Cars, physics, thought experiment

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.

Some Kinda Souped-Up Diesel

If you recognized that quote (without fancy Internet cheating), then I will buy you dinner. I’m not kidding. I’ll buy you some chicken fingers or a curry or something. Because the quote comes from Steven Spielberg’s slightly-obscure first feature film: Duel. You should watch it. It’s the only example I’ve personally seen of a made-for-TV movie that doesn’t suck. It’s a great car movie. It’s a great car chase movie. It’s a great road movie. It’s a great psychological thriller. Here’s the villain of the film:


That is a Peterbilt 281. If you ask me, we should go back to this body-style for all our semis. The new ones all have that stupid over-designed over-smooth overly-generic SUV-crossover-station-wagon-thing look. Even though I was born long after this style fell out of favor, it’s this kinda thing I think of when I think “truck.”

That aside, the protagonist of Duel discovers the hard way that the Peterbilt 281 has a fair bit of power under the hood. That power is provided by a Cummins NTC 350 turbo diesel. (Thanks, Internet!) According to some slightly moody diesel enthusiasts, the NTC 350 produced about 1,100 foot-pounds (1,491 newton-meters) of torque at 1300 RPM. Plus, it looks like I might actually be able to fit that engine somewhere in my car, if I got rid of unnecessary things like passenger seats, spare tire, trunk, upholstery, safety, et cetera. The math is the same: how could my souped-up diesel Toyota perform? It could deliver 8,623 newton-meters at the axles, which would provide a force of 24,250 Newtons against the road. Apply the reverse drag equation, and I get 259 m/s (579 mph, 931 km/h).

Erm…huh… That’s an awfully large number. That’s larger (if I remember rightly) than the current land-speed record for a wheel-driven vehicle… When the silliness of your answer is this much greater than the silliness of your question, that suggests you did your math wrong. I checked it, and nope. Did it right for a change. But there’s another factor at play here: RPMs. My car doesn’t have a tachometer (one of the annoying things about it), but being a standard gasoline-powered production car, its redline is probably somewhere near 5,000 RPM. But the torque measurement I used above wasn’t taken at 5,000 RPM. It was taken at 1,300 RPM. Diesels, as a rule, run a lot slower than gasoline engines. And it doesn’t matter if you’ve got the theoretical torque to push your car to 579 mph, if your drivetrain can’t make the wheels spin fast enough to keep up.

Luckily, I don’t have to do a lot of complicated math and research to figure out what angular velocity the souped-up diesel could deliver to my wheels. I can just use the same gear ratios as before, but instead of multiplying the torque with all of them, I can just divide the torque by all of them. Without adding extra gears to the drivetrain, the souped-up diesel couldn’t spin my wheels any faster than 225 revolutions per minute. The circumference of my wheels is 14 inches * 2 * pi, or 87.965 inches. This is one of those awesome situations where I can just do what my instinct tells me and multiply 225 revolutions per minute by 87.965 inches per revolution and actually get the right answer.

And here’s where I admit that I screwed up. In my calculations, I made the naive assumption that every part of the drivetrain behaved like a gear. But, funnily enough, only the gears do that. I would guess that the torque converter, while it does multiply torque, doesn’t actually multiply and divide rotational speeds, or if it does, not in the same way gears do. Because when I included the torque converter in the math, it told me that, with the stock engine, at 5,000 RPM, my car couldn’t go any faster than 71 mph. I know that’s not true. So I took the torque converter out of the equation (fully admitting that I don’t understand its contribution to final wheel speed), and got a more reasonable answer for both the stock Yaris’s top speed and, more importantly, the souped-up diesel Yaris’s maximum speed: 36 mph (59 km/h).

“But,” I hear you say, “Diesel semis can go way faster than that.” You’re right. They can. That’s because they have different gearboxes with different gear ratios. To nobody’s surprise, when you just drop a random engine into a car with no other modifications, you don’t get ideal results.

Also, watch Duel. Seriously. It’s a good movie.

Er läuft!

Speaking of good movies, here’s a picture of a sweaty German man:


That’s Johann (played by Erwin Leder), from Das BootDas Boot is one of those movies I kept hearing about and kept failing to watch. But I finally watched it, and I’ve been slightly obsessed with it ever since. I keep wanting to tell people to see it. I keep re-watching it. Of the war movies I’ve seen, it’s rapidly become one of my favorites (alongside Saving Private Ryan and The Hurt Locker).

