In what has to be a win for science communication, AMC—the cable network behind the wildly successful zombie comic adaptation The Walking Dead—has decided to partner with an online instruction company and the University of California-Irvine to create a zombie-themed interdisciplinary course. Entitled Society, Science, Survival: Lessons from AMC’s The Walking Dead, the free online course will run for eight weeks alongside the show and use the premise of a zombie apocalypse to teach students about everything from viral infections to the nutritional value of survival foods (read: should I eat a squirrel?).

The collaboration has also decided to include a physics module in the course, focusing mainly on the gruesome interactions between projectiles and brains. For any science and pop culture geek, it seems like too good an opportunity to pass up if you have the time. Personally, I felt compelled to try the physics portion of the zombie-themed course before the new season of the show airs this Sunday night. I did.

Thanks to an email exchange with Professor Michael Dennin, professor of physics and astronomy at the University of California-Irvine, and one of the professors for the AMC course, I got my hands on one of the physics questions you could expect. Yes, it involves crossbow bolts and zombie brains.

So, time to enrich your brrrraaaaaaiiiiinnnnnssss with a zombie pop-quiz! Try your best; I’ll put the answer to the first question at the bottom of the post.

PART 1: Imagine that while scavenging a dilapidated convenient store for supplies, Daryl Dixon happens across a walker. He fires his trusty crossbow. Daryl's crossbow bolt—traveling at 120 meters per second and weighing 0.025 kilograms—comes to rest with one end sticking out the back of the walker's head. Estimate the average force on the bolt from this interaction with the zombie’s brain. (Assume the length of the walker’s head is 0.25 meters long and ignore the skull).

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Professor Dennin made clear to me that this will be a fairly introductory course, and so he can’t assume much knowledge of physics on the part of the students. I won’t either.

To solve PART 1, we first need to know how much energy a crossbow bolt carries with it in flight. In physics, the energy associated with motion is called kinetic energy, and is dependent on an object’s mass and velocity. You could derive the equation for kinetic energy if you really wanted to, Newton-style, or you could simply look it up like any engineer would do. Kinetic energy is equivalent to one-half the mass of an object multiplied by the square of that object’s velocity, or:

K e =(1/2)*Mass*(Velocity)2

Once you have the energy of the bolt, you need to find the force it will impart to the zombie’s brain. To go from energy to force, you could take advantage of the fact that energy can also be defined as a force acting over a distance, or work. Work is equal to force times distance, so, by dividing the bolt’s kinetic energy by the distance it traveled through the zombie’s head, you get back the force imparted to the bolt by the squishy zombie brain. It’s quite a bit—about the same as getting hit in the face with a baseball going 40 miles per hour.

That was pretty simplistic as far as physics is concerned, and contains more than enough material than can be suitably covered in one module in an online course. However, dear readers, here the quiz can get more complicated—much, much more complicated.

PART 2: Does a crossbow like the one Daryl Dixon uses on the show have enough energy to completely pass through a zombie head? Model the brain as a dense fluid and consider the skull in your calculations. Consider the bolt to be a simple cylinder with a drag coefficient of 2. Explain your answer. (Take your time, you’ll need it.)

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This question is substantially more complicated, ate up about five hours of my afternoon, and rigorously tested my googling skills. But the answer does tell us something interesting—it’s unlikely that even the most powerful crossbow could send a bolt all the way through a zombie’s head. It would probably get stuck.

To solve (or at least approximate) PART 2, we first need some initial conditions. The most powerful crossbow that I could find puts out 165 foot-pounds of energy, or 223 Joules. Next, based on the morbid task of assessing how fast a musket ball needs to go to pierce a human skull, it might take around 120 Joules to get through a zombie’s. A musket ball has a larger diameter than a crossbow bolt, but the value does give us a comparison (it’s harder to find skull-piercing values than you think).

We already have the bolt’s mass from PART 1, so next we have to model what happens when the bolt passes through a zombie brain. Like a car moving down the highway, the main resistance to the movement of the bolt through the brain will be fluid, in this case the brain itself (air in the car’s case). Therefore we have to calculate how much the brain will slow down the bolt. Ultimately that will tell us if the bolt can make it all the way through a zombie head. To do this, we could find the drag force acting on the bolt. This force is dependent on how dense the brain tissue is, how fast the bolt is going, how massive the bolt is, how much the shape of the bolt resists movement, and how much of the bolt is in contact with the brain (equation here).

If a brain is about as dense as water, the bolt has a drag coefficient of an arrow [PDF], and the bolt has a projected surface area about the size of a postage stamp, as I assumed, we can go ahead and calculate the drag force as the bolt travels through the brain. Of course, this is easier said than done as the bolt velocity, and therefore drag force, changes over time. I’ll save you the tedious numerical integration using Newton's second law, but here’s a sample of what I did:

You’ll note that the bolt only carries 101 Joules with it on brain entry—that’s because it had to make it through a layer of zombie skull first. Also, because the bolt travels through the brain almost instantaneously, I had to use a timestep of hundredths of a millisecond. I won’t bore you with the other 20,000 data points.

As you can see in the force column above, the bolt experiences huge drag forces as it travels through brain matter. That’s because the tissue is so dense. Firing a bolt into a brain is like shooting a high-powered rifle into water--hypersonic bullets basically explode. The rapid deceleration is what makes the forces so large (the bolt is pulling 3,600 Gs when it enters the undead head).

The portions of the table we are really interested in are the position and kinetic energy columns. If the crossbow bolt makes it to the back of the skull (0.25m from PART 1) with enough kinetic energy to pierce it, it will go right through. Assuming the bolt first expends 120 Joules of energy to make it through the front of the skull, here is a chart of how much energy it has by the time it makes it to the back:

Looking at the graph, by the time the bolt makes it to the back of the skull, it has less than 20 Joules of energy still bound up in its motion. If it will take another 120 Joules to make it out the back of the head, there is no way the bolt will make it. It will get stuck.

Even if you play around with the assumptions I made—the density of brain matter, the energy needed to enter the skull, the drag coefficient—the numbers never really make a crossbow through-and-through seem plausible. Indeed, in the highly scientific zombie research that I found on YouTube, even with an expert bowman, a powerful crossbow, and an analog head complete with skull and brain matter (seen above), the bolts almost always get stuck. This conclusion also agrees with how zombie heads on The Walking Dead typically experience crossbow bolts.

Sadly, real-world examples also back up the numbers. Most crossbow-related head injuries are self-inflicted in suicide attempts [NSFW], and in those the bolt too remains in the skull.

Of course, because I had to make a ton of assumptions, I can't definitively answer whether or not a crossbow bolt could really make it all the way through a zombie head. A decomposing zombie may have “squishy” brain and skull material, making it much easier for a bolt to make its way through. A clever crossbowman like Daryl Dixon may aim for the eyes or the back of the mouth, meaning less skull to penetrate and a greater chance of a through-and-through. Daryl might even use different types of bolt tips that make penetration easier. As for the rest of the assumptions I’ve made, I can leave that up to you sharp quiz-takers to decide if they are plausible.

No matter how complicated you want to get, the message is simple—physics is way more fun with zombies. I think this is science communication at its best. AMC and California-Irvine have developed a Trojan horse of pop culture filled with science, ready to come for the brains of eager students. Time will tell as to how many students sign up and are satisfied, but as far as getting them interested in science via their favorite fandoms, it seems like a no-brainer…because a crossbow bolt destroyed most of it.

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ANSWER:

Part 1: 720 Newtons

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