As you might have guessed, this post is heavily influenced by xkcd’s brilliant weekly What-If blog.
While pondering meteorites striking the earth with absurd velocities, I got to wondering whether or not you could actually kill the human race with a single BB. Because physics is a frightening place, the answer to questions like this is usually yes.
To simplify the first stage of calculations, we need to know how much energy is required to kill the human race. I will call this constant the “ohgod,” and I will set it equal to the kinetic energy of a 15-kilometer-wide stony asteroid traveling at 22 kilometers per second, which would be more than sufficient to cause a mass extinction which would almost certainly wipe out the human race. One ohgod is approximately 1.283 x 10^24 joules, or about 2.6 Chicxulubs (or, as people who fear the awesomeness of Mesoamerican words put it, 2.6 dinosaur-killers).
The mass of a BB is surprisingly hard to find, although there is a very handy chart listing the masses of high-end BBs in grains, which can easily be converted to grams. By my reckoning, a standard 4.5-millimeter (0.177 caliber) BB should weigh about 0.4 grams. In order to figure out how fast a 0.4-gram BB would have to be moving to have 1 ohgod of kinetic energy, we must solve the relativistic kinetic energy equation for velocity. The relativistic kinetic energy equation is a little unwieldy:
E = [(1/sqrt(1-v^2/c^2)) + 1] * m * c^2
I actually had to get out pen and paper to solve this equation for v. Here’s my math, to prove that I’m not a lazy cretin:
When I calculated v and plugged my numbers back into the relativistic kinetic energy formula on WolframAlpha, I was greeted with one of the most satisfying things a nerd can ever see: I got back exactly 1.283 x 10^24 joules, which means I didn’t have to do all that algebra again.
As it turns out, in order to have a kinetic energy of a species-killing asteroid, a BB would have to be traveling quite fast. I would have to be traveling at 0.9999999999999999999996074284163612948528545037647345 c, in fact. That speed is slower than the speed of light by only a few parts in 10^22, which is to say by only a few parts in 10 billion trillion. A few parts in 10 billion trillion equates to a bacterium-sized drop of water added to an Olympic-sized swimming pool. Feel free to insert your own joke about homeopathy here.
Our lethal BB would be traveling almost as fast as the fastest-moving particles ever detected. I’m thinking here about the Oh-my-God particle, which was (probably) a proton that hit Earth’s atmosphere at 0.9999999999999999999999951 c. The Oh-my-God particle is faster by a hair. But our BB is still traveling a ludicrous speeds. Light is fast. A beam of light could circle the Earth in 133 milliseconds, which, if you look at reaction-time data, is about half the time it takes a human being to detect a stimulus and for the nerve impulse to travel down their arm and make a muscle contract. Very few physical objects can hold their own against light. But our BB could. If you raced our BB and a beam of light head-to-head, after a thousand years, the BB would only be lagging 3 millimeters behind, which is about the diameter of a peppercorn, which is ridiculous.
But the question here is not “How fast would a BB have to be traveling to have the same kinetic energy as a humanity-ending asteroid.” The question is “Could you actually use a hyper-velocity BB to kill the human race,” which is much more interesting and complicated.
First of all, the BB would have an insane amount of kinetic energy. E = m * c^2, of course, and from that, we know that our BB’s kinetic energy (its energy alone) would have a mass of 1.428 * 10^7 kilograms, which is about the mass of five Boeing 747 jet airliners. I would have to be a very gifted physicist to tell you what happens when you’ve got atoms with that kind of energy, but I suspect that there would be very weird quantum effects (aren’t there always?) which would conspire to slow the BB down. Because of quantum randomness, I imagine the BB would constantly be emitting high-energy gamma rays, which would decay into electron-positron and proton-antiproton pairs. Which is to say that our BB would be moving so fast that, rather than leaving behind a wake of Cherenkov radiation, it would leave behind a wake of actual physical matter, conjured seemingly from the ether by the conversion of its kinetic energy.
As for what would happen when the BB actually hit the Earth, that’s beyond my power to calculate, on account of I don’t have access to a fucking supercomputer. But we can assume that the BB would pass straight through the Earth with no physical impact: all of its interactions with our planet would probably be on the level of ultra-high-energy particle physics. And from that, we can estimate its effects.
The BB would cut a cylindrical path 4.5 millimeters across and 1 earth diameter long. If it deposited all of its kinetic energy along this track, it would raise the temperature of the rock by 10^15 Kelvin, which would make it one million times hotter than a supernova, which would most certainly be more than enough to kill all of us and vaporize a significant fraction of the Earth.
But the BB would only spend 44 milliseconds passing through the Earth, and somehow I doubt that regular matter would stop it entirely. Let’s assume instead that only one tenth of its energy got deposited in its track. We’re still talking a temperature a hundred thousand times that in the center of a supernova, which is ridiculous and would, once again, kill us all and peel the skin off the planet.
What if the BB only loses one one hundredth of its energy as it passes through the Earth? Same result: the Earth is replaced by a ball of radioactive lava.
But if, because of its ridiculous speed, it only loses one one millionth of its kinetic energy interacting with Earth, it still heats its needle-thin track to 3 billion kelvin, which is hot enough to fuse Earth’s silicon into iron and produce a violent explosion that would spawn earthquakes and firestorms and might, in spite of the energy losses, kill us all anyway.
But when you consider how much energy even the mighty Oh-my-God particle (which was, let me remind you, moving so fast that light was having trouble staying ahead of it) deposited just by hitting the atmosphere, I’d say the BB would lose quite a bit more than one one-millionth of its kinetic energy on impact. And I’d say that that kinetic energy would be spread over a fairly wide area. I’m thinking it would leave behind a column of hydrogen-bomb-temperature fusion plasma in the atmosphere, then hit the crust and fan out within an ice-cream-cone-shaped volume of the mantle. The nastiness of the results depend entirely on how big an ice-cream cone we’re talking, but it’s likely to be fairly narrow and fairly long, so we’re probably looking at a near-supernova-temperature column of fusing rock plasma with a length measured in kilometers. The explosion would be worse than anything the Earth has ever seen and would, yes, almost certainly kill all of us. If the immediate radiation didn’t get us, then the explosion would expose the mantle and lift enough dust to darken the sky for years.
You know, at the start of this, I thought I had an idea I could pitch to Daisy Outdoor Products. Now that I think about it, I think I’ll put the proposal in a drawer and forget about it.