Short: Zeno from Coast to Coast

There’s an old joke. A mathematician and an engineer die in a car crash and go up to Heaven. They meet Saint Peter up there, and he leads them to a mile-long hallway with a big pearly door with a gold handle at the other end. Saint Peter says “Both of you have done good things and evil things in your lives. To test if you’re inherently good-natured and worthy of getting into Heaven, I’m going to put you through a challenge. Whenever I sound my trumpet, you’re allowed to walk half the remaining distance to the door. If you get to the door, you can open it and go to Heaven. But if you move more than I tell you to, or try to cheat, you’ll go to Hell.”

The two guys confer for a second. The mathematician says “It’s a trick. We’re going to Hell anyway: no matter how many times you divide the distance in half, it’ll never be zero. We’ll never actually reach the door.”

They don’t get a chance to say anything else: Saint Peter sounds his trumpet. Both guys walk half a mile, to the middle of the hallway. The mathematician is getting antsy. Saint Peter sounds his trumpet again, and they walk a quarter-mile. He sounds it again: they walk an eighth of a mile. Again: 330 feet. Again: 165 feet. By now, the mathematician’s shaking and sweating and turning beet-red. The engineer starts to say something to him, but the mathematician screams and starts running back the way he came. He doesn’t get ten paces before a hole opens beneath him and he falls into a pit of fire and brimstone.

After Saint Peter’s sounded his trumpet ten times, the engineer’s only a foot or so from the door. He turns back and looks at Peter and says “Is it all right if I reach out and turn the handle?” Saint Peter says “Of course.” The engineer opens the door. On the other side is Heaven, full of angels on clouds and such. Saint Peter says “Well, you opened the door. Go ahead and walk through.” The engineer walks through Saint Peter goes with him and says “One warning, though: all our rulers are warped, our T-squares are crooked, and our compasses are made out of rubber.” The engineer thinks for a second and says “I see. Look, is it still possible for me to go to Hell? Have they got an opening?”

…

That joke is one of the many warped forms of Zeno’s Paradox. It should be impossible to reach any destination, since first you have to cover half the distance, and then you have to cover a quarter of the distance, and then an eighth, and so on, and you never actually arrive. Well, I’m going to take that literally. I’m going to imagine traveling from the west coast of the United States (specifically, from The Riptide bar and honky-tonk on Taraval and 47th, in San Francisco, California) to the east coast (specifically, to the parking lot of the Holiday Inn in Wrightsville Beach, North Carolina). That’s a distance of 4,014 kilometers. I’m going to imagine covering that distance in the same fashion as in that joke: covering half the remaining distance each time. I’m going to move once every ten seconds.

1/2

I travel 2,007 kilometers in 10 seconds. I’m traveling at about 201 kilometers per second, meaning I carve a ram-heated plasma trail through the air, setting a swath of the United States on fire as I travel. If I were human (which I’m clearly not, if I’m doing this kinda shit), I’d be pulped by the acceleration required to follow the curvature of the Earth at these speeds. I stop not too far from Dodge City, Kansas. Good thing I already have to move again, because I’m pretty sure there’s an angry mob gathering.

1/2 + 1/4 = 3/4

I travel 1,003.5 kilometers in 10 seconds. I’m still setting fire to every object I pass, blinding bystanders, and knocking down trees and buildings with my shockwave. I’ve broken every window in Wichita, Kansas and Springfield, Missouri. I pause, momentarily, in the westernmost tip of Kentucky. If I were human, I’d still have been pulped by the acceleration.

1/2 + 1/4 + 1/8 = 7/8

I travel just under 502 kilometers in 10 seconds. I’m moving as fast as some of the fastest shooting stars, but at ground level. I’m wrecking everything I pass in a way the Chelyabinsk meteorite could only aspire to. My shockwave and fireball cause burns, injuries, and structural damage in Knoxville and Nashville. I scream over the Blue Ridge Mountains and stop just short of Asheville, North Carolina, which I’ve been to, and which is a nice town.

1/2 + 1/4 + 1/8 + 1/16 = 5/16

I travel 251 kilometers this jump. The centripetal acceleration required to follow the curve of the Earth is an almost-survivable 10 gees. I’m still meteoric, though, blasting through Asheville, narrowly missing Gastonia, and ruining the lives of everyone in south Charlotte. I stop not far past Charlotte. I didn’t plan it this way, but I’ve visited my hometown on my accidental rampage.

