A trebuchet is a type of catapult that converts the potential energy of a counterweight into the kinetic energy of a projectile. The simplest version operates like a see-saw, with the counterweight suspended from a hanger attached to the short arm and the projectile held in a sling attached to the throwing arm. When the short arm is raised and then released, the throwing arm rotates faster because it is longer and the sling rotates even faster as it whips around the end of the throwing arm.
At their inception during the Middle Ages, trebuchets were used to hurl boulders at or over castle walls in an attempt, usually successful, to batter them down. Modern trebuchet designers, lacking castles to besiege and fair maidens to rescue, must content themselves with hurling cooking pumpkins for distance. Competition, however, is fierce, and the winner of the contest can expect any fair maidens present at the pumpkin festival to hurl themselves at him. Thus, it behooves us to put as much study into the ballistics of pumpkins as old-time mathematicians put into the study of cannonballs.
Ron Toms, the owner of the Catapult Message Board, dares me to go talk to an engineering or physics professor regarding that nonsense I’ve been peddling about 45° being the optimal launch angle.
Very well. I will take that dare. Are there any engineering or physics professors on this forum who would like to comment on the accuracy of my paper?
You’d need one Hell of a trebuchet for air resistance to be significant for a pumpkin, I’d bet. But if you insist on including air resistance, approximate it as a sphere, and call the coefficient of drag a half.
Mr. Toms is correct: a 45-degree launch angle is optimal only for a zero-drag projectile.
Instead of trying to logic it out a la Hobbes, I’d suggest playing with a simulation in Excel.
As it happens, I did this a couple of years ago when a friend and I were experimenting with a pototo cannon. If you’d like to meddle with it, click here to download a zip file containing an excel spreadsheet. You can tweak any quantity highlighted in yellow, including:
-initial firing altitude
-firing angle
-muzzle velocity
-projectile diameter
-coefficient of drag (note: this is not a “ballistic coefficient,” just a coefficient of aerodynamic drag)
-projectile density
The spreadsheet will do the rest, showing you a plot of the trajectory, as well as indicating maximum downrange distance before impact and peak elevation.
You will find that the only way 45 degrees is optimal is if the drag coefficient is reduced to zero.
In my experience, playing with similar simulations and my own pumpkin-launching trebuchet, the optimal angle is absolutely less than 45 degrees, but not THAT much less. Something more like 40 degrees I’d say (ETA: that’s for my relatively modest trebby that can chunk around 300 feet. More powerful trebs will have a lower optimal angle). Also that optimal peak is pretty broad so plus or minus two or three degrees doesn’t make much of a difference. You missed that peak in your calculations, by sampling two angles that are way to shallow, skipping from 31 degrees to 45 degrees without sampling the points in between.
Also, in the real world, it’s pretty freaking impossible to accurately determine the release angle when everything is moving that fast, even if you’ve got a good camera to take videos. That’s why we went with an adjustable release pin – interchangeable pins with a range of angles at first, and a genuinely adjustable pin in the current revision.
I ran a few pumpkins through my simulator. Assuming a spherical (C[sub]d[/sub] = 0.47) water-filled pumpkin (density 1000 kg/m[sup]3[/sup]) 0.2 meters in diameter, for various muzzle velocities the optimal angle is as follows:
Air cannons are required for truly ludicrous range; Wikipedia claims one has fired up to 1351 meters. According to my calculator, assuming the projectile is as described earlier, this requires a muzzle velocity of 650 meters and a firing angle of just 24 degrees. Official Pumpkin Chunkin rules require a projectile to weigh 8-10 pounds, so this is probably about right; the calculator shows just over 4 kilos for the 20-centimeter solid pumpkin.
In my freshman engineering physics class, we did some actual experiments with shooting rubber bands. Everyone came up with 45 degrees, though our simple protractors mean that’s about +/-5 degrees in accuracy. The professor felt that 45 was the right answer.
Looking at the numbers other people have, it looks like you need a really high velocity to make something less than 45 preferable.
Cecil is a very busy man, and as such very very rarely actually posts to these fora.
However, there is at least one actual rocket scientist (Stranger on a Train) that posts here. Hopefully he or his compatriots will be along to assist you with this…
How fast/far were you shooting these rubber bands? If you were barely stretching them, then yeah, the answer is going to be something close to (but necessarily less than) 45 degrees. It sounds like experimental error was a very large factor here, especially if you were shooting them from your fingers instead of some kind of rig that gives a repeatable firing angle and stretch length.
That’s a really odd statement. You’re not supposed to “feel” what the right answer is in physics.
It is not the OP’s question, but I will note that such siege engines were normally fired at MUCH less than 45 degrees, because the arrival angle at a fortress wall will be very close to the launch angle, and the vertical component of velocity is rather less damaging to a wall, than the horizontal. They built them as powerful as they could manage, and moved back just beyond the range of the defenders.
It was not a fancy experiment, but we did have some control. Basically, we took a ruler, a protractor and a rubber band. We set the angle of the ruler by holding it against the protractor, hooked the rubber band over the end of the ruler and stretched it down to a fixed length. Then we released it and measured distance. We did a few trials and averaged the results. Experimental error was definitely a factor.
When I say the professor “felt” it was the right answer… We all knew that the standard ballistics equation assumes 0 friction and 45 degrees optimal angle. The experiment was phrased as a test to verify that equation, and there was no instruction to correct for drag or find flaws with the basic equation. All of us came out with numbers that matched the 45 degree conclusion (or matched it to within the error ranges of our measurements) and the professor gave us full credit.
Mostly, I offered the story up because my freshman physics instructors certainly did not tell us that lower firing angles might be better due to drag.