More Punkin Chunkin Centrifugal Machine

This year's episode of Punkin Chunkin is coming up (I think tomorrow). Discovery just showed a teaser commercial with the specifications for one team's machine. If you are not familiar with Punkin Chunkin (World Championship Punkin Chunkin), the basic idea is to project some pumpkins. (note, if you waiting for the Discovery Channel show for the 2009 Punkin Chunkin, don't click on the previous link, it has the results already).

One of the categories for Punkin Chunkin is the centrifugal machine. These are machines that spin pumpkins around really fast in circles to shoot them. They are basically like a giant-sized stone and sling. The one that caught my attention was the one from Greg Wolfe's team Captain Inertia II. Here is a basic diagram of most of the centrifugal type machines.

Really, the only reason for the centrifugal machine to move the pumpkin around in a circle is so that it can speed up over a longer time. The air powered launchers do this by having a longer barrel. There are a couple of issues though. In order to move something around in a circle (even at a constant speed) there needs to be a force on the object. The faster it goes, the greater this force needs to be. (Here is an introduction to objects moving in a circle). The relationship between the total force need and the speed and size of the circle is: (this is just the magnitude)

So, this is really the limit to the centrifugal machine. If the acceleration of the pumpkin is too high, it will lose its structural integrity (go splat). In a previous post, I estimated the maximum acceleration of a pumpkin. A typical compressed air launcher accelerates a pumpkin to about 600 mph in about 100 feet. This would be an acceleration of around 1000 m/s².

For Captain Inertia II, what would be the circular acceleration?

Here is what I know. They claim a launch speed of 692 mph. I don't know the radius of the circle, but he did claim that it will launch 80 feet above the ground. This would mean the radius of the machine would probably be less than 40 feet (not sure what angle he thought it would be released at). Let me just assume r = 40 feet = about 12 meters. Also, 692 mph would be 309 m/s. Putting this into the above formula and solving for F_net/m, I get:

Ok - there's your problem. A couple of points. First, the machine may in fact be smaller than 40 feet radius. This would make the acceleration even larger. Second, the maximum acceleration of a pumpkin could be greater than this, but here the acceleration is way too large.

Well, how fast would it shoot at if it had a max acceleration of 1000 m/s²? Using the same expression and solving for the velocity, this would give a launch speed of just 110 m/s or 246 mph. Not good enough, not nearly good enough.

What about the range?

The other claim was that this launcher would shoot the pumpkin 1 mile. I have a suspicion that they calculated the launch speed that would give that range. Here is my short tutorial on projectile motion. This assumes that there is no air resistance, which clearly there would be some. If I use the numbers from the commercial and ignore air resistance, the pumpkin would go 27,000 feet when launched at 30 degree angle. Ok, then they did include air resistance.

I can't tell exactly how they calculated the range for their speed. In my previous post, I estimated a pumpkin would have to be launched near 800 mph in order to achieve the mile range. I should probably go back and look at my calculations. It seems these guys might be using pumpkins that are a little more dense that I assumed.