31/32

I travel 125 kilometers. Still fast enough to cause a hell of a shockwave and probably a bit of a plasma trail, but now I’m only going as fast as a high-velocity railgun projectile. To stick to the Earth, I have to accelerate downwards at 2.5 gees, which is very much survivable, especially for only ten seconds. I stop not far north of Lumberton, North Carolina, which I’ve never visited, but I’ve heard is another nice little town. We’ve got a lot of those in North Carolina, actually. It’s kinda cool.

63/64

I travel roughly 63 kilometers in 10 seconds. I’m moving at the speed of a very ambitious bullet. My centripetal acceleration is just over half a gee. I stop in a track of farmland with not much around me.

127/128

This jump covers 31 kilometers at the speed of a sniper-rifle bullet. My acceleration is 0.15 gees, which is the kind of acceleration people experience in cars on a regular basis. I’m not far outside the city limits of Wilmington, NC, in the woods between a church and a water treatment plant (according to Google Earth.)

255/256

I’m still moving like a bullet, covering 15.7 kilometers in 10 seconds. I stop over a river in Wilmington’s northern outskirts, near a drawbridge and what looks like an oil-tank complex.

511/512

I travel 7,839 meters in 10 seconds, which brings me down to the muzzle velocity of an ordinary handgun. I come to a stop on the roof of a Home Depot in Wilmington.

1023/1024

This jump is 3,920 meters at around Mach 1. I’m still probably bursting eardrums wherever I pass, but those newly-deaf people should count themselves lucky: there’s a lot of people burned to ash on the West Coast.

2047/2048

1,960 meters at the speed of an ordinary airplane. I stop in a marsh on the bank of the estuary that separates Wrightsville Beach from Wilmington. People are no longer being injured by my passage, but they’re probably pretty horrified to see a human being moving this fast. Plus, it’s been over a minute and 45 seconds since my accidental rampage began. That’s probably fast enough for people to post pictures of my path of destruction on Twitter.

4095/4096

I only travel 979.911 meters this time at an almost-sensible 219 mph (352 km/h). I manage to cross the estuary, although I’m still in the damn marsh, not yet to Wrightsville Beach. I stop near what, on Google Maps (on November 27th, 2016, anyway) looks like a horrifying half-mile long translucent river-worm:

Some may say it’s just a boat wake. I say they’re just blinding themselves to the truth that Big Google Data Brother Government Conspiracy (LLC) is trying to hide from the people. That’s what you’re supposed to say when you find weird shit in Google Earth, right?

8191/8192

I’m still going way over the posted speed limit as I cover the next 489.956 meters. Still stuck in this damned marsh, too.

16383/16384

A 244.978-meter jump at a sensible 54 mph (88 km/h). I’m starting to feel a little like the mathematician in the joke: all of a sudden, things are moving painfully slow.At least I’m out of the stupid marsh, and passing through a fairly pleasant-looking seaside housing development.

32767/32768

122.489 meters covered at a very pleasant 27 mph (44 kph). I’m almost within the ordinary residential speed limit! I’m only a block from the Holiday Inn, so tantalizingly close. I must persevere! I wanna be the engineer in the joke, not the mathematician! Hell sounds terrible!

65535/65536

A nice neat number. 2^16. 10000000000000000, in binary. I travel 61.244 meters. Less than the length of any sort of football field. I’m moving at 6.1 meters per second. A decent sprinter could manage that. Usain Bolt could most certainly manage that.

131071/131072

30.522 meters this time. I’m jogging across the parking lot, almost a literal stone’s throw from the Atlantic.

262143/262144

15.311 meters covered this jump, and 15.311 left to go. I can look into the windows of the Holiday Inn, and see all the employees on their phones reading about the carnage in California.

524287/524288

7.656 meters. A man passing me in the Holiday Inn parking lot wonders why I’m walking so damn slow.

1048575/1048576

3.828 meters. I’m walking at less than 1 mile per hour, which feels really weird when I do it in real life. I’m starting to regret many decisions. Plus, people are starting to get suspicious of me.

