I don't know why they call it "Punkin" instead of pumpkin, but this contest is awesome. Surely you know what I am talking about, right? Ok. Here is a quick overview:
- Teams get together and build a machine to throw pumpkins.
- Really, that is it.
And yes, it is that awesome. Here is the official Punkin Chunkin stie. Why is this awesome? It just seems that Punkin Chunkin is a perfect combination of ingenuity, throwing stuff, beer, and a little bit of physics. I suspect that you could teach a whole introductory physics course just using Punkin Chunkin for the examples.
Here is one such example. One division of Punkin Chunkin is the human powered machines. For this competition, the team as 2 minutes for one person to "power up" the machine. All of the energy to launch the pumpkin must be from this person during this 2 minutes. Of course the machine can store this energy - like in a spring or a spinning flywheel or compressed air or whatever.
The Question: How much power does a person need to produce?
Of course, there isn't an answer to this question about power. Why? The more energy I put into the machine, the farther the pumpkin will go. So, that is the first thing to look at. How much energy do you need? I could go with the simple answer: you need the amount of energy needed to get the pumpkin up to its launch speed. This would be the kinetic energy of:
But this misses all the fun. First, how do you know the launch speed of the pumpkin? You don't. Second, no one really cares about how fast the pumpkin is moving at launch. Everyone cares about how far the pumpkin goes. This is a more complicated problem. Why? Because of air resistance. Here is a diagram of a pumpkin at some time during its "flight".
The air resistance is the problem. Without this force, the pumpkin throwing becomes a simple projectile motion calculation. You could easily determine where (how far) the pumpkin would land. The air resistance makes things tricky because it is a force that changes with the speed of the pumpkin. The faster the pumpkin, the greater the air resistance. Surely, you have seen this yourself. Stick your hand out of a moving car window. The faster you go, the greater the air resistance pushing on your hand.
How do you model this air resistance force? The most common way is with this expression for the magnitude of the force:
Here ρ is the density of the air, A is the cross sectional area of the object and C is the drag coefficient (depends on the shape of the object). If the air resistance is proportional to the square of the velocity, you can perhaps see one of the big problems that pumpkin chunkers have. As you launch a pumpkin faster and faster, you get an extremely large air resistance force. So doubling the launch speed does not double the distance the pumpkin will go.
Perhaps the best way to estimate the pumpkin range is with a numerical calculation. Instead of setting up one (or several) equations to solve, we cheat. We break the pumpkin flinging problem into many small steps. During each one of these steps, I can approximate the air resistance as being a constant force. Then this small step because a very easy problem. Oh, but if I break this into thousands of steps it will take me forever to do these tedious calculations. And this is where the computer comes in. Computers can do these tedious steps really quickly and they don't even complain.
Before starting with a calculation, I need to guess some stuff. Here are my assumptions for pumpkins (stolen from my previous post on punkin chunkin):
- Spherical pumpkins with a drag coefficient of 0.2.
- Pumpkin mass of 9 pounds (Punkin Chunkin rules require pumpkins to be between 8 and 10 pounds).
- A pumpkin diameter of 20 centimeters. This is really just a guess.
- Density of air with a value of 1.2 kg/m3.
Using that data, this is what the trajectory of a typical pumpkin.
But one trajectory isn't what I want. I want a plot of pumpkin range vs. launch speed. To do this, this is one other slight wrinkle. What angle should you launch the pumpkin at for maximum range? Hint: it isn't 45° (that is the maximum range for no air resistance motion). The correct answer is: who knows? Anyway, there is a different answer for different launch speeds.
This means that to find the relationship between launch speed and range, I will do the following:
- Start with some reasonable launch speed and launch angle.
- Adjust the angle of launch until the maximum range is found. Record this range and speed.
- Start over with a new launch speed.
This will give me the following plot of maximum range vs. launch speed. Simple, right? Here is that plot.
Maybe I should have used units of "mph" and "feet", but I didn't. Also, how about for fun (and to make sure things are working) let me plot the best launch angle vs. launch speed.
Interesting. First, why is it not a smooth curve? Well, I only changed the angle by 1 degree at a time. This means the launch angle could be 31° or 30°, but not in between. Next, does this angle plot make sense? I think so. Look at the lower launch speeds. The optimal launch angle is getting closer to 45° which is what you would expect for the case with no air resistance. Actually, this pumpkin starts 5.4 meters above the ground (my estimation of how high the pumpkin would be at launch).
Why smaller angles at higher speeds? Think of it this way. With no air resistance, a higher angle (up to 45°) gives the object more time to move horizontally. When you add air resistance, the object ends up just falling straight down after getting to the highest point. This doesn't give you much extra distance.
But wait. I didn't start talking about optimal launch angles, I was trying to look power.
Human Power
When talking about power, the first thing I need is the energy. Why? Because here is the definition of power:
The power comes from the person and goes into the kinetic energy of the pumpkin. Kinetic energy is:
Now you can see why I needed the launch speeds. Oh, and what about the time? The Punkin Chunkin rules set this at 120 seconds. What about efficiency? The power above would be the power put into the device (like into the rubber band). Of course, these things aren't perfectly efficient. Let me just say the efficiency is 80%. Total guess, really. There are so many different types of systems that a Chunkin team could use to "charge up" the machine.
So, here is the plot I really want to make. What is the human power needed to get the pumpkin range you want?
Looking at the Punkin Chunkin results for 2011, it looks like the winning team had a range of about 520 meters. According to my plot, this would take a person producing 114 watts for 2 minutes. Of course this assumes an 80% efficiency, remember? But nonetheless, it is possible.
Actually, I asked my brother about this. He just happens to be a bike-geek. That means he really really loves his bike (and he isn't too slow either). His bike has this device that records his power and he claims that for a 2 minute stretch, he could do about 400 Watts.
So 100 Watts should be fine. If you could do 400 Watts, you might be able to get the pumpkin to go over 1000 meters - again assuming very high machine efficiencies.
