The Mathematics of Speed

Ok y'all. I went into no-no land that is the comments section of Youtube again and this person I'm arguing with has me questioning my knowledge of math a little bit. So I'm here presenting my knowledge and seeking affirmation or negation. Here goes.

We'll use SAE units. The power output of an engine is Power = Torque x RPM / 5252. That is a simple, well-known equation, with which I am very familiar. This relationship of power and torque means that there are some consequences for engine output. Then, to increase an engine's power output, you can increase the torque that an engine makes at a given speed (e.g. better exhaust), increase the speed at which that torque is produced (e.g. better cam/valvetrain) or some combination of both methods. This also means that to maintain constant power output, torque must decrease as
Torque = Power x 5252 / RPM as engine speed increases. This looks like a scaled graph of 1/x, x > 0.


Here's the root of the argument that we're having: A big, lumpy torque curve is better. No, a flat one is.

I just can't get through to the guy, because he keeps trying to complicate the argument by adding in variation in gearing.

That said, yes I do know how torque multiplication works, and that high-revving engines use this to their advantage.

Now, in a real engine, you can't have a torque curve that changes to keep power at maximum, unless you do some really fancy, complicated things with valve timing.

So, let's assume that there are 2 engines, both with a certain power rating. One has more torque, one has less torque. In reality, the one with more torque would max out its power at a lower speed and trail off as it approaches its redline. The one with less torque would max out later, maybe even at its redline.

Which one is better? It depends on how high each engine can rev and produce viable torque. Take the C6.R GT1 and the C5.R as examples. Both redline at 7000 rpm. Both make in the ballpark of 600bhp. The C6.R's engine produces much more torque, so it has the mid-range oomph to get it out of corners but can't make maximum power near redline, so I end up short-shifting by quite a bit. The C5.R, on the other hand, can't leave corners quite as quickly but produces maximum power higher up in the rpm range; that is, it can theoretically go faster, letting the drag of both cars be equal.

Now, take a 900hp 426 Hemi producing 800lb-ft of torque in a modified drag 'Cuda vs a 900hp Ferrari V12 producing 400 lb-ft of torque in a F1 car. The Hemi will rev to, say, 7500 rpm. the V12 will rev to, say, 17,500 rpm. For argument's sake, let all other quantities between the cars be equal. The V12 can be geared 2.33 times lower than the Hemi and have the same shift points. Then, the effective difference between the two can be generalized as the F1 engine can produce another 133 lb-ft. compared to the Hemi, because of gearing.

Here's the thing, though: if two cars' redlines are different but close, the difference in thrust due to the difference in gearing is not quite as noticeable. Then, we need to do some math to figure out which one is better, depending on the situation.

On shift points: after some head-scratching and pencilwork, I've come up with this: the best point to shift, combined with the seperation in gear ratios (aka how far down the revs your engine goes from one gear to another) is determined by the integral of average power from lowest operatingrev to highest operating rev. This yields the most work done per time over the course of the engine's time spent in that gear.

As a grain of salt: I know that there are, in some cases, tens of variables to be included in my simulation of physics. I just don't have 300 pages to work all that out and explain how I reached those conclusions.

If any of you guys know more about this than I do (assuming that this is indeed the case), please let me know if this is all reasonable stuff or if I'm crazy.

In short.... As my father always said... You want to be in at your peak as long as possible.

I think everything you said is correct. On shift points: "integral of the average" doesn't really make sense to me. I think just the integral is enough. Even just the average would get you really close unless you had a really wavy powerband.

As easy as what you said is to follow, you are indeed a very smart guy. I had to consult my resident car guy of 40 years. His long winded answer boiled down to this. He said that with all variables as similar as possible the HEMI with the "lump" of a power band (as he put it) would drive and act as if it had shorter gear ratios (even if they were identical). He said that because of that in a quarter mile the lump would be better suited because of the explosive burst of torque and power. He did say that in say a half or full mile drag that the flat torque band would be better. As for the argument that your youtube adversary made... Well that is where my father's argument went very fast. He basically said that while its possible with lots of testing and R&D to make the lumpy banded car compete with the flat one in the longer run it would not be as easy and would be very costly. Gearing would matter a lot and it would be interesting to see them preform side by side. He says that the flatter band would win in any place other than a quarter mile drag.

It really does depend on the application doesn't it? The higher the rpm, the sooner parts wear out and the shorter the operating life span. It is a mis-application to use a high revving motor for torque dependent work loads. This is why you'd rather have a diesel truck more than a gasoline truck to haul things.

I would think that in a comparison of power efficiency and optimum torque/horsepower curves, you wouldn't use an F1 motor to power a Peterbuilt no matter what gearing you used. F1 motors are not designed to pull anything, and aren't going to be much good outside of their intended purpose.

If I had to guess, in theory, you could make a high rpm engine with little torque, the same as a low rpm big torque engine if you had to, but why try? In actuality low rpm torque rich motors are still superior because they have shown much wider performance capabilities in many more applications, on and off road, two-wheel and four.

Size does matter of course, and the size of a motor will have an enormous effect on the size of the car it rests in. Smaller motors with less torque are going to produce smaller car designs with less weight. Less weight is in effect more torque. But weight savings can only go so far, whereas an engines torque output can go farther even when max weight reduction has been reached.

This is why you'll never see a flat-plane V12 Ferrari, Lambo, or McLaren motor used for drag racing. No matter what you do to it, it will never have the torque or horsepower figures to compete in the unlimited Top Fuel and Funny Car classes. For this you'll need a much bigger, slower revving, torque rich engine, that can produce earth moving amounts of torque and horsepower. You'll need a 500cid cross-plane Hemi V8, running nitromethane through a supercharger producing 8,000hp and 6,000lb-ft of torque.