2097151/2097152

1.914 meters left to go. If I fell flat on my face (which is starting to seem like a good idea…), my head would push the front door open.

1 – 2^(-22)

Less than a meter to go. As I stand almost within arm’s reach of the hotel’s front door, people are giving me looks usually reserved for those who start talking about lizard people at bus stops.

1 – 2^(-23)

Half a meter. Finally–finally–I can reach out and push the door open. And thus, my bizarre trip comes to an end. I’ve covered 4,013,716.5 meters in about 3 minutes and 45 seconds. I’ve killed many, many people and injured countless others. There will be an investigation. Even though this isn’t exactly the sort of thing the FBI is used to, they’ll probably do their damndest to figure out just who or what set the west coast on fire, smashed millions of windows, and made a sonic boom over North Carolina. There’s probably enough security camera evidence to find me. Until then, I’ll just catch some sun on Wrightsville Beach.

Supersonic Toyota? (Cars, Part 2)

A while ago, I wrote a post that examined, in much greater and (slightly) more accurate detail what speeds my 2007 Toyota Yaris, with its stock drivetrain, could manage under different conditions. This post is all about Earth at sea level, which has gotta be the most boring place for a space enthusiast. Earth at sea level is what rockets are built to get away from, right? But I can make things interesting again by getting rid of the whole “sensible stock drivetrain” thing.

But first, since it’s been quite a while, a refresher: My Yaris looks like this:

Its stock four-cylinder engine produces about 100 horsepower and about 100 foot-pounds of torque. My drivetrain has the following gear ratios: 1st: 2.874, 2nd: 1.552, 3rd: 1.000, 4th: 0.700, torque converter: 1.950, differential: 4.237. The drag coefficient is 0.29 and the cross-sectional area is 1.96 square meters. The wheel radius is 14 inches. I’m totally writing all this down for your information, and not so I can be lazy and not have to refer back to the previous post to get the numbers later.

Anyway…let’s start dropping different engines into my car. In some cases, I’m going to leave the drivetrain the same. In other cases, either out of curiosity or for practical reasons (a rarity around here), I’ll consider a different drivetrain. As you guys know by now, if I’m gonna do something, I’m gonna overdo it. But for a change, I’m going to shoot low to start with. I’m going to consider a motor that’s actually less powerful than my actual one.

An Electric Go-Kart Motor

There are people out there who do really high-quality gas-to-electric conversions. I don’t remember where I saw it, but there was one blog-type site that actually detailed converting a similar Toyota to mine to electric power. That conversion involved a large number of batteries and a lot of careful engineering. Me? I’m just slapping this random go-kart motor into it and sticking a couple car batteries in the trunk.

The motor in question produces up to 4 newton-meters (2.95 foot-pounds). That’s not a lot. That’s equivalent to resting the lightest dumbbell they sell at Walmart on the end of a ruler. That is to say, if you glued one end of a ruler to the shaft of this motor and the other end to a table, the motor might not be able to break the ruler.

But I’m feeling optimistic, so let’s do the math anyway. In 4th gear (which gives maximum wheel speed), that 4 newton-meters of torque becomes 4 * 1.950 * 4.237 * 0.700 = 21 Newton-meters. Divide that by the 14-inch radius of my wheels, and the force applied at maximum wheel-speed is 59.060 Newtons. Plug that into the reverse drag equation from the previous post, and you get 12.76 m/s (28.55 mph, 45.95 km/h). That’s actually not too shabby, considering my car probably weighs a good ten times as much as a go-kart and has at least twice the cross-sectional area.

For the electrically-inclined, if I was using ordinary 12 volt batteries, I’d need to assemble them in series strings of 5, to meet the 48 volts required by the motor and overcome losses and varying battery voltages. One of these strings could supply the necessary current of 36 amps to drive the motor at maximum speed and maximum torque. Ordinary car batteries would provide between one and two hours’ drive-time per 5-battery string. That’s actually not too bad. I couldn’t ever take my go-kart Yaris on the highway, but as a runabout, it might work.

The Land Speed Record that Will Never Be Broken

As I’ve noted more than once, human beings like to make things go really fast. Part of me thinks that’s because we’re hunter-gatherers by nature, and somewhere deep in our limbic systems, we think that if we can make it to Mach 3, we’ll finally catch that damned antelope. The other part of me thinks we like it because it’s AWESOME!

