical. (and, yes, it’s not my first language either).

        1. I see, makes sense.
          (Btw, a sense cannot be “made” in some of the other European languages. In German, something “results” into a sense.)

          I assume the term “rate” itself is the problem of confusion here. Something was lost in translation, maybe.

          If I understand the Wikipedia article in German correctly, “Rate” comes from “rata pars”, a calculated part or a dimension/size unit in relation to speed (from ratum= calculated). It’s also used as a percentage, it seems. “Rate” also is commonly used to describe how often events do occur.

          I assume the article simply ment to say “at a high amount of speed” in a more fancy way. 🙂

        2. High rate of speed is pretty commonly used to mean “fast”. From the use of extra words to make it sound more important or impressive. Police and video narrators describing car chases or accidents say it frequently. I think we are too late to this party. To the same people all “flips” are back-flips.

        3. But speed itself is almost always expressed as a ratio of distance divided by time. The exceptions being things like walking speed, a snail’s pace or the speeds of sound or light through a medium.
          You could be traveling at a high rate, or a high speed. People mess it up by saying rate of speed much like saying ATM machines.

    2. I understand where you’re coming from in that the word rate implies a ratio between two things.
      As a native English speaker, though, I can tell you it’s not an uncommon phrase. You can effectively just remove “rate of” from the sentence and its meaning does not change. Perhaps the original phrase was “a high rate of distance over time.” But since v=d/t, at some point, they just substituted “speed” for the last portion, and nobody corrected them.
      Don’t think about it too much. English is full of quirks like that…

    3. It’s a slightly idiomatic figure of speech that simply means “at a high speed”, and you are correct, it is not a strictly accurate phrasing. It’s very often heard from police trying to sound technical and authoritative when describing an accident (e.g., “…the vehicle was traveling at a high rate of speed when the driver swerved…”) and it seems to be drifting into common usage.

      The argument for its use is usually that you’re saying it’s a high rate, and then clarifying that the rate is speed. Perhaps if you knock a hole in a boat there might be a “high rate of water leaking”, or if your business is doing well there’s a “high rate of income”. But speed is by definition a rate, so it’s just extra words. There’s definitely a high rate of annoying this pedant. :)

      At this point it’s about like people using “literally” to mean “figuratively”; you try to correct it and you just get people arguing “languages evolve, you know what I mean so it doesn’t matter if it’s the dictionary definition”.

    4. It depends on how the word “rate” is intended to be understood. A rate can be many things per unit time: peanuts per hour, gallons per second, etc., and more loosely time can be discarded: a rate of 50 pounds fertilizer per acre. Having used the word “rate”, the type of rate being considered must be specified, and in this case it’s speed.

      It’s not wrong, it’s just superfluous. “Rate of” is purposeless filler and should be removed.

  1. >first plane ever to shoot itself down

    I was always told that before the invention of the interrupter gear there were planes that did try to shoot (with things like metal props) and did shoot themselves down. Also that the interrupter gear wasn’t exactly perfect and accidents did happen. Is that not true?

    1. To my knowledge no aircraft were actually lost entirely to such occurrence, but it may have happened. You are most certainly correct shooting through the prop has harmed the shooting aircraft before – and to an extent you could call shooting down – but turning yourself into a very under powered aircraft or glider and making something like a landing is a little different to smashing and burning…

      1. Blowing a blade off your prop is usually followed by the engine shaking the motor mount apart. Moving the CG back to the point the plane is uncontrollable.

        I was watching a bunch of vids about Rusky wwii aircraft.

        Apparently Stalin was fixated on big guns for fighter aircraft. Nobody dared disagree with him (not after the first one did anyhow). A whole lot of them shot themselves down.

        This was hardly the first plane to ‘shoot itself down’ if you include the gun damaging the airframe with recoil, the engine ingesting powder fumes or the recoilless port blowing off the tail.

        1. Interesting and depressingly true to form for the Russian’s…

          Never been interested in Russian airpower in that era really, as you don’t need to know much to know they were rather behind the times and didn’t make a great deal of use of airpower – so nothing in my other studies sucked me down that rabbit hole.