About the drag racing engines: that's what I had always thought, because in a torque-rich engine the extra thrust you receive in the low/mid range means that the other car is playing catch-up until the next gear. About trucks: these are pretty much my thoughts exactly.

Sainteds car guy is correct (to me). I prefer low revving engines in any situation but that is a matter of preference. Some people prefer to stretch the power and rpm out and receive power later. Its mostly up to what you like as to which is better. Only reason I prefer the torque happy engine better is the feeling of shorter gears even if they are in fact longer. I like getting power early and being able to run a long gear set to keep in the torque or power, which ever I am using at the time. It differs for everyone though and thats why the argument will continue.

Edit: woops, thought I was signed in as me. This is PiiHB if you couldnt tell.

Polycyclic wrote:

Sainteds car guy is correct (to me). I prefer low revving engines in any situation but that is a matter of preference. Some people prefer to stretch the power and rpm out and receive power later. Its mostly up to what you like as to which is better. Only reason I prefer the torque happy engine better is the feeling of shorter gears even if they are in fact longer. I like getting power early and being able to run a long gear set to keep in the torque or power, which ever I am using at the time. It differs for everyone though and thats why the argument will continue.

Edit: woops, thought I was signed in as me. This is PiiHB if you couldnt tell.

I was soooo thinking "wow Poly knows her booboo"

@duty: Whoops I guess I typed the average thing backwards. I meant the average power from the lowest to highest rpm you'd be revving over (for which I used the [integral(power curve)]/(upper limit - lower limit)).

Then, it seems that lumpy torque curves are more suited for lower-speed applications like for heavy trucks, where speed limits exist, and for 1/4 mile runs, where you don't have to spend much time near the redline; and long, flat torque curves are more suited for high-speed runs, where having lots of thrust near the redline means holding a shorter gear for longer is preferable. Hence, a 458 can go as fast as a ZR1 but with a lower power rating; however, the ZR1 accelerates more quickly in the mid-range. That seems right.

Polycyclic wrote:

Sainteds car guy is correct (to me). I prefer low revving engines in any situation but that is a matter of preference. Some people prefer to stretch the power and rpm out and receive power later. Its mostly up to what you like as to which is better. Only reason I prefer the torque happy engine better is the feeling of shorter gears even if they are in fact longer. I like getting power early and being able to run a long gear set to keep in the torque or power, which ever I am using at the time. It differs for everyone though and thats why the argument will continue.


Edit: woops, thought I was signed in as me. This is PiiHB if you couldnt tell.

I would add a frame of reference here when it comes to driver preferences on torque delivery. You can tune any engine and move the torque curve somewhat, but as a rule drivers are looking for superior traction and smoother drivability which means getting the power to the ground more efficiently. This is why 4-stroke bikes have taken the lead over 2-stroke machines. The lower rpm greater torque 4-stroke is easier to control and ride faster, because it gets the power to the ground easier, sooner, and with greater control and throttle modulation. Once the extra weight of the 4-stroke design was roughly brought down close to the lighter weight 2-stroke, the power delivery and torque advantage of the 4-stroke then made it superior.

In cars we see a similar situation in endurance racing with the P1 Audi TDi diesel cars that have dominated at LeMans. The low rpm high torque of the diesel race engine makes it very track efficient getting the power to the ground in a controlled way that is superior to traditional gas powered cars. The Corvette too, with its small block American cross-plane V8 has done exceedingly well throughout the decades in its many road racing configurations. Compared to its European competition the venerable Chevy V8 is almost primitive in design, a design that hasn't changed in 50 years. Yet it is ultra reliable for any kind of endurance racing, and puts out good horsepower and more torque at a lower rpm. With the use of aluminum blocks and heads weight isn't even a factor compared to the once lighter Euro flat-plane engines. As I understand it team Corvette and at one time team Viper have a good deal more restrictions to abide by then their competition from Ferrari and Porsche. Speaking of the Dodge Viper it too was able to dominate endurance racing with its massive V10. Not the zingy high reving 18,000rpm V10's of Formula 1, but again a slower reving high torque cross-plane configuration of about 7,000rpm easily won the day. Until more restrictions were demanded and the car lost its initial advantage that is.

Well i wont argue that a small block chevy is a well rounded motor. that is for sure. If im in forza i like the flatter porsche or ferrari torque bands bc i dont like how a american v8 drops off.

Side note on the small block. I have heard of people putting them into 70's and 80's 911's and i would love to be able to do that. They sound fantastic and its just something i would want to do as a project.

SaintedPlacebo4 wrote:

Well i wont argue that a small block chevy is a well rounded motor. that is for sure. If im in forza i like the flatter porsche or ferrari torque bands bc i dont like how a american v8 drops off.

That's exactly why I like the American muscle cars in this game. You can turn massive power early and (not that it matters in the game) don't even have to stress the engines. That's why they work so well in real world applications also I think. As far as in game goes, most people don't bother to look at the torque curve when tuning and think power always comes later. I've had people send me tunes and say the car has so much power but feels slow. They don't take the time to realise that they are turning Tue engine at seven or eight thousand when max power is around five. Common mistake and also the reason why some people don't like them. Even the caddy is this way to an extent. The only reason that car runs so well pulling out of a corner is it has a damn near perfect torque curve. Holding massive torque numbers all the way into the power band. Late power is also why its so dominate at high speed courses. Having power at sixty five hundred and redline at seven makes it perfect for high speed and great shift points. The stock gearing works well because it runs long gear ratios over a long final drive. Most cars don't like this because of a lack of torque but a good ol American V8 thrives with this setup.