As of this writing, the world land speed record stands at a hair over 763 miles per hour (almost 1228 km/h, or 341 m/s). The record is held by Andy Green and his ThrustSSC. This is the first land speed record to break the sound barrier. I must also note that when I looked at that page, the entry at the top of the “Related Records” section was the world’s thinnest latex condom. I’m starting to wonder what exactly Andy Green gets up to when he’s not cruising through Nevada at Mach 1.002…

But that’s just the record for the fastest land vehicle with a person inside it. The ultimate land speed record, as far as I can tell, is held by a multi-stage rocket sled at Holloman Air Force Base, which deals with creepy secretive things. Their rocket sled reached Mach 8.47, or 6,453 mph (or 10,385 km/h, or 2,885 m/s; why are there always so many damn units…). They’re not saying why, exactly, they’re accelerating a rocket sled to railgun velocities, but they’ve done it.

There’s no theoretical reason a human being couldn’t go that fast. (There are lots of practical reasons, but I’ve never let that stop me.) In fact, there’s no theoretical reason a land vehicle couldn’t go much faster. Technically speaking, if we ignore aerodynamic effects (which we theoretical types always do, which is why there are engineers to explain to us that astronauts don’t like burning up in the atmosphere), the fastest a land vehicle could ever go is 7.91 kilometers per second. That’s orbital speed at sea level. It’s Mach 23. This is the speed at which the centrifugal acceleration from traveling around the circular earth exactly balances the acceleration due to gravity. To put it another way, this is the speed where your vehicle becomes weightless, and if you go any faster, you’re going to leave the ground.

7.91 km/s is fast. Here’s a good way to understand just how fast it is. Say you’ve got a really good reaction time (around 100 milliseconds; let’s say you’ve had a lot of coffee). If you were trying to time this ultimate land-speeder on a 1,000-kilometer track (about 10 football fields end-to-end) with a stopwatch, the speedy bugger would have traveled from the beginning to the end of the track by the time your brain noticed that it had entered the track, processed the fact, and sent the signal to your finger to press the button on the stopwatch. It wouldn’t matter, of course, because you’d be obliterated by a superheated shockwave a moment later.

But even 7.91 kilometers per second isn’t the ultimate limit on land speed. As a matter of fact, if you have a vehicle that can reach that speed anyway, it’s going to have to have some aerodynamic surfaces on it to keep it from lifting off the ground and turning into the world’s fastest plane. But, while we’re adding downward thrust (in the form of aerodynamic lift, or perhaps I should say anti-lift), why not go all the way? Why not put some rockets on this thing and make it stay on the ground?

The fastest a human being could reasonably expect to travel across flat ground and survive is 23.7 kilometers per second. Before I get into explaining just how horrifically fast that is, and why you can’t go faster than that without killing the pilot, I want to paint a picture of the vehicle we’re talking about.

In all likelihood, it looks more like a plane than a land vehicle. It’s got some sort of massive engine on the back that burns sand to glass behind it. It’s got enormous wings to keep it from bounding into the stratosphere. It’s got rocket motors mounted on the tops of those wings. And we’re not talking wimpy JATO motors. We’re talking ballistic-missile-grade motors. Motors powerful enough that, if you just strapped a human to them, the human would have a hard time staying conscious through the acceleration.

The cockpit’s weird, too. It’s a sort of pendulum, with a reclined seat aligned along the axis of rotation. Because of the pendulum arrangement, the seat rotates so that the occupant always feels the acceleration as vertical. You could be forgiven for thinking this is some kind of Edgar Allan Poe torture device.

I’ll explain all that in a minute. But for right now, I want to convey to you how fast 23.7 km/s is. It’s the speed of extinction-triggering asteroids. It’s Mach freakin’ 71. It’s twice as fast as the crew of Apollo 10 (the holders of the ultimate human speed record) were moving on their way back to Earth. It’s faster than both the Voyager probes and New Horizons. Matter of fact, there are only two human-constructed objects that have ever gone faster than this: the amazing Galileo atmospheric probe, which dropped into Jupiter’s atmosphere so fast that all the speeding bullets in the world momentarily blushed (47.8 km/s, for those who don’t like overwrought metaphors), and the equally amazing Helios 2 probe, which holds both the record for the fastest human-built object and the human object that’s gotten closest to the sun (at perihelion, it was moving at 70.2 km/s; hopefully, NASA won’t can Solar Probe Plus and we can break that record).