          Very much sounds like its still a define ‘shot down’ though – most if not all of those examples are the plane can make it to the ground in a state it is potentially repairable, just likely not worth the effort, as the design is clearly flawed…

          1. Ever read about the Ilyushin IL2? They made at least 35,000 of them, and the aircraft made an enormous difference on the eastern front.
            Russia didn’t spend much energy on air superiority. They imported airplanes for that. Instead they made a flying tank and used it to kill tanks.

          2. Indeed I have read about it, doesn’t make Russian aircraft on the whole interesting, use in really unusual interesting ways/locations (which may have happened and I just never bumped into it) or cutting edge engineering (which it really isn’t) are far more interesting than the one single aircraft that really isn’t very good or all that unique overall – its decent enough, certainly a bit more workable being better armored in ground attack roles than the more general purpose very competent aircraft overall were, and at least reportedly got put to some good use but for me just not interesting enough to change the overview on Russian aircraft of this era for me.

            Just making lots of something doesn’t make it good, or worth making nearly that many of – though something is better than nothing… If they had gone a little more all in on such a ground attack concept along the lines of the Warthog – something with some more real firepower to go with the tougher shell so it really would be very effective rather than mostly a nuisance I’d be singing a different tune…

    2. At first steel plates were put on props at the radius of the gun barrel. Lets just say that the planes with the guns on the upper wing had fewer problems. I think Eddie Rickenbacker describes all this in a book.

  2. Car engineers / designers could also stand to learn a similar lesson as well, and that lesson is called “design for serviceability” (and practicality, too boot).

    Why should changing the battery in certain models of cars (PT Turbo) require me to remove the driver’s side wheel? Same model of car also put the alternator at the lowest point of the engine, exposing it to water ingress which does not play nicely with the charging and electrical systems. Oh, and you have to remove one of the half-shafts to replace it. (WTF, Chrysler?!?!)

    And cars that require you to drop out the engine to change the spark plugs. Or remove large amounts of the front trim to change headlight bulbs.

    1. Cost. Design time. And, of course, once the car is sold & the manufacturer has their money, they really don’t care if you have to drop the engine to change the oil filter, do they?

      [Lily Tomlin as Earnestine: “We don’t care. We don’t have to. We’re The Phone Company!]

      1. CAD. The designers are no longer mechanics or machinists, so they don’t know the practical points of construction. They make lots of errors, like designing structures that cannot be physically built, until they pick up just enough practical skills that their designs can be manufactured – by robots because people would be far too slow.

    2. Every automotive integration engineer should be forced to work on a PT Cruiser at least once. Dang ol’ waterpump replacement is an engine-out procedure. Gotta pull the front wheels to change the headlights. Hateful.

      1. 1997-2004 Dodge Dakota. To change the oxygen sensors you’re supposed to either lower the transmission or lift the cab, because the connectors are tucked up in the tight space between the upper side of the transmission and the underside of the floor.

        I got a strong piece of wire, bent it just right to hook onto the connectors, then ripped them off their brackets. New O2 sensors and header pipe with cats got installed and the connectors tie wired up where they’d be away from the exhaust pipes.

      2. New ‘bug’.
        First step to access most firewall mounted parts. ‘Remove front bumper’.

        PT Cruiser is just a tarted up, overweight Neon.
        People that own them deserve it.
        Same for new bugs.

        Front wheels out to change the headlights isn’t uncommon. Every car that requires the wheel well liners out, which is most newer cars.
        Also timing belt driven water pumps are genius compared to timing chain driven water pumps. Timing chain driven water pumps never need replacing, because when they fail, they wreck the whole damn motor. Saving you from fixing them.

    3. The problem is Design for Manufacture takes priority, because it has a direct impact on cost (and thus sticker price). Designing a more easily maintained vehicle (even using the exact same quality components!) will increase the cost due to changes made at the expense of manufacturability, which means either a less profitable vehicle or a higher sticker price.

      And buyers of cars have so consistently and repeatedly chosen “cheaper” over “more maintainable” for so long that we are now in the current state of affairs.

      1. In contrast to this problem, my wife’s Toyota Camry (2002) was made to be worked on.

        I had to change a power steering hose. They placed a hole in the wheel well on the passenger side. A crows foot and 3 ft of extensions allows reaching that exact fitting. I looked, it doesn’t seem likely that hole is for anything else.

        I love working on that car.

        I think they used this chassis as taxi cabs in Japan long before it became the camry.

