Cannonballs do not travel on parabolas.
If you've ever driven a golf ball (I have, many times) you may also observe that from the tee, it looks nothing like a parabola. It appears to fly off into the distance, then plummet. I dare say cannonball flights look much the same, viewed from the cannon.
Just who is demanding that "reality should behave this way"?
It appears to fly off into the distance, then plummet. I dare say cannonball flights look much the same, viewed from the cannon.
Cannonballs weigh a lot more than golf balls relative to air resistance. More importantly much of the phenonemon you describe is due to the Magnus effect. That is, when you hit a golf ball with a golf club you impart a whole lot of backspin which allows the ball to maintain elevation beyond that of a pure parabolic arc. As far as I know cannons do not consistently impart backspin on their projectiles.
Cannonballs weigh a lot more than golf balls relative to air resistance.
Let's look at the numbers.
Cannonballs are a lot bigger and go a lot faster than golfballs, and resistance is proportional to the square of velocity and to the cross-sectional area, hence drag deceleration is proportional to (vel*diam)^2 / mass.
Cannonball (example): 590mph, 20 pounds, 5.5in diameter, giving ((263*0.14)^2) / 9.09 = 149.
Golfball: initial speed 70 m/s, diameter 1.68in, mass 1.62oz, giving in SI units a deceleration proportional to ((70*0.0426)^2) / 0.046 = 193.
So there's not much in it. Despite weighing two hundred times as much, the initial deceleration is only about 25% less. Solving the equations numerically gives clearly asymmetrical trajectories. When fired at 30 degrees elevation, the cannonball peaks 60% of the way through its flight and lands at an angle of 50 degrees.
You are correct about the Magnus effect.
A golf ball is a good example of something light enough that air resistance has a high impact.
I doubt people could follow the trajectory of a cannonball with their eyes. Ever tried to follow the trajectory of a bullet?
Oo! Oo! [Raises hand] I've followed the trajectory of a bullet with my own eyes! Admittedly, it was with tracers.
But, I bet in the history of firearms, there have been projectiles of such size, fired at such low velocity, that you could follow them with your eyes in the daytime.
Still, if you can't follow the exact trajectory, all the more reason to assume that the conventional wisdom is correct.
Agreed.
If you're standing directly behind (or in front of, I suppose) the cannon/gun it gets a lot easier, since the angular rate is much lower.
At night with the flood lights behind you, it's quite easy to watch the arced trajectory of handgun bullets.
But, I bet in the history of firearms, there have been projectiles of such size, fired at such low velocity, that you could follow them with your eyes in the daytime.
Some book I've read lists the ability to watch the arrow in flight as one of the advantages of the bow compared to the rifle (i.e. every round is a tracer round).
Not Sun Tzu, surely. Dates are uncertain, but it appears that he lived at least seven centuries before the probable invention of gunpowder.
I was moderately bothered by that as well but I went ahead and trusted my memory. Continued research is not finding the section I was thinking about. This is bizarre because I don't thing I've read anything else that that could have come from- maybe I had a copy that was embellished? Maybe I've forgotten the name of some samurai text that would have included that as a tip? The last seems most likely, since I believe that was the only time people were actually comparing the two.
But if you launch a stream of water out of a slanted tube or sleeve, even if you know nothing about paraboles, you can observe that the curve it follows in the air is symmetrical.
That's a very good point. Particluarly since most of the experiments were done by males. Come on guys, you've been aiming with full stream based visual feedback for years. It doesn't look anything like that!
My understanding is that in Aristotle's thinking, there's no reason to think a stream of fluid should move in the same way as an arrow that has left the string. They're made of different elements, for one thing.
Not for me it doesn't. I've heard the pressure varies from person to person, and decreases as you age. I can get a nice parabola still.
Early firearms had projectile velocities somewhere in the region of the speed of sound. I suggest that it is unlikely anyone would see the trajectory of the ball. The next most obvious thing to check is, does this model give correct-ish predictions of where the ball lands? And then I'd have to say, it's not obvious to me that the prediction would be wrong. There's such a thing as air resistance. It might well be correct to within 10% for the part of parameter space they cared about, namely firing at castle walls at less than a mile, noting that gunpowder varied, windage varied, and the position of the gun after being hauled back into battery varied. Observe that siege guns weighed multiple tons and were shifted by muscle power, human and oxen. Exactness of positioning was not in the cards.