“What,” I hear you ask, “the hell are you talking about?” Well, your rhythm may be strange, but your question is valid. Because now, we’ve moved from the almost-reasonable to the completely ridiculous. Johann there is, as you’ve probably seen, standing in front of a gigantic diesel engine. The engine’s pretty hard to see in the picture, so here’s an amazing clip (containing FAIRLY MASSIVE SPOILERS, mostly after 0:43) of that engine in action:

Er läuft!

The submarine in Das Boot is a World War II German U-boat: a class VIIC, to be exact. It was powered by two supercharged six-cylinder diesels. Here’s a better view of one of those engines (which, apparently, was meant for a U-boat, but got stuck powering a generator instead):

Another long detour. Sorry about that. Like I said, I keep wanting to talk about Das Boot, but the people I watch movies with don’t like subtitles, which is the only way to watch Das Boot, if you ask me.

Either way: as small as the engine room on U-96 is, it could still fit my whole car two or three times over. If I actually wanted to power my car with U-96’s powerplant (and I would insist on saying Beide Dieseln every time I started it up), I’d have to tow it on a trailer or something and have a very complicated drive-shaft setup. Nonetheless, if I did it, I’d be getting about 3,000 horsepower at 500 RPM. That gives me a horrifying 31,5oo foot-pounds (42,700 newton-meters) of torque. The equivalent of resting a half-ton (or half-tonne, almost) weight on the end of a 10 meter (32 foot) wrench. That is twisting power. That makes for 511,500 newtons of pushing force. (Theoretically, of course. In reality, the second the motor started, it’d either just make my car spin like a drill bit or yank all the guts out of it. But this blog isn’t about reality.) Do the usual reverse-drag thing. I get a maximum speed, at sea level, on Earth, of 1,188 m/s. That’s over Mach 2. That’s 2,657 mph, or 4,276 km/h. Faster than a rifle bullet.

But I’m not even going to pretend to get all excited, because as we saw with the previous example, the low speed of a diesel engine is the real upper limit here. At 500 RPM, through my standard drivetrain, that’s only going to give me 6.3 meters per second (14 mph, 23 km/h, and damn am I getting sick of specifying speed in three different units…)

So, as cool as it would be to have a German U-boat motor powering my Japanese economy car down American highways (it’s a weird World War II salad!), if I did it, I’d be getting passed and honked at by everybody else on the road. Even the alternate-world version of myself with a go-kart motor for an engine.

The Overcompensator

Here’s a picture of an awesome and terrifying truck:


That monster is the BelAZ 75710. If you’re Belarusian and you’re tired of hearing people say “What the fuck is Belarus?” just show them that picture and say “That’s Belarus.” As of this writing (late June/early July 2016), it’s the largest mining truck on Earth. A quick back-of-the-envelope calculation suggests that it could probably carry a small office building (450 metric tons, almost 500 U.S. tons). As is tradition with ridiculously large vehicles, one engine just isn’t enough. That yellow behemoth is packing two V-16 diesel engines, each rated at up to 5,700 horsepower. Each one looks like this:


(Gentleman in flatcap added for scale.)

The U-boat engine room was big enough to hold my car. I think this single engine is big enough to hold my car. But what kind of speeds will those 5,700 horses get me? We’re dealing with another diesel engine. That means the limiting factor isn’t going to be torque, but wheel speed. The BelAZ engines run at a surprisingly brisk 2,100 RPM. That gets me 59 miles per hour (95 kilometers per hour). That’s not very impressive, you say? Well, BelAZ doesn’t build 75710’s to race them (although, if they ever have a 75710 race, somebody fly me to Belarus!). The 75710 is built for hauling. Attached to my car’s wheels, that’d give me about 231,750 Newtons (in reality, of course, it would blow the wheels off). That’s almost 1/5th the thrust of a Space Shuttle solid rocket booster. Here’s a bit of perspective: with that kind of force (and ignoring things like wheel-slip), my car could tow a bulldozer. A bulldozer with the parking brake on. Through sand. 42 tons of towing capacity. If you put the load on a decent trailer, my car would be able to haul 500 tons or more: one or two BelAZ 75710’s. (That makes sense, of course: a truck that can’t tow the weight of itself isn’t going anywhere empty, let alone loaded.)