23.7 km/s is one of those speeds that just doesn’t fit very well into the human mind, unless it’s the kind of human mind that’s accustomed to particle accelerators or railguns, and frankly, those minds are a little scary. At this speed, our peculiar death-trap vehicle could circumnavigate the Earth in 28 minutes and 9 seconds. It could travel from New York to Los Angeles in 2 minutes and 47 seconds.

“But hell,” I can hear you saying, “we’ve already got a ridiculous impractical land-speed vehicle. Why not crank it up all the way? Why not go as fast as Helios 2? Or faster!” The problem is that I specified a vehicle being driven (or at least occupied) by a human being. Before I explain, here’s a video of a person making a very funny face.

That’s a pilot in training being subjected to 9 gees in a centrifuge. You’ll noticed that he briefly aged about 60 years and then passed out. But he was being trained for practical stuff (that is, not blacking out when making a high-speed turn in an airplane). That’s boring. And, more relevant to our speed record, he was almost certainly experiencing gees from head-to-foot. Humans don’t tolerate that very well. The problem is that human beings have blood. (Isn’t it always?) When gee forces get very high, it takes a lot of pressure to pump blood to levels above the heart. Unfortunately, when you’re dealing with vertical gees, the brain is well above the heart, and all the blood essentially falls out of the brain and into the legs. (There are some ways to compensate for that, like with the weird breathing technique the trainee was doing and the pressure-compensating suits most high-gee pilots wear, but there are limits).

But even if the gees were from back to front (that is, you’re accelerating in the direction of your nose), 9 gees would probably still be the upper limit. Because, even lying down, that acceleration is going to make the blood want to pool below the heart. It’s going to flood into places where you don’t really need it like your buttocks, your calves, and the back of your head. In fact, at 9 gees, you run a pretty good risk of rupturing blood vessels in the back of your brain from the pressure. But human beings can tolerate 9 forward gees for a few seconds, so we’ll pretend they can tolerate it for the 2 minutes and 47 seconds it takes to blaze from New York to LA.

And that’s why we’ve got the weird pendulum recliner in our hypothetical ultra-hypersonic land vehicle: at 23.7 kilometers per second, the vehicle’s going to have to accelerate towards the ground at 9 gees just to keep from flying off into space. The pilot’s seat will be upside-down, relative to the ground, with the pilot all smashed down and funny-looking for the duration of the flight. If we try to go any faster, our pilot isn’t going to be able to survive the acceleration for more than a few seconds at a time. According to this nifty graph

(Source.)

whose source material I unfortunately couldn’t verify, a human being can’t tolerate 10 gees for more than 10 seconds. A human can tolerate 20 gees for 1 second (this I know to be true, because lunatic rocket-sled pilot John Stapp did it; actually, he pulled 25 gees for a full second, and in spite of all his insane rocket-sled stunts, lived to be 89). And human beings have been known to survive 30 gees or more (up to about 100 gees) for very brief periods in car crashes.

But trust me, the weird French organization that certifies land speed records (and air speed records, and altitude records) probably isn’t going to be very impressed by your traveling 50 km/s for a tenth of a second. If you want to go that fast long enough to actually get anywhere, you’re limited to 8 or 9 gees, and even then, you’d damn well better make sure your life insurance is up to date.

So, unless you use weird technologies like liquid respiration (in which you breathe oxygenated liquid fluorocarbons instead of air, and which is a real thing that actually exists and is sometimes used for hospital patients with burned lungs) and those creepy full-body gee-tanks from Event Horizon, the 23.7 km/s land speed record can never be broken. Partly because of the gee-forces involved, but mainly because trying to go that fast on land is absolutely, certifiably insane.

Tune in next time, where I get all gory and try to imagine what would happen to a human body exerted to much larger gee forces.

Supersonic Submarine

I seem to be obsessed with submarines, which isn’t something I realized until now. There’s nothing worse than a stealth obsession.

