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A Toyota on Mars (Cars, Part 1)

I’ve said this before: I drive a 2007 Toyota Yaris. It’s a tiny economy car that looks like this:

(Image from RageGarage.net)

The 2007 Yaris has a standard Toyota 4-cylinder engine that can produce about 100 horsepower (74.570 kW) and 100 foot-pounds of torque (135.6 newton-metres). A little leprechaun told me that my particular Yaris can reach 110 mph (177 km/h) for short periods, although the leprechaun was shitting his pants the entire time.

A long time ago, [I computed how fast my Yaris could theoretically go]. But that was before I discovered Motor Trend’s awesome Roadkill YouTube series. Binge-watching that show led to a brief obsession with cars, engines, and drivetrains. There’s something very compelling about watching two men with the skills of veteran mechanics but maturity somewhere around the six-year-old level (they’re half a notch above me). And because of that brief obsession, I learned enough to re-do some of the calculations from my previous post, and say with more authority just how fast my Yaris can go.

Let’s start out with the boring case of an ordinary Yaris with an ordinary Yaris engine driving on an ordinary road in an ordinary Earth atmosphere. As I said, the Yaris can produce 100 HP and 100 ft-lbs of torque. But that’s not what reaches the wheels. What reaches the wheels depends on the drivetrain.

I spent an unholy amount of time trying to figure out just what was in a Yaris drivetrain. I saw some diagrams that made me whimper. But here’s the basics: the Yaris, like most front-wheel drive automatic-transmission cars, transmits power from the engine to the transaxle, which is a weird and complicated hybrid of transmission, differential, and axle. Being a four-speed, my transmission has the following four gear ratios: 1st = 2.847, 2nd = 1.552, 3rd = 1.000, 4th = 0.700. (If you don’t know: a gear ratio is [radius of the gear receiving the power] / [radius of the gear sending the power]. Gear ratio determines how fast the driven gear (that is, gear 2, the one being pushed around) turns relative to the drive gear. It also determines how much torque the driven gear can exert, for a given torque exerted by the drive gear. It sounds more complicated than it is. For simplicity’s sake: If a gear train has a gear ratio greater than 1, its output speed will be lower than its input speed, and its output torque will be higher than its input torque. For a gear ratio of 1, they remain unchanged. For a gear ratio less than one, its output speed will be higher than its input speed, but its output torque will be lower than its input torque.)

But as it turns out, there’s a scarily large number of gears in a modern drivetrain. And there’s other weird shit in there, too. On its way to the wheels, the engine’s power also has to pass through a torque converter. The torque converter transmits power from the engine to the transmission and also allows the transmission to change gears without physically disconnecting from the engine (which is how shifting works in a manual transmission). A torque converter is a bizarre-looking piece of machinery. It’s sort of an oil turbine with a clutch attached, and its operating principles confuse and frighten me. Here’s what it looks like:

(Image from dieselperformance.com)

Because of principles I don’t understand (It has something to do with the design of that impeller in the middle), a torque converter also has what amounts to a gear ratio. In my engine, the ratio is 1.950.

But there’s one last complication: the differential. A differential (for people who don’t know, like my two-months-ago self) takes power from one input shaft and sends it to two output shafts. It’s a beautifully elegant device, and probably one of the coolest mechanical devices ever invented. You see, most cars send power to their wheels via a single driveshaft. Trouble is, there are two wheels. You could just set up a few simple gears to make the driveshaft turn the wheels directly, but there’s a problem with that: cars need to turn once in a while. If they don’t, they rapidly stop being cars and start being scrap metal. But when a car turns, the inside wheel is closer to the center of the turning circle than the outside one. Because of how circular motion works, that means the outside wheel has to spin faster than the inside one to move around the circle. Without a differential, they have to spin at the same speed, meaning turning is going to be hard and you’re going to wear out your tires and your gears in a hurry. A differential allows the inside wheel to slow down and the outside wheel to spin up, all while transmitting the same amount of power. It’s really cool. And it looks cool, too:

(Image from topgear.uk.net)

(Am I the only one who finds metal gears really satisfying to look at?)

Anyway, differentials usually have a gear ratio different than 1.000. In the case of my Yaris, the ratio is 4.237.