Given all the other variables, a rule-of-thumb approach didn't have to give you a whole lot of accuracy to be good enough. I observe also that you don't want to hit a wall with your projectile going straight down; you want to hit it in the flattish part of the trajectory, preferably low on the wall. So what happened at the end of the trajectory might not be that interesting to an actual gunner, it might be there mainly for decoration, like dragons at the edges of maps. Perhaps all they needed to know was that the trajectory was reasonably flat for X yards, and then outside of that range - well, don't set up your gun that far out, you'll waste your ammunition. Don't you know that gunpowder is expensive?
As for streams of urine, come now. You don't have to accept Greek elements to think that a stream of liquid does not behave like a discrete hard object, especially when the initial speed is so different. The kinematics are the same but the aerodynamics are not. And who pisses at an initial upwards angle, anyway?
Finally, let me point out that Galileo's great invention for studying trajectories was not any sort of mathematics, but an inclined plane that slowed things down so he could observe them with the naked eye. This suggests to me that nobody here was going to do any better in trying to look at cannonballs going half the speed of sound, or for that matter streams of urine.
This model was not proposed by Aristotle, but by Albert of Saxony in the 14th century. (Added attribution to the main post now, thanks.) I mentioned Aristotle when talking about his completely different idea that a flying arrow is kept in motion by air that was set in motion by the bowstring releasing it, which predates the impetus theory.
but what kind of possible thought process could lead one to publish theories so grossly out of touch with everyday observations, even those that you see without making any explicit experiments? Did the authors think something along the lines of "Well, reality should behave this way, and if it doesn't, it's its own fault"?
Honestly, if I personally had never heard of Newton, I doubt that I would have found my own casual observations out of line with Aristotle's theory about how projectiles move or Albert of Saxony's diagram of the path of motion.
While this model is closer to reality than the original prediction, I still cannot help but think... How could they deviate from observations so strongly?
The trajectory where the object ends falling in a straight line is the observed trajectory for many objects, because air resistance slows them until forward motion is imperceptible. Cannonballs probably flew too quickly to be easily observed. The only person who knew that a cannonball was about to be fired and thus would be ready to watch it, would be standing at the cannon, and thus unable to perceive its trajectory in any case.
I agree, though, that if people played catch with balls, somebody watching from the side should have noticed that the ball never fell straight down at the end.
Some factors:
Thinking of the whole "pursuit of knowledge" process in terms of "retriveing what the ancients knew", something in which the concept of "experiment" doesn't feature
(possibly) a religious context in which truth == revealed truth. By definition, what's true is what the ancients wrote.
No concept of paraboles - it seems "right" (per Occam's Razor) that nature should only use simple shapes like circles and straight lines. Any observation that didn't seem to be an exact circle could be considered an "imperfect circle", distorted by things like wind.
Not that many occasions to actually witness a parabola, at least until fountains shooting water became widespread (yes, OK, men can regularly observe streams of liquid, but not from an angle that gives you a good view of the trajectory.
(Possibly) a social context that doesn't reward improvements to theory; if an apprentice to an engineer noticed that Aristotle was kinda wrong, what would he get by pointing that out to his master? If two rival engineers are vying for a juicy catapult contract, and one of them was known for criticizing Aristotle,who would the Lord choose? (Situations like that haven't disappeared, it's just that now Aristotle lost his prestige)
Note that the ancient Greeks studied parabolas in detail. They were consider the next most nice shape after circles (essentially tied with the other conic sections, ellipses and hyperbolas). So your third suggestion doesn't hold water. I suspect that Constanza's remark above is closer to the truth; there's a fair bit of hindsight bias in seeing the impetus theory as obviously incorrect.
Note how this obsession with "perfect shapes" led the ancient Greeks to postulate hypotheses that were, in fact, not empirically founded and ran contrary to Occam's Razor (which, granted, was not explicitly formulated back then):
I see a pattern here....
Seems analogous to the concept of elegance in modern math and science. I'm not sure if we should interpret the characteristic Greek speculation about symmetries as a violation of Occam's Razor -- it's certainly not empirically founded, but it's not clear to me that it'd increase the K-complexity or any of the other usual complexity measures when applied to the rather fuzzily defined Greek models of the world.
