Dot Physics

I already attacked the 2008 Punkin Chunkin Show. So, now I going to give the chunkers some tips. In case you aren't familiar, the Punkin Chunkin contest has teams create devices to launch a pumpkin. They have different categories, but I am going to focus on the air-powered devices. The basic idea is to make an over sized pneumatic potato gun. Here are the things I was inspired to think about.

• It seems all the canons were aimed at about the same angle. Did they guess at the angle? Or is this trial and error? What would be the best angle for a pumpkin launch?
• Does the optimal angle of launch depend on the launch speed?
• The teams all try to obtain the goal of launching a pumpkin over a mile. Is there a theoretical pumpkin range limit?

Where do I start? Obviously, (if you know me at all) I am going to do this with a numerical calculation. Here is the basic idea (as a refresher). First, think about the forces on the pumpkin after it is launched. For now, I will just assume that it starts with a certain initial speed.

The complicated thing is F_air. This force depends on the velocity, making things tricky (unless you do it numerically). The model I will use for air resistance is:

The key thing is that 1) this force is proportional in magnitude to the square of the magnitude of the velocity. 2) it is in the opposite direction as the velocity vector. For the numerical model, the plan is:

• Calculate the forces on the pumpkin (air and gravity)
• Use the net force to calculate the change in momentum in a small time interval
• Use this average momentum (or the initial momentum - it doesn't really matter if the time interval is small) to calculate the new position.
• Keep repeating the above until I get to where I want the pumpkin to be.

Simple, no? Yes. Well, I do need to make some assumptions. Here they are:

• v-squared air resistance force. Yes, this an assumption. There are some cases where this doesn't work too well, but if the pumpkins are going slower than the speed of sound, it should be ok.
• Spherical pumpkins. This will allow me to calculate the cross sectional area and guess at the drag coefficient (C). Wikipedia lists some values for C. I am not sure about a pumpkin, is it a smooth sphere or a rough sphere? C for a smooth sphere is 0.1 and a rough sphere is 0.4. I will change these up, but start with a guess of 0.2 for C since it is more smooth than rough.
• Starting speeds. The Discovery Show 2008 Punkin Chunkin said they launch the pumpkins at 500 to 700 mph (actually different people said different things). That gives me a range to consider.
• The pumpkin mass. The official rules say that the pumpkin must be between 8 and 10 pounds in weight, so that is a good start.
• Pumpkin size. Really this is going to be a guess. I saw on the show that they use these white pumpkins that are more dense that a normal pumpkin. Totally guessing here - diameter from 20 - 30 cm.
• Density of air. I am going to assume that these pumpkins don't get too high that I have to worry about the change in density of the air (although I could take that into account).

Seems like a lot of assumptions, but I have a trick. The show said that the pumpkins are in the air for about 9 seconds and go 4100 feet. If I can get my numerical calculation to give me something like that, I will be golden. Here is the something that is close enough for me:

I didn't get it exactly the same. For this trajectory, the pumpkin went 4100 feet and was in the air for 14.7 seconds. I don't still have the original video and I can't remember what the guy said. My notes said it was in the air for 9 seconds, but maybe the guy actually said it was in the air for something like 9 seconds. To get these values, I played around with the initial values of the pumpkin some. This is good enough for me. (I will include my python code at the bottom if you want to play with it some more)

Now I can play around. First, what is the best angle to shoot this at? Here is some data. I ran the calculation at a variety of angles (from 1 degree to 89) and at three different launch speeds (550 mph, 600 mph, and 650 mph). This is a plot of range vs. launch angle.

So, for these ranges of initial speed it seems the launch angle doesn't change that much. About 33 degrees seems like a good choice. Next, how about we pump it up? How far can it go? I am going to launch at a 33 degree angle (although this might not be the best for long ranges) and then I will keep increasing the launch speed. One assumption I will make is that the coefficient of drag is independent of speed. This is certainly not the case especially when going faster than sound. Let me just see what happens anyway.

Even if the drag coefficient is not constant, I would expect something like this. At some point, starting faster really doesn't do too much for you because the drag will be so huge. But, this graph suggests that maybe a mile range could be achieved with a launch speed of around 800 mph. And here is the problem. How do you get a pumpkin up to that speed? The expanding gas exerts a force on it. If the force is too great on one side of the pumpkin (the other side does not have compressed air pushing on it), then the pumpkin will not survive the launch. A smashed pumpkin probably has a much greater drag coefficient.

Well, how do you get it up to the 800 mph speed with the same force? Make the launch tube longer. This will allow the force to act over a longer distance and not smash the pumpkin. You can do that calculation as a homework assignment (or maybe I will put it on a test) - hint: think work-energy principle.