Title: Chased by zombies

When I heard word about the ScienceBlogs Zombie Day, I knew I wanted to participate with a post – but I had no idea what to do. My first thought was to somehow talk about living off the electric grid in the case of a zombipocolypse – you know, like how big of a solar panel would you need?

But you know what? Physics is difficult – but modeling is easy. How about I model something? How about a model for the motion of a zombie horde? This will be great.

## Zombie motion model

What do I want in my model? What are the constraints? What real-life situations can I use to test my model? I don’t know. Let me start with some guesses:

- Zombies move at a fairly constant speed. A speed that is slower than a human.
- Zombies like to eat brains. They will aim towards a living brain.
- Zombies are dumb, they will not take the best path to intercept. This should be a lot like the “dog chasing a rabbit problem”

I want to make a numerical model. Let me make up two fake forces that act on the zombie (just to see if this will give me a motion that I like).

- Propulsion force. This is a force that drives the zombies towards some brains. Really, this is just the frictional force (I will always assume the zombie will try as hard as possible to get to the brains). The direction of this force is towards the brain (person).
- Drag force. This is a force that keeps the zombies moving at a constant speed. If there was only the brain attracting force, they zombie would just keep speeding up. Just for simplicity, I am going to assume this drag force is proportional to the velocity and in the opposite direction as the velocity vector for the zombie.

How about a simple test. I will have a person with a brain walking at a constant speed (same speed as the zombie). The zombie will start off near the brain person and hopefully turn to follow. Oh, I am just using my normal numerical calculations techniques here.

This looks sort of like the classic “chasing a rabbit” problem – so that is good. Note – this is NOT the chasing a rabbit problem.

## Stuck in a room with a zombie

Seriously, this could happen. How could you move to keep the zombie away? Before I try to solve this analytically, let me suppose that move in a circle of radius 4 meters with zombie starting stationary at the center of the room. If you have the same speed as the zombie, this is what would happen.

No point making it a movie. The zombie would catch you. What if you went twice as fast as the zombie (not unreasonable, right?)

Interesting. This seems to work. The zombie starts moving in a circle of radius 2 meters with you (or me) going twice as fast in a 4 meter circle. I wonder what would happen if I move in a circle of radius 5 meters? I will just tell you, the zombie will “orbit” at 2.5 meters. Here is a fun graph. This is the distance from the dude (me or you) to the zombie as a function of time.

So, the motion of the two gets into a sort of equilibrium. Oh, I know what you are thinking – hey, it is just like orbital motion. Well, no. In orbital motion, there are some key differences:

- For gravitational orbits, the force depends on the distance. In my model, the force is constant.
- The zombie does not start off moving – so even if it was a gravitational force, the zombie would just “fall” into the dude.
- There is a “drag” force that would slow down the zombie even if there was a distance-based force.

But it is stable. How about I look at the forces and see if I can see something about this motion. Here is a diagram of the forces for the zombie.

This looks weird. Shouldn’t there be a force pushing the zombie towards the center of the circle? Well, there is. Those two forces aren’t exactly in opposite directions. There is a small component of the friction force pointing towards the center of the circle. Let me see if I can make this more obvious. What if the dude runs faster in a smaller circle? I just tried that, it still looks about the same.

When the zombie-runner combo is stable, they are both moving in a circle at constant speed. This means they have the same angular velocity. If I know the speed and radius of the runner, and the speed of the zombie, then I can find the radius of the zombie’s track:

Does this agree with the data from the simulation? Well, if I start with the zombie at 0.5 m/s and the runner (also known as the dude) at 1 m/s going in a circle of radius 5 m, this says the zombie should have a radius of 2.5 meters – which is just what I get from vpython.

Next question, how small of a circle can I move in to be safe if I only move at zombie speed? It appears the answer is: none. No matter how big the circle you would move in, the zombie would continue to “cut corners” of the circle ever so slightly. The closer the zombie is to you, the less this would gain the zombie, but eventually the zombie would be just close enough to eat you. Maybe you can see this better with another movie.

What if I just go slightly faster than the zombie (say 10% faster)? What is the smallest circle? I guess this depends on how close you want the zombie to get. Suppose you don’t want the zombie to get any closer than 0.5 meters. This is the same as saying:

Hopefully by now you realize *r*_{d} is the radius of the circle the dude moves in and *r*_{z} is the radius of the zombie’s circle. No wait. I can’t do this. This is not the distance between the two circle runners. The zombie will be lagging behind the human.

Fine, I will do it by trial and error. It looks like about 3 meters is the smallest radius circle. Any smaller than that and the zombie will get close to the human just at the beginning before settling into a stable pattern.

This is enough zombie calculations for now. But hear me now and listen to me later. I am not finished with zombies. First, I am adding a zombie tag to this blog. Second I propose APS open a new division of physics: zombie physics. This way we can devote more time to studying the motion of zombies and humans. That way we will be prepared.

Already, I can think of the following questions to be answered:

- What is the lag angle for the zombie? What does it depend on (well, clearly it depends on the speed and radius of the human dude) – but how?
- What if there are two or more zombies? How would the zombie interact with each other? Could you move in a circle still to keep them away?
- What if there are two zombies blocking a door? Could you get past them without going super-fast?
- Are there any other patterns of motion (other than a circle) that could keep one or more zombies away?

**Note:** Do you like my zombie profile picture? These were made for ScienceBlogs by Joseph Hewitt. He is also working on a scifi RPG game – check it out at http://www.gearheadrpg.com/