Dot Physics

Some Tips for Punkin Chunkin

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.

i-8a66e9c28b48a06419222209628d04d1-untitled.jpg

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

i-7c6c95388a9648241f075e8774ef6f40-la_te_xi_t_1.jpg

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-e20360d4066079836fc9fc3894b86f89-pumpkintrajpng.jpg

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.

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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.

i-efa20eec139f8edc1c106221ccbc7b5c-range_1png.jpg

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.

Comments

  1. #1 Leif Segen
    November 7, 2009

    What software do you use for your diagrams and graphs? I would guess OmniGraffle, but I don’t think it can integrate mathtype.

  2. #2 Rhett
    November 7, 2009

    @Leif

    For graphs, I mostly use matplotlib module for python. If I want to draw on top of a graph, I put the graph in Keynote (by Apple) and then use the drawing tools. I do the same for the other diagrams. I use Keynote and equations are imported from LaTeXit.

  3. #3 Uncle Al
    November 9, 2009

    http://www.punkinchunkin.com/rules.htm

    Rules ban deep pockets for innovation (18-wheeler of laser capacitors, heater bank, hogshead or five of liquid helium). If you already bought the capacitors… a pumpkin railgun? Grow the projectiles in square frames and apply for DARPA funding re solid state switching and low impedence cabling.

    The US Naval Surface Warfare Center Dahlgren Division railgun accelerates a 7-lb projectile to Mach 8 (6090 mph). A 2000 mph pumpkin is entirely resonable.

  4. #4 Andy
    November 18, 2009

    800 mph is close (past?) Mach 1 at sea level, so the simple v-squared drag is no longer accurate. I suspect that a supersonic pumpkin would not last very long at all – certainly not 5000 feet.

  5. #5 a mile will fall
    November 29, 2009

    Uncle Al :http://www.punkinchunkin.com/rules.htm
    Rules ban deep pockets for innovation (18-wheeler of laser capacitors, heater bank, hogshead or five of liquid helium). If you already bought the capacitors… a pumpkin railgun? Grow the projectiles in square frames and apply for DARPA funding re solid state switching and low impedence cabling.
    The US Naval Surface Warfare Center Dahlgren Division railgun accelerates a 7-lb projectile to Mach 8 (6090 mph). A 2000 mph pumpkin is entirely resonable.

    1. There are no rules banning expensive technology. Just mechanical (including compressed air) means of acceleration are allowed.

    2. Growing pumpkins in square frames violates general rule number 4.

    A mile chunk will happen – we’re already 85% there. A big enough machine and enough R&D / build money are the only obstacles.

    what if you pulled a vaccuum in front of the pumpkin and compressed air behind?

    what if you could impart a nice backspin like a golf ball?

    what if you bred / engineered a more aerodynamic pumpkin?

    and so on.

    a mile will fall one way or another

  6. #6 Jim
    November 26, 2010

    While I love the physics, I am childishly amused at the use of the term “supersonic pumpkin” in one of the comments.

    It makes me want to purchase an orange car and get a personalized license plate. SSNCPNKN.

  7. #7 Edward
    November 29, 2010

    A rough punpkin should go futher, shouldn’t it? Dimpled golf balls go further. Aerodynamically, a pumpkin if fairly rought.

    To obtain further flight of a spherical surface at high Reynolds number, as is the case with a pumpkin moving at about 100-600 miles per hour through air near sea level, we want a roughened surface. This is counter intuitive, however, a rough surface will have the airsteam traveling further along the body of the pumpkin before breaking away into turbulence. The further the air flows, the less overall drag. This occurs at a small expenditure in increase drag. Laminar flow (for a smooth ball) will have less surface drag (in my recollection) but more turbulent (wake drag) drag, resulting in greater overall drag.

  8. #8 Anthony
    December 1, 2010

    The rail gun was the option that had me wondering. If you could somehow build a rail gun that released the pumpkin at high speed without any piece of the machine crossing the firing line, would the pumpkin…

    - crush at acceleration?
    - explode in mid air due to air pressure?

    Would breaking the sound barrier have any effects on the pumpkin?

  9. #9 Pierce Nichols
    December 6, 2010

    I found this blog post while looking for data on the acceleration tolerance of pumpkins. I think that’s the real limit on the range of a chunk, because it limits your firing speed. When the pumpkin is released from the firing device, it will be subjected to a large deceleration by air resistance. The velocity at which that becomes larger than the acceleration tolerance of the pumpkin is the ultimate limit on firing speed, and therefore range.

    Anyone know what the range of acceleration tolerance for pumpkins is?

    I have some obvious ideas for getting more range out of the cannons — optimize the air path, improved valving, use a captured piston to improve bore sealing, etc.