So let’s say I’m in first gear. The engine produces 100 ft-lbs of torque. Passing through the torque converter converts that (so that’s why they call it that) into 195 ft-lbs, simultaneously reducing the rotation speed by a factor of 1.950. For reference, 195 ft-lbs of torque is what a bolt would feel if Clancy Brown was sitting on the end of a horizontal wrench 1 foot (30 cm) long. There’s an image for you. Passing through the transmissions first gear multiplies that torque by 2.847, for 555 ft-lbs of torque. (Equivalent to Clancy Brown, Keith David, and a small child all standing on the end of a foot-long wrench.) The differential multiplies the torque by 4.237 (and further reduces the rotation speed), for a final torque at the wheel-hubs of 2,352 ft-lbs (equivalent to hanging two of my car from the end of that one-foot wrench, or sitting Clancy Brown and Peter Dinklage at the end of a 10-foot wrench. This is a weird party…)

By this point, you’d be well within your rights to say “Why the hell are you babbling about gear ratios?” Believe it or not, there’s a reason. I need to know how much torque reaches the wheels to know how much drag force my car can resist when it’s in its highest gear (4th). That tells you, to much higher certainty, how fast my car can go.

In 4th gear, my car produces (100 * 1.950 * 0.700 * 4.237), or 578 ft-lbs of torque. I know from previous research that my car has a drag coefficient of about 0.29 and a cross-sectional area of 1.96 square meters. My wheels have a radius of 14 inches (36 cm), so, from the torque equation (which is beautifully simple), the force they exert on the road in 4th gear is: 495 pounds, or 2,204 Newtons. Now, unfortunately, I have to do some algebra with the drag-force equation:

2,204 Newtons = (1/2) * [density of air] * [speed]^2 * [drag coefficient] * [cross-sectional area]

Which gives my car’s maximum speed (at sea level on Earth) as 174 mph (281 km/h). As I made sure to point out in the previous post, my tires are only rated for 115 mph, so it would be unwise to test this.

I live in Charlotte, North Carolina, United States. Charlotte’s pretty close to sea level. What if I lived in Denver, Colorado, the famous mile-high city? The lower density of air at that altitude would allow me to reach 197 mph (317 km/h). Of course, the thinner air would also mean my engine would produce less power and less torque, but I’m ignoring those extra complications for the moment.

And what about on Mars? The atmosphere there is fifty times less dense than Earth’s (although it varies a lot). On Mars, I could break Mach 1 (well, I could break the speed equivalent to Mach 1 at sea level on Earth; sorry, people will yell at me if I don’t specify that). I could theoretically reach 1,393 mph (2,242 km/h). That’s almost Mach 2. I made sure to specify theoretically, because at that speed, I’m pretty sure my tires would fling themselves apart, the oil in my transmission and differential would flash-boil, and the gears would chew themselves into a very fine metal paste. And I would die.

Now, we’ve already established that a submarine car, while possible, isn’t terribly useful for most applications. But it’s Sublime Curiosity tradition now, so how fast could I drive on the seafloor? Well, if we provide compressed air for my engine, oxygen tanks for me, dive weights to keep the car from floating, reinforcement to keep the car from imploding, and paddle-wheel tires to let the car bite into the silty bottom, I could reach a whole 6.22 mph (10.01 km/h). On land, I can run faster than that, even as out-of-shape as I am. So I guess the submarine car is still dead.

But wait! What if I wasn’t cursed with this low-power (and pleasantly fuel-efficient) economy engine? How fast could I go then? For that, tune in to Part 2. That’s where the fun begins, and where I start slapping crazy shit like V12 Bugatti engines into my hatchback.

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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!

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Return of the submarine-car!

About a month ago, I tried to figure out how fast my 2007 Toyota Yaris would be able to drive underwater. I came up with 15 MPH (23 km/h, 6.52 m/s), which is respectable for a submerged economy car, but not that impressive for an underwater vehicle. Plus, as commenter Azure James correctly pointed out, propelling a car underwater isn’t as simple as having the horsepower to overcome drag: you have to make sure your drivetrain, from engine to gearbox to wheels, can generate enough torque to overcome the drag force.

As usual, I took this as an opportunity to imagine something ridiculous and draw a stupid picture. I present to you, the Hobo Sullivan Cruiser, the world’s most impractical underwater vehicle!

It would be about the size of a large van or a small delivery truck, and that’s mostly because it has to accommodate a spherical pressure hull. Because if I’m going to make an underwater car, I’m for damn sure going to make it able to drive into the Challenger Deep! A lot of the other stuff is self-explanatory. The outer hull has a half-torpedo shape (Or if you prefer, half-cigar shape. Or, if you’re really picky, half a Sears-Haack body.) That’s to minimize drag. I chose to power it with a fuel cell and electric motors because those allow a greater range of rotation speeds (which, as my commenter correctly pointed out, would limit my speed on Mars, since a normal gearbox can’t get the wheel speeds up high enough to propel an ordinary car at 300+ MPH). Plus, with electric motors, exhaust isn’t as serious a problem.

