As I mentioned in Part I, the 7th and 8th graders were making a trebuchet.1 As you see in the picture above, it was parked in my classroom for a while. The public demonstration was at the Medieval Feast, which I was sorry to miss. Me and the missus were offered the positions of King and Queen, but I had an MIT Free Speech Alliance Exec Comm meeting at that time.
We talked in Part I about how to mathematically model the path of a trebuchet missile. I went through three models:
The Triangle Model, in which the missile travels in a straight diagonal line up and out. This is good for modelling the first part of the flight. It is what is known as “a linear model”, because the path is a straight line. It does imply that the missile goes up forever, into outer space, and so it work badly for most of the flight unless you launch from the asteroid Nemesis or a space station. But it is a simple model, easy to work with, which is good. “Make your model as simple as possible, but no simpler,” as the saying goes.2
The Gravity Model, in which the missile travels in a smooth curve first up and then down. This is good for modelling the flight until the missile hits the ground or a wall. It is an example of “a nonlinear model”, since it is not a straight line and the equation is not something like y = 4 + 80x, but rather something like y = 4 + 80x - 3x^2. This particular nonlinear curve is called a parabola.3 Such a model is also called a "quadratic model”, because of the x^2. Squaring something is related to quadrangles (four-sided shapes); hence the term, “quadratic”.
The Stopping Model, which is the gravity model plus a restriction that the missile has to stop when it hits the ground. This is a good model for if there is no castle wall that would stop the missile before it hit the ground. We could go on to create a Castle Model to add a castle wall that is hit if the trebuchet is close enough to the castle.
VirtualTrebuchet.com, a nifty site for estimating range and for lots of other information on trebuchets, has an even more intricate model. The next figure shows how the output on the website looks:
The equations the site uses are hard to decipher:
Much of the complexity is because the program doesn’t require you to know the horizontal and vertical speeds in advance, as our three models did. Instead, it calculate hose speeds using things like the angle of release, the weight of the missile, the dimensions of the trebuchet, and the size of the weights that propel the missile. In effect, it figures out the diagonal speed at release and then uses trigonometry— the SINE and COSINE functions— to figure out the horizontal and vertical speeds. But my 7th graders don’t know trig yet, which is why I start by assuming we know the horizontal and vertical speeds at release. Still, it’s interesting to see the shape of those functions, since they’re very useful in mathematical modelling. They look exactly alike, just shifted, and they are good for when you need to understand things that repeat over and over like ocean waves or radio waves.
The VirtualTrebuchet.Com model allows you to add in the effect of the wind, which could help or hurt the distance. I suspect it even adds in the friction of the air, which tends to slow down both the horizontal and the vertical speed. So it’s a more precise model, but a more complicated one.
Is it worth the extra precision? That depends on which you like more, simplicity or precision. The tradeoff between simplicity and precision comes up lots of places. In fact, all of thinking is about that tradeoff. There are lots of ways to think about any idea. Most of them are wrong. But there are even lots of right ways to think about any idea, and which of those right ways is the best depends on your situation, on what decision you are trying to make or what problem you are trying to solve. You might be thinking hard about the first 10 feet of a missile’s travel or you might be thinking about the entire path. You might be thinking of shooting it on the Earth, or on an asteroid.
What we’ve been talking about is an example of the science of “ballistics”. Ballistics is essential to thinking about shooting a gun or a missile. If you are shooting a revolver at a crazed shooter twenty feet away, you can use the Diagonal Model, because gravity doesn’t matter. The bullet will travel in almost exactly a straight line and it would be silly to spend time figuring out how much gravity will pull down the bullet in those twenty feet or what direction the wind is blowing. If you are shooting a sniper rifle at a crazed Islamist half a mile away, you do need to use a more complicated model. If you are shooting an ICBM (an InterContinental Ballistic Missile) at Shanghai, it’s worth using an even more complicated model than Trebuchet.com has. ICBM’s aren’t like cruise missiles, which are powered throughout their flight; they are more like bottle rockets, which are powered for a while but then keep going even after the gunpowder is all used up; they are a combination of airplane and bullet.
As the Center for Arms Control tells us:
Ballistic missiles have three stages of flight:
Boost Phase begins at launch and lasts until the rocket engine(s) stops firing and the missile begins unpowered flight. Depending on the missile, boost phase can last three to five minutes. Most of this phase takes place in the atmosphere.
Midcourse Phase begins after the rocket(s) stops firing. The missile continues to ascend toward the highest point in its trajectory, and then begins to descend toward Earth. This is the longest phase of a missile’s flight; for ICBMs, it can last around 20 minutes. During midcourse phase, ICBMs can travel around 24,000 kilometers per hour (15,000 miles per hour).