The Speed Demon


That’s a pretty nice-looking car. That is Poteet & Main’s Speed Demon. As of this writing, it holds the land speed record for a wheel-driven car: 440 mph (707 km/h). “That’s fast,” I hear you saying (I’m hearing you say a lot of things. Maybe I need to adjust my meds.) “but the overall land-speed record is over Mach 1.” True. But the Speed Demon broke the wheel-driven land speed record using nothing but a turbocharged small-block Chevy engine. As in, an almost-ordinary V8 engine. And when you consider that the previous record-holder, the Goldenrod, used four supercharged V8’s end-to-end to reach similar speeds, it’s damned impressive. (It should be noted, of course, that apparently the track conditions were a little kinder to the Speed Demon than the Goldenrod, but it’s still damned impressive.) Here’s the engine in question:


Of course, since we’ve been looking at absolutely ludicrous engines, the Speed Demon‘s isn’t going to seem all that impressive by comparison. After all, it only (“only”) produces 2,600 horsepower at (thanks, Internet!) 8,800 RPM, producing 1,313 foot-pounds of torque. Putting that thing in my car (which is, oddly enough, one of the most feasible things I’ve considered so far), would get me to 632 mph (1017 km/h). Theoretically. There’s also maximum engine speed to contend with. And this is where your engine’s red-line and your gearbox become important. Because, even though the maximum speed is nowhere close to 632 mph, it’s still a respectable (suicidal) 247 mph (398 km/h). That’s not bad, and it’s one of the few thought experiments here that might actually be doable in real life.

The Reason I Wrote This

This is the main event. This is the motivation for the whole post. Because, as I said in the previous post, I wanted to find out just what my car would be able to do if I put a a helicopter turboshaft engine under the hood. A turboshaft is a jet engine that uses its exhaust turbine to turn a shaft (often connected to a propeller), rather than just belching it out the back for thrust. The turboshaft in question is the Honeywell/Lycoming T55. Here it is, with serious-faced people for scale:


The T55 is a proper workhorse engine. Among other things, it drives the rotors of the famous dual-rotor heavy-lift Chinook helicopter.  As you can see, it’s small enough that I might actually be able to fit it in my car (or, at the very least, in an ordinary production car). There are a couple of things that please me about the T55. The first is the power: according to Honeywell, the T55 can deliver a shade over 4,000 horsepower continuously. The second is the shaft speed: 15,000 RPM. That’s both of the sticking points in the previous experiments dealt with.

The same math applies. Considering only torque (1900 newton-meters) and air resistance, my car could make 390 mph (627 km/h). Considering shaft speed and ignoring air resistance, my turboshaft Yaris could make 421 mph (678 km/h).

But there is one tiny (massive) catch: fuel consumption. The efficiency of turbine engines is measured as “specific fuel consumption” (SFC, in the document I linked you to). The efficiency of the T55 is 0.5, which, when you do the math, comes down to 2.2 kilograms (over 4 pounds or 3 liters) of aviation kerosene every second. My fuel tank holds 11 gallons. It’d run out in under 14 seconds. That’s the price of power.

Abandon all Logic ye who Enter Here

The world’s largest diesel engine (the Wärtsilä-Sulzer RTA98-C) probably wouldn’t fit in my house, even if you hollowed out all those troublesome floors and walls. It looks like this:


It’s built to power the world’s largest oil tankers. The individual cylinders are so big that, when the engineers are maintaining them, they just climb inside and sit down at the bottom. To even consider attaching that monster to my car is to descend into madness. Luckily, I’m already down here. The RTA98-C produces a horrifying 46,000+ horsepower, and 7,600,000 newton-meters of torque at 102 RPM. That torque is ridiculous. Here’s how ridiculous: if you took the engines on a 747 and attached them so that they pointed up and down (respectively), to make the plane spin like a pinwheel, the plane would be experiencing the same torque that the RTA98-C puts out. It produces so much torque that, if you attached an I-beam 150 meters long to the crankshaft, it could lift me even if I sat on the end. And even if you include the mass of the I-beam.