The power for the electric motors is provided by a methanol-oxygen fuel cell. I chose methanol for one reason: space. Hydrogen, even compressed and cooled to cryogenic temperatures, takes up a lot of room. If you look at the hydrogen-burning stages of the Saturn V, the tanks are mostly hydrogen with a little spheroid at the end for the oxygen. Methanol fuel cells don’t provide as much power, but methanol’s a hell of a lot easier to transport. I’d still have to do something about the exhaust from the fuel cell, but for now, let’s assume I run the fuel cell hot enough that the pressure pushes it out a one-way valve.

You know, the more I look at it, the more “screw wheels” makes it look like I’ve got some weird grudge against wheels. In reality, those funny-looking objects are screw-drive wheels, which are so good at all-terrain they make tanks drive home to their hangars and weep transmission fluid into their berths.

That glorious contraption is the 1929 Fordson snow machine. It’s got huge pontoon-shaped wheels that turn on an axis parallel to the direction of travel, rather than perpendicular, with the force provided by many small segments of the helical screw pushing against the snow as the wheels turn. Because the pontoon wheels have such a large footprint, they don’t sink even in deep snow. And there is, of course, the cool factor, which is almost off the scale here. And now, let me send it careening completely off the scale by showing you the Russian incarnation of the screw-drive all-terrain vehicle:

When they say this is an all-terrain vehicle, they mean all terrain. Including the surface of a fucking lake. It’s hard not to be impressed.

Okay. So screw-wheels are cool. Is that reason enough to complicate the design of my submarine car to throw four of them on there? Well, yes. The seafloor is unpredictable. Without vegetation to anchor it or ordinary algae and fungi to bind it, the seafloor can have a wide variety of textures, and almost all of them would be a nightmare for a wheeled vehicle: silt six feet deep, loose sand, gravel rocks, weird clay, mud, sulfide sludge spewed from seafloor volcanoes… A wheel is very likely to sink into a morass like that. I could make the wheels larger, but a pneumatic wheel will deflate at seafloor pressures. I could make the wheels non-pneumatic, but they’re still big and bulky and taking up a lot of room, and they’re going to increase drag, which I’m already spending all my energy fighting. Tracks might work, but they contain an awful lot of joints and moving parts, and would probably get silted up.

Of course, screws as small as the one in the illustration probably wouldn’t fare too well in soft silt, either, so I’d have to make them larger, but nonetheless, I’d say screw wheels are the optimal solution. And besides that, they’re fucking awesome!

Of course, all this is really moot. There’s a very good physics reason we don’t turn all cars into planes (And plenty of other reasons, if you see how people drive on the ground.): it’s hard to generate lift in air. For a long time, many believed heavier-than-air flying machines were impossible, and they only became really successful a century ago. Air isn’t all that dense. An average 80-kilogram (172-pound) human could comfortably stand in a box 2 meters on an edge. That box would be seriously oversized, but the air in it would still only weigh 10 kilograms.

A box of water 2 meters on an edge, though, would weigh 8,000 kilograms, or about the same amount as an African elephant. Water is dense. Denser fluids resist being pushed around, and therefore more readily generate force. In my earlier article, I figured out that my car would theoretically be able to go 135 miles an hour in air, but only 15 miles an hour in water. But when it comes to flying, that resistive force comes in handy, because you don’t need very big wings to turn your submarine car into a submarine…submarine. Here’s a picture to illustrate how much better wings work in water than they do in air:

(Source.)

That is a boat. Being lifted almost completely out of the water by wings that look to me to be smaller than the wings on a Cessna.

And there’s another thing that makes an underwater vehicle confined to a surface kind of silly: buoyancy. Like I said, air’s not very dense, and for a bag full of gas to lift anything, it has to be less dense than the surrounding medium. There aren’t many gases less dense than air. There are hydrogen and helium, of course, but both are made of such small atoms that they leak out of containers no matter what you do, and hydrogen is flammable. There’s hot air, but that’s a lot denser than hydrogen or helium, and after a certain altitude, it gets hard to heat it up enough to provide useful lift.

Now here are some things that are less dense than water: Myself. Wood. Warmer water. Air. Compressed air. Methanol. Ethanol. Propanol. Butanol. Low-density polyethylene. In other words, around half the matter (by volume) in a submarine is less dense than water. Submarines actually have to take on extra water to sink. I had to stick depleted uranium ballast weights in my ocean car just to make sure it wouldn’t float.

But really, why shouldn’t it float? It’s not like I couldn’t stick landing skids or water-filled pontoons on a regular submarine and then have a submarine that can go to the bottom. And with a little creativity and plenty of power (provided by a wide variety of sources), you can adjust your buoyancy on-the-fly.

So forget the submarine car. I just want a submarine.

Actually, that’s a lie. I want a submarine car with screw-wheels. But I’m going to use it on land. Because, holy shit, look at those motherfuckers in the videos up above!

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