Terminal Phase begins when the detached warhead(s) reenter the Earth’s atmosphere and ends upon impact or detonation. During this phase, which can last for less than a minute, strategic warheads can be traveling at speeds greater than 3,200 kilometers per hour (1,988 miles per hour).
Cruise missiles are unmanned vehicles that are propelled by jet engines, much like an airplane. They can be launched from ground, air, or sea platforms
Friction is a pretty serious concern for an ICBM. It is going a long way through air, and that is going to slow it down as it pushes through. Worse, as far as complexity, the air is getting thinner as the missile gets higher, and eventually it is negligible when the missile leaves the atmosphere and is in outer space. Unlike in the Triangle Model on an asteroid, though, the Earth is so big that the missile does get pulled back down eventually. A further complication, though, is that since the Earth is a ball, not flat, you have to figure out how the missile is curving around the Earth’s surface to figure out where it lands.
Like the ICBM, however, we need to depart from outer space and get back to earth. All of these models are ways to understand something about what goes on in the world.4 But we can’t know what the world is *really* like. All we know is what we can see and touch and hear, and what other people tell us from what they can see, and what we can understand by learning models to organize our thoughts about what people see. Getting at true reality is tough. And sometimes the best model isn’t even mathematical; it may be something quite different. For example,
When the sun rises, do you not see a round disc of fire somewhat like a guinea? O no, no, I see an innumerable company of the heavenly host crying Holy, Holy, Holy is the Lord God Almighty.
-- William Blake 'A Vision of the Last Judgement' (1810) in 'MS Note-Book' p. 95
Blake is talking about two models of the Sun. One strips things down to appearance in the sky. The other strips it down to a monument to the glory of God. There are more model: as a hot ball of gas, or as a point in the Solar System, or how it crosses the sky from morning till evening from east to west.
Footnotes
A trebuchet is a kind of catapult pronounced, to delight and surprise, equally accurately as trebyoushett as as traybooshay. (I could have written that to avoid “as as”, which usually would be better style since it wouldn’t focus attention on itself, but part of the style of Ras-Stack is to have fun with words.) Just because the word came from France doesn’t mean we need to pronounce it froggishly. And they are right to pronounce "un hashtag” as they do.
“Hyperbole” in rhetoric is the practice of intentional exaggeration, obvious to the listener, intended to state something very strikingly. This different from the elegant lying Pooh-Bah describes in The Mikado as “Merely corroborative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative.” Like many aphorisms, the modelling aphorism is hyperbole, or, as one might say, “hyperbolic”. The very simplest model would be that the missile goes nowhere. The next simplest would be that the height and speed never change and it just goes horizontally in a straight line forever. Those are useless models. The aphorism is not to be taken literally; rather, when you hear it, you think, “Yes, I should try really hard to make my models very simple, but of course they can't be *too* simple, or they just fulfill their intended purpose.” In a later footnote you will read about the mathematical “hyperbola”, which is also hyperbolic, but in a very different way.
The modelling aphorism is quite likely from Alfred Einstein, as “Everything should be made as simple as possible, but not simpler,” but it first appears as quoted from him in 1950. I have made my own version in the text above. It is a dramatic version of Ockham’s Razor: “Entia non sunt multiplicanda praeter necessitatem” (Entities must not be multiplied beyond necessity), which comes from John Punch, who said it was a "common axiom" (axioma vulgare) of the Scholastics in his 1639 commentary on Duns Scotus's Opus Oxoniense, book III, dist. 34, q. 1; in his Johannus Poncius, Scotus Opera Omnia,. Volume 15. Ockham’s Razord was around before Ockham, e.g. in Aristotle and Aquinas. Ockham himself published it in a different version, “Numquam ponenda est pluralitas sine necessitate” ("Plurality must never be posited without necessity") in his commentary on Peter Lombard, Sentences of Peter Lombard (Quaestiones et decisiones in quattuor libros Sententiarum Petri Lombardi; ed. Lugd., 1495, i, dist. 27, qu. 2, K).
A “parabola” has an equation like y = 4 + 30x - 2x^2, as opposed to a “hyperbola”, which has an equation like y = 78 + 1/x and curves down from infinity towards zero but never reaches zero. I remember how back in Uni High in the 1970’s Mike Mueller would talk about a third curve, the “superbowla”, which, however, had no equations associated with it, though it does have numbers (I, II, III, IV, . . . ).
Well, except when we want a model to understand outer space. It *is* hard to leave that subject, isn’t it?)