But you’re not hear to hear about bizarre man-lifting contraptions. You’re here for speed! We already know that the limitation isn’t going to be the force the engine can put out. It’s the motor’s low speed. The RTA98-C could propel my car at all of…2.87 mph (4.61 km/h). Even when I’m out of shape, I can walk faster than that. But I can’t do what the Wärtsilä would let me do: pull with almost ten times the force of a Shuttle solid rocket booster. I could haul the whole Titanic. Over land. On a trailer, I could probably haul a couple of office buildings. When you’re working with this much excess power, you pretty much have to hang rubber testicles from your trailer hitch.

But What About the Gearbox?

My car’s smallest gear ratio is 0.700. That’s because 0.700 can get me going as fast as a car like mine will ever need to go. But although it’s not practical to build, it’s feasible to build a much more powerful gearbox. All you do is stack up sets of gears. The first level consists of the input gear of radius 1 meshed to an output gear of radius 0.5. That smaller gear is bonded to a radius-1 gear on the second level.This second 1-radius gear drives an 0.5-radius gear bonded to another radius-1 gear on the third level. Repeat as much as you want. The first 0.5 gear will output at a 2:1 gear ratio. The next, at 4:1. Then 8:1. 16:1. 32:1. And so on.

So what if I had ten 2:1 gearings? That gives me a 1024:1 gear ratio, and unless I made the gears really massive, it would fit just fine inside my car, even if you include the complicated shifting system you’d need to access all the different ratios. And, leaving the rest of the drivetrain alone, that could spin my wheels at 20,000 revolutions per second. The wheels would make an awful dentist’s-drill sound, and would theoretically propel my car at 45 kilometers per second, far beyond Earth escape velocity. My car would be the second-fastest machine ever made.

Except, of course, it doesn’t work that way. Gear ratios are amazing and powerful, but they’re not magic. My bizarre new gearbox would multiply the angular speed of my engine by 1,024 (times the speed ratios of the differential and torque converter). But it would divide the torque by 1,024. And when you consider that my car only puts out about 100 newton-meters of torque, dividing by 1024 makes things very feeble and depressing: a ruler attached to the output of my gearbox wouldn’t be able to press a computer key, let alone lift anything.

But let’s go crazy. I’m talking batshit. I’m talking “I’ve just spent four hours writing a blog post about engines and I’ve had too much coffee and no food and my ears are starting to ring” crazy. Let’s attach this fantasy gearbox to the Wärtsilä engine! Cut out the differential and torque converter. Who needs ’em?

With the new gearbox, the Wärtsilä engine could propel me to 537 mph (864 km/h), considering only drag (and only the drag from my own car, not the engine, which is ridiculous). A house-sized engine traveling at half the speed of sound. There’s a nightmare for you. Here’s the real nightmare, though: my car could reach that speed and hold it, because with the 1024:1 gearbox, its wheel-speed would allow a speed as fast as 3 km/s, which is faster than any rifle bullet and a quarter of the way to orbital speed.

Why don’t we use 1024:1 gearboxes, then? Well, there’s always friction to contend with. And friction in rotating parts produces torque. Let’s consider an example case. Let’s say that, at a particular speed, the frictional torque is 10 newton-meters. That’s a fair bit of torque: as much as a 12V power drill can produce. Here’s the thing: if that friction is downstream of the gearbox (somewhere between the gearbox and wheels), it’s going to get multiplied by 1,024 on the engine side. Suddenly, you need eight Bugatti Veyrons worth of torque just to resist friction. A little wobble or a bearing that sticks for a second, and your driveshaft twists like Play-Doh and then shears off and probably ends up in your chest cavity.

One More Little Bit of Madness

You know how nuclear fission releases a ridiculous amount of energy? Well, there exist nuclear reactors small enough to put on wheels. Here’s an example: the kind used to power US Los Angeles-class nuclear submarines:


The nuclear reactor and turbine generator section is about 50 meters long and 10 meters across. It would only be as wide as the BelAZ. And it would produce 70,000 horsepower. Put three or four or five BelAZ axles under it, with those 4-meter tires. Run the turbines at 1600 RPM, and even with terrible aerodynamics, you could be driving a hundred-ton reactor on wheels at 176 mph (284 km/h). With decent aerodynamics, you could manage almost 400 mph (644 km/h). That is, if you got lucky, you could break the wheel-driven land-speed record with a nuclear reactor the size of a house.