    4. Amen.

      I can’t find the make and model but it was a low end “luxury” car with a direct impingement engine and a belt drive for the cam shafts. So you can imagine the bent/smashed valves, poor timing, and cracked piston heads that came of this after the belts stretched. And of course the belt was hidden under layers of components.

      But wait there’s more! Its not just cars.

      If you’re truly masochistic, join the Navy or Coast Guard and work your way into the shop that maintains the Mk38 Mod II 25mm. Take all the logic used in bad car designs and use it on a gun routinely exposed to salt spray and ceaseless vibration, then abandon the software suite halfway through the first build after you reach a minimum viable product.

    5. I was trying to fix a work boat used on offshore vessel. Yannmar engine – all electrical wiring done in braid manner with cables tight to engine. Engine under deck with whole barely larger than engine. To replace rpm pickup – remove engine or deck. I figured out one relay was broken but could not find find it. Finally I found it but was positioned in somewhere under engine and socket was directly mounted on engine and fastened by warping cables around structure. Finally had to cut cables and reposition the relay (far from bilge water) and left old one where it was ( over bilge water).

    6. Elon Musk buying Twitter is a great recent example of forward-thinking by an automobile manufacturer. New Telsas will be able to automatically tweet denials if they happen to run over small children while on autopilot.

    7. Car dealers make a lot of money in their service departments. Car makers like to support the dealers and, if possible, drive people to buying new cars when the cost to make otherwise minor repairs exceed the used-car value.

      Sure, some of this is from making the drive-train a one-piece module that has accessible parts during its assembly and is a drop-in (well, lift up) part on the final assembly where all the remaining components box it in, but the previous reasons make sure there is no further consideration needed.

          1. Your problem is “or density and pressure”. I suppose you are thinking PV=nRT settles it (Ideal gas law under adiabatic conditions.). But the temperature goes down with altitude until you hit the troposphere/stratosphere boundary at which temperature increases with altitude. That boundary varies from roughly 20,000 to 70,000 feet depending on where you are in the World at any given time and is caused non-adiabatic sources.

      1. There is a relationship between the 3 when the air moves from one location or condition to another, not when the air stays in place and the airplane does. For example, air pressurized in a tank to 2000 psi can be at the same temperature as the air outside of the tank. Likewise a closed volume of air when heated maintains a constant density while the temperature and pressure change. Otherwise the three are entirely independent variables

  3. “So when the bullet leaves the plane it would take the plane over three seconds to catch up with it, by which time it has moved ever further away, right?”. Uhm, why exactly? When the bullet leaves the plane, there’s zero distance to catch up to.

    My only guess is that the author meant that the plane will take ~3s to catch up with the point where the bullet was 1s after firing. But that makes almost as much sense as the rest of the article…

  4. How GLaDOS might describe the incident…

    Speedy things go out, speedy things slow down, less speedy thing catches up to speedy things and hits them. Less speedy thing makes fiery crash.

  5. Minor point of note is that the Dutch F-16 incident mentioned was not a complete round hitting the aircraft but a practice round (Reduced-Ricochet-Risk training munition) designed to break into separate parts on impact that had split immediately after leaving the muzzle due to damage incurred before or during loading.

    (Summary, in Dutch here: and full report (has some pictures of the damage, equipment used and the 20mm reduced ricochet practice rounds here:

    Turns out that if you use worn/old manual equipment to load and unload rounds designed to break apart it’s a bad idea to reuse ammunition once it’s been loaded into the aircraft. It’s also not the only incident of F-16s returning with extra holes after firing the 20mm canon, but it’s one of the few where damage was more extensive than damaged plating.

  6. “Common sense isn’t always engineering sense.”

    But the fact that an aircraft could run into its own projectiles and missiles is BOTH common and engineering sense since the projectiles immediately start decelerating and the missiles do so after motor burnout while the aircraft doesn’t because it is continuously powered. Thinking otherwise is probably just a case of bureaucratic stupidity.

  7. “Trust your instincts, but verify with solid mathematical models that take everything into account.”
    That’s exactly the issue the article started with, incomplete mathematical models. So it’s a weird conclusion. Intuition actually often helps fill gaps in a model and vice versa.

    Testing systematically makes more sense, like in software engineering, because it’s too complex to model or prove the whole system. Validate your assumptions, know common pitfalls, try to test corner cases, “unlikely” combinations.

    None of this is really math related.

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