Anyway: sound has a speed. That speed depends on the properties of the medium. The speed of sound in air is about 330 meters per second. The speed of sound in olive oil is 1430 meters per second (Yes, somebody measured that, and here’s the proof, along with some other handy tables of speeds of sound in other materials). The speed of sound in aluminum is 6320 m/s. The speed of sound in beryllium is an amazing 12,900 m/s, which is not only faster than the International Space Station’s orbital velocity, it’s actually faster than Earth escape velocity.

The speed of sound in seawater is a much tamer 1500 m/s (the exact speed depends on depth (meaning pressure), temperature, and salinity). That got me thinking that, since I’ve abandoned the submarine car in favor of an actual submarine, why not make it a supersonic submarine?

There’s nothing in the laws of physics to stop me. There’s no physical reason that makes it impossible to move through water faster than the speed of sound in water. There are plenty of engineering reasons, but we’ll get to those in a second.

The interesting thing about moving supersonically in water is that water isn’t a gas. Air isn’t very dense, it’s compressible, and it doesn’t have many phase transitions readily available. It can liquefy if you compress it while keeping it cool, and it can turn to plasma if you compress it and let it heat up. But when you’re talking about supersonic vehicles, the air heats up rather than cooling down. It heats up a lot. The air around re-entering spacecraft turns into plasma.

Water, on the other hand, is much denser (pure water is about 1,000 kilograms per cubic meter), and compared to air, is almost incompressible. Water is about five orders of magnitude less compressible than air. This means that a whole slew of new phenomena happen in supersonic submarines that don’t happen in supersonic aircraft. The coolest one is cavitation.

Cavitation is what happens when, for one reason or another, the pressure on a volume of water drops below that water’s vapor pressure, or when something moves through the water so fast that the cavity in the water doesn’t have time to close around the object. There are all sorts of cool videos of cavitation on the Internet, but I think this is my favorite:

Ain’t that beautiful? Many thanks to The Slow Mo Guys and Smarter Every Day for filming that, and for doing exactly what I would have done if I had access to one of those slow-motion cameras.

Notice the large cavity that opens behind the bullet as it travels. The spherical cavity around the gun’s muzzle is from the blast of hot, escaping gas, but the sort of sausage-shaped bubble attached to the bullet is pure cavitation. The bullet slams the water aside so hard that, even though water is usually very good at closing voids within itself, it has no choice but to stand aside for a fraction of a second. For the brief period that it exists, that cavity is full of a little water vapor that evaporated from the surface and not much else, and as soon as the moving water has deposited its inertia in the stationary water around it, pressure wins out and makes the bubble collapse again.

But a cavitation bubble isn’t the same thing as a sonic boom. The bullet in that video was fired from a revolver. Since I don’t know the make of the revolver or what kind of ammunition it was using, I don’t know the muzzle velocity, but if we assume it was in the same class as a Ruger firing .357 Magnums, then the muzzle velocity would have been around 450 meters per second. Not faster than the speed of sound in water. Barely faster than the speed of sound in air.

Either way, we know that our supersonic submarine would cut quite a large hole in the water as it flew. (Flew? That doesn’t sound right. What is the right verb for a submarine’s movement? Somebody let me know. That’s gonna bother me now). It would also, true to acoustics, generate a sonic boom. I would guess that this sonic boom would be more than enough to rupture the eardrums of unlucky divers who happened to get in its way, and that the drop in pressure after the shock would probably create a whole swarm of smaller cavitation bubbles in its wake. And because the water that evaporated from the surface of the cavity would be moving roughly in the same direction as the cavity (relative to the submarine), the submarine would likely create a second, much slower-moving sonic boom in the water vapor. After the submarine passed, the cavity would expand to a maximum size, then slam closed, possibly heating the gases inside enough to glow. This is called sonoluminescence, and is very impressive:

After the collapse, you’d have a soup of very hot bubbles and very hot water vibrating and rising to the surface. The water would be hot from the collapse of the cavity. Here’s about what our supersonic sub would look like:

And, from a practical perspective, it would be hot for another reason. To break the speed of sound in water, you’d need the engine power of 4 Saturn V moon-rockets.