And that’s why I love writing this blog.


6 thoughts on “Supersonic Toyota? (Cars, Part 2)

  1. Ah yes, the old torque vs horsepower vs traction trick. I had wondered, years ago, about dropping a Merlin from what was until then the last flyable Spitfire in the country, into my Ford Cortina… Apropos such matters, it’s always intrigued me that for all the ingenuity being poured into the wheel-driven LSR these days, everything since Donald Campbell’s Proteus-engined Bluebird has been essentially small-team ‘garage’ projects. Whereas his car was a million-pound (in 1960 money) effort, literally built like a jet fighter, that drew from the British aero industry. The result? Even today, if anybody took that car out of the museum, it could certainly get the Poteet & Main record and potentially go for the Vesco Turbinator record (470.444 mph) – depending on source Bluebird was designed either for 450 or 500 mph. Campbell’s peak speed on Lake Eyre was 445 mph Eyre despite the slushy track. I suppose the ultimate limit for these things will be where drag = available traction, wherever that speed might end up (my guess is around 500 mph depending on aerodynamics, but human ingenuity is always going to push that…)

    • From the little bit I’ve read, it seems like the wheels are the limiting factor. You sound like you have more mechanical knowledge than me, but I’ve been running some simulations in preparation for Part 3, and it seems to be the wheels that go wrong before anything else. I’ve seen some German tractor-pull tractors with up to four turboshafts, or as many as 6 standard piston engines, and one with an aircraft radial engine. Even the best performers can only pull so far, because the wheels start slipping. I remember reading something about Campbell’s Bluebird, where he said that wheel slip was the main thing that kept him from managing 500 mph. My simulations say just about the same thing: at 300+ mph, wheels start to lose traction, or wobble, or fly apart. Part of that is the limitation of my simulation engine, but I think the wheels are gonna be the real killer in the end.

      Also, the thought of a Merlin engine in a Cortina makes me drool a little. I live in the US, so I’d never seen one until I looked it up after reading your comment. A very attractive European-style body. I’ve got a soft spot for that. Also, I have a soft spot for V12s, and for the beautiful madness of sticking high-power aircraft engines in cars.

      • That Cortina was a very tough, but also very rough car – plenty of ‘go’ for its day (it actually had the standard 2-litre crossflow motor…I might possibly have welded a Merlin to the roof, but that would have been about it…) but it needed a LOT of maintenance. Apropos wheelspin – my knowledge is more theoretical than actually doing the maths, but people I know in the field tell me it’s a real issue and, indeed, 300 mph is where things start to get awkward. George Eyston dealt with it in 1937 using 8 wheels and a 7-ton vehicle – Thunderbolt – which hit nearly 360 mph in the end. It ended up for the duration of WW2 where I live, Wellington NZ, and was destroyed when the storage facility went up in flames. The hulk was left on the roadside and apparently taken to a local dump. Nobody knows exactly where it’s buried, but a local car museum wants it for restoration. The same museum, incidentally, has one of Campbell’s Proteus-Bluebird wheels, presumably a spare. Odd where some of this stuff has ended up.

      • Like I mentioned in the post, I’ve often wondered about the possibility of fitting cars with ridiculous engines by putting the engines on trailers towed behind the cars. A precision-machined railcar-type coupling between the trailer and the car, to eliminate play. A driveshaft running to the car, with a wide-range CV joint, connecting to a heavily-modified back axle. But if I was a good enough engineer to design something like that, I’d be a millionaire by now.

        I’m not sure how I’d deal with wheelspin, but more wheels seems like a good way to do it. Big, wide, low-pressure tires with huge contact patches, and lots of them.

        Thinking about where things end up… That reminds me for one thing of all the cool cars that’ve been found in barns. But beyond that, it reminds me of the best-preserved German U-boat motor I’ve ever seen prowling the Internet, which survived WW2 because it never got put in a U-boat, and ended up as a generator.

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