Monday, December 24, 2012

Physics of the Apocalypse

Since we all just survived another apocalypse, let's take a look at what could have been. I've always associated impending apocalypses (apocalypsi?) with feline and canine precipitation. A quick googling shows that there isn't any particular reason for me to make this association. I think I'm probably conflating two things: raining animals generally is seen as an apocalyptic portent, and Bill Murray believed dogs and cats living together was a disaster of biblical proportions. I've learned over the years to trust Bill Murray on these sorts of things.

Anywho, just how bad would a rain of cats and dogs be? To answer that, we need to know how much energy a falling animal packs. The basic equation here is U = mgh, where U is the gravitational potential energy of a falling object, m is its mass, g is the acceleration due to gravity at the Earth's surface (~9.8 m/s2), and h is the height of the falling object above the surface. When the object falls, it loses potential energy as its height decreases. But conservation of energy tells us that this energy cannot simply be lost; it is transformed into other forms. Ideally, all of the potential energy has been turned into kinetic energy by the time the helpless animal hits the ground.

So, wikipedia says that cumulonimbus clouds, which are the kind of clouds we often see during thunderstorms, range from 2,000-16,000 meters in elevation. Now, I don't really know what sorts of clouds you'd have during the apocalypse, or what sorts of clouds precipitate animals, but let's just say your dogs and cats are falling from a nice round 10 km up.

If we have a fat cat, or a smaller breed of dog, the animal may have a mass in the neighborhood of 10 kg. This gives us a potential energy of 980,000 J. Let's just call it one megajoule. So, on a perfectly spherical, airless ball resembling the Earth, a falling animal hits you with the kinetic energy of one megajoule. What's that like? Well, that's about the same as getting hit by a car going 70 mph. That is to say, it kills you dead.

But a car is significantly more massive than a dog or cat, which suggests that the falling animal is going much faster than 70 mph when it hits. Indeed, we can calculate the speed of the animal's descent from the energy equation. The kinetic energy of a moving object is the familiar K = &frac12mv2. A megajoule of energy and a mass of 10 kg works out to a speed of 450 m/s, or about a thousand miles per hour. That is very, very fast. It's also entirely unrealistic.

(Readers may notice that the initial energy, mgh, is equal to the final energy, &frac12mv2. Since both terms contain mass, the mass cancels out. Consequently, the final speed of a falling object turns out to be √2gh, which is independent of mass, just as Galileo and Newton told us.)

We're all familiar with the concept of terminal velocity, and it would seem to play a role here. Objects in an atmosphere can only fall so fast, because there comes a point at which the upward force from air resistance is equal to downward force of the object's weight. At that point, there are no net forces, which means no acceleration according to our good friend Newton. Modeling air resistance turns out to be somewhat hard (and involves differential equations, which your humble physics student hasn't gotten to yet), but one of the interesting things about air resistance is that it increases as the speed of the object increases. In fact, it generally increases with the square of the speed. The upshot here is that falling objects reach their terminal velocity very quickly, so the height from which our animals are falling doesn't really matter all that much.

While figuring out the specifics of air resistance can be difficult, the formula for terminal velocity pops right out if you have a model for air resistance. The basic formula is v = √(2mg)/(&#961AC), where &#961 is the density of air, A is the cross-sectional area of the animal, and C is the drag coefficient, a constant that has to do with the shape of the animal. Working this all out, the terminal velocity for our 10 kg animal is around 25 m/s (56 mph). This is lower than the terminal velocity of a human-like object, and this fact has been put forward as one of the reasons why cats seem particularly capable of surviving falls from great distances.

So, with a maximum speed of 25 m/s, the kinetic energy of our raining animal is a measly 3 kJ, or roughly equivalent to the kinetic energy of a bullet. Getting shot is a bad thing, I'm told, but bullets are much smaller than cats and dogs, which means that the force exerted by the falling animal at any one point is significantly lower than that of the bullet. Getting hit by a falling cat or dog would certainly hurt, but it would be unlikely to kill you. This turns out to be a fairly mild apocalyptic event. The real danger probably arises from the fact that, once the rain has ended, there will be a surprisingly large number of living, and very pissed off, cats and dogs roaming the streets.

The bigger the animal, of course, the more dangerous it gets. Because terminal velocity is dependent on mass, the kinetic energy of a falling animal is proportional to mass3/2. Cloud-bound elephants would be quite deadly, for example.

Now, some people might be wondering what happened to the other 9,997 kJ our falling furry friend was supposed to have. After all, energy is conserved, right? The answer can be framed a couple different ways. One is to say that the air does negative work on the falling animal. If this negative work is leaving the air, then the air gains energy. While this is a valid interpretation, it leaves one wondering just what negative energy is supposed to mean. The other point of view is that the falling animals push the air out of the way as they fall -- that is, they do work on the air. From this vantage point, the energy isn't lost but transferred to the air. Where it goes from there is beyond the scope of this example, but thanks to Emmy Noether, we know the energy will do just fine on its own.

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