Yes, really. This comes from the basic drag formula I’ve been using all along:

drag force = (1/2) * (density of medium) * (velocity of object)^2 * (drag coefficient (depends on shape and texture of object)) * (projected or cross-sectional area of the object)

I have no idea where we’re going to get a rocket four times as powerful as a Saturn V. I guess we could just make the end of the submarine a parabolic reflector and drop antimatter out the back and ride the blast of steam, but I hear people get pretty upset if you go dumping antimatter in the ocean. Especially if they happen to be swimming behind you.

But that’s the least of our worries. At 1500 meters per second, the front of the submarine would be experiencing pressures ten times greater than at the bottom of the Mariana Trench. Not unsurvivable, but between the pressure of the water against the front of the hull and the cavitation going on around the back of the hull, the whole thing’s going to need to be a pressure vessel. That’s going to be one heavy submarine. While we’re pretending that a submarine-sized craft could produce 141 million Newtons of thrust for an extended period, why not just turn the bastard into a rocket? Besides, I’m afraid that if I tooled around underwater making watery sonic booms, I might upset an octopus, and I have a deep and inexplicable affection for octopuses.

But before we stop doing weird things underwater, there’s a question that demands to be answered: if our supersonic submarine would need four times the thrust of a Saturn V to travel through the water, how fast would the Saturn V itself be able to go underwater. Well, input some reasonable values into the drag equation, set the drag equation equal to 35 million newtons (the Saturn V’s first-stage thrust), and we have:

41.2 meters per second

or

92 miles per hour

or

148 kilometers per hour

The Saturn V is one of the most powerful rockets ever built. And, under ideal conditions, it could manage 92 miles an hour underwater. I have driven my car faster than that. A good baseball pitcher or cricket bowler can throw faster than that. I guess the people at NASA weren’t planning for the possibility that the space between the Earth and Moon might inexplicably be filled with seawater. The fools.

But, although 92 miles an hour is not a very impressive speed, especially by rocket standards, you have to admit, it’d be one hell of a sight to behold:

Now that‘s a fucking torpedo!

Fun with logarithms 2: A race between a snail and a beam of light.

In my last post, I mentioned how cool logarithmic scales were. A logarithmic scale, for instance, takes two numbers that differ by a factor of one trillion (say 0.001 and 1,000,000,000, which differ by 999,999,999.999) and reduce their difference to a much more manageable number: 12.

Logarithmic scales are handy in the universe we happen to live in, because, as you might have noticed, this universe contains a lot of very tiny things and a lot of ridiculously large things. It contains both bacteria and galaxies, which differ in scale by a factor of 950,000,000,000,000,000,000,000,000. But today, we’re not talking about distances. Today, we’re talking about speeds. Some things in the universe move very quickly. Others move very slowly. Continents drift together (and apart) at speeds of around 1.585×10^-9 meters per second, which is less than the diameter of a DNA helix every second. Light, on the other hand (which, as far as we know, moves as fast as it is physically possible to move) travels at 299,792,458 m/s. This is the perfect place for a logarithmic scale.

Therefore, I present to you, the Bolt, a logarithmic scale named in honor of Usain Bolt, who is the fastest Jamaican in the world. He’s also the fastest human in the world. To get a measurement of an object’s speed in Bolts, divide that speed by our reference speed (1 meter per second, which is about walking speed), and take the base-10 logarithm of the result.

But actually, the Bolt is kind of a large unit. Ironically, we’ve gone from too large a scale (0.0000000015 m/s to 299,792,458 m/s) to too narrow a scale. So let’s take inspiration from the decibel, and multiply the result of our logarithmic calculation by 100, which gives us a measurement in centiBolts (cBo).

Let’s work an example before I get to the list, which is the fun part. The land speed record for a garden snail (which record, apparently, you can only challenge at the World Snail Racing Championships in Congham, England) is 0.002752 meters per second. To get the speed in centiBolts, we calculate 100 * log10((0.002752 m/s) / (1 m/s)), which works out to -256.035 centiBolts. Now that you know how the math is done, let’s compute the centiBolt rating for some ridiculously low and ridiculously high speeds!

With a helium-neon laser and a Michelson interferometer, you could (probably) measure a change in distance of 300 nanometers over the course of an hour, which is 0.00000000009 m/s or -1005.61 centiBolts.

Continents drift apart (or together) at a speed of about 3 centimeters per year, or -902.2 cBo.