The addition of an extra planet, empirically unobserved and claimed to be hidden from observers, just to have the celestial body count add up to a "good" number, seems like a pretty clear Occam's Razor violation to me.
Conceded, but only because the specific mechanics of the Counter-Earth proposal were rather far-fetched.
In Classical times, so little was known about the actual mechanics underlying natural phenomena that an emphasis on fitting those phenomena into mathematical symmetries would be productive, even if there were some holes in the data. There simply wasn't that much rigorous data to study, and even fewer well-understood analytical tools to do it with, so I'd expect some real symmetries to look awkward in practice thanks to sampling bias. I think the Greek philosophers had some idea of this, too.
Hi - I would like to have permission to reprint the image of the trajectories for a book I am working on. Do you know who owns the publishing rights to the image? many thanks Simon
Your Google skills could use work.
If you copy the image URL and punched it into Google Images, the first hit is http://curvebank.calstatela.edu/harriot/harriot.htm which immediately says "Earlier contributions from Tartaglia's "Nova scientia . . ." (1537):"; as a 2D scan of a public domain work, by Corel v Bridgman, it is also in the public domain, so no one can own the right to it.
Late to the table, I know, but it just occurred to me that humans have an innate and intuitive understanding of projectile motion, which is why we can accurately throw and catch objects. Juggling, as an example, has been around for thousands of years, and it's a uniquely human activity (other apes can't be taught to juggle, and it's an incredibly difficult task to teach a machine).
Even if medieval theoreticians had never come across parabolic curves before, there was some gooey parabola-like shape somewhere in their brain.
Even later in replying, but oh well.
some gooey parabola-like shape somewhere in their brain
This does not seem obvious to me. The ability to make a rock go roughly where you want does not translate to the ability to accurately draw its trajectory on paper. Granting that there is clearly some part of the brain that does calculations (which may not involve parabolas because of the air resistance, as noted in earlier comments) you have no introspective access to those calculations. Besides which, they might well be wrong for cannonballs; humans do not throw half-ton weights at a good fraction of the speed of sound.
Yeah, I've re-addressed that line of reasoning. I became briefly fascinated by how a human brain could plot a quadratic shape, until I discovered the Gaze Heuristic.
I'm still convinced there's some sort of parabola heuristic though, simply through my own experience of juggling, which doesn't seem to conform to the gaze heuristic. I also cite the popularity of Angry Birds as weak but hilarious evidence.
Scratch that. I've put a bit more thought into it.
Primates can throw but not catch. More to the point, they can accurately hit things with projectiles, but they are considerably worse at anticipating the trajectory of an incoming projectile, (c.f. "chimps can't juggle").
Being a quadratic shape, calculating an intercept of a projectile's parabolic trajectory would require a square root operation. Our brains can't natively do that, so they need to cheat.
In order to anticipate where a parabolic projectile is going to land, you need to see it in motion a fraction of a second before or after the apex of its curve. (Disclosure: I've read this fact somewhere I trust but for the life of me can't remember where). Then you can just treat it as a falling object with a constant horizontal velocity, and no square root is required. Since the parabola's symmetrical about its apex, you can plot its entire path provided you know where its apex is, what gravity's like, and how fast it's going horizontally.
That explains catching, but how the hell does throwing work? That requires a complete projection of the parabola, which is presumably some sort of spandrel from the catching heuristic, so how come chimps can throw but not catch?
I'm reading a popular science encyclopedia now, particularly chapters about the history of physics. The chapter goes on to evaluate the development of the concept of kinetic energy, starting with Aristotle's (grossly incorrect) explanation of a flying arrow saying that it's kept in motion by the air behind it, and then continuing to medieval impetus theory. Added: The picture below illustrates the trajectory of a flying cannonball as described by Albert of Saxony.
While this model is closer to reality than the original prediction, I still cannot help but think... How could they deviate from observations so strongly?
Yes, yes, hindsight bias.
But if you launch a stream of water out of a slanted tube or sleeve, even if you know nothing about paraboles, you can observe that the curve it follows in the air is symmetrical. Balls such as those used for games would visibly not produce curves like depicted.
Perhaps the idea of verifying theories with experiments was only beginning to coalesce at that time, but what kind of possible thought process could lead one to publish theories so grossly out of touch with everyday observations, even those that you see without making any explicit experiments? Did the authors think something along the lines of "Well, reality should behave this way, and if it doesn't, it's its own fault"?