Because it loses orbital energy by stretching the Earth (making tides) and slowing Earth’s rotation, the Moon’s orbit is very gradually getting larger. The moon, therefore is receding at about 3.8 cm/year, or -891.9 cBo.

Human hair grows at about 15 cm/year, or -832.3 cBo.

Bamboo grows at a rate of about 14 microns per second (meaning it can grow several feet in a day), giving it a speed of -485.4 cBo.

Jakobshavn Isbræ glacier, in Greenland, reaches a maximum speed of 12,600 meters per year, which comes out to -339.8 cBo.

As I mentioned before, the land speed record for a garden snail is 0.00275 m/s or -256.07 cBo.

A wind speed of 1 mile per hour (MPH) would just barely be detectable by the drift of smoke. It wouldn’t even move weather vanes. It gets a speed rating of -34.97 cBo.

1 m/s, being our reference point, gets a rating of 0 cBo.

Usain Bolt, for whom we named this unit, ran at an average speed of 10.438 m/s when he broke the 100 m world record. He was running at 101.9 cBo…

…but his maximum speed was 12.42 m/s, or 109.41 cBo.

In the United Sates, most non-residential roads have a speed limit of 45 MPH. In my experience, no matter the speed limit, people usually drive around 50 MPH, which is 134.9 cBo.

I once drove my car at 105 MPH (don’t tell anybody). I was moving at 167.2 cBo.

The Bugatti Veyron, the world’s fastest street-legal production car, can get up to 267.856 MPH (431.072 km/h), or 207.825 cBo.

The fastest wheel-driven car on record (as of May 2014) got up to 403.10 MPH (648.73 km/h), which is a terrifying 225.58 cBo.

The land speed record (again, as of May 2014, set by the ThrustSSC, the first land vehicle to break the sound barrier) is 760.343 MPH (1223.657 km/h), or 253.1356 cBo.

The F-22 raptor can supposedly reach Mach 2 (the actual top speed is probably classified), which is 277.093 cBo.

The awesome-looking (and sadly retired) SR-71 Blackbird could manage 2,193.2 MPH, or 299.1 cBo.

The even more awesome rocket-powered X-15 could do 4,160 MPH (6,695 km/h), or 326.9 cBo.

At this point, we’re moving from the realm of really fast aircraft to the realm of really slow spacecraft.

The International Space Station orbits at 7.656 km/s, which is 388.4 cBo. It moves so fast, that it only takes 14 milliseconds to travel its own length. Curious about what 14 milliseconds sounds like? So was I. It sounds like this: https://soundcloud.com/hobo-sullivan/the-iss-passes. Which is just barely long enough for my ears to recognize as an actual sound.

The human speed record (relative to the Earth) was set by the crew of Apollo 10, who reached 24,791 mph, or 404.5 cBo. At these speeds, which are frankly quite ridiculous, the spacecraft was covering 1 kilometer every 90 milliseconds. To give you an idea how fast that is, here’s a 90-millisecond tone: https://soundcloud.com/hobo-sullivan/apollo-10-travels-1-kilometer.

The light gas gun at Sandia Labs (which, it has been noted, is essentially a high-tech BB gun) can fire projectiles at a silly 36,000 MPH, or 420.7 cBo. At these speeds, a small plastic projectile can make a hand-sized crater in a block of aluminum.

The Helios 2 solar probe holds the record for the fastest human-made object. When it swung by the sun (getting slightly closer to the Sun than Mercury), it hit 157,100 MPH, or 484.7 cBo.

But, as all physics geeks know, 157,100 MPH isn’t all that fast. It’s only 0.00023 times the speed of light, or 0.00023 c (as science-fiction authors put it). There are much faster things in this crazy-ass universe.

Like, for instance, the brightest component of a lightning bolt, the return stroke, which, according to some measurements, reaches 220,000,000 MPH, or 0.3281 c, or 799.3 cBo.

The Large Hadron Collider can accelerate protons much faster. It can get them up to 670,615,282 MPH, 0.999999991 c, or 847.7 cBo.

Which is not far from the maximum speed any object can attain, the speed of light itself: 670,616,629 MPH, 299,792,458 meters per second, or 847.7 cBo.

Sorry, Mr. Bolt. You’re going to have to train harder. You’re still running 29,979,246 times slower than you could be. Are you even trying? Come on!