Horsepower and Torque: A Primer

Almost invariably, when car nuts get together to bench race, or just talk shop, sooner or later they get to talking about power. BMW enthusiasts seem to be at least as interested as anyone else in the topic, and yet it's been my experience that many folks are a little confused about the terms, and how they relate to automotive performance. This article is intended to explain horsepower and torque from a driver's perspective, with some basic arithmetic mixed in for illustration.

OK. Here's the deal, in moderately plain English.

The Basics - Force, Work and Time

If you have a one pound weight bolted to the floor, and try to lift it with one pound of force (or 10, or 50 pounds), you will have applied force and exerted energy, but no work will have been done. Work requires movement. If you unbolt the weight, and lift it one foot, then one foot pound of work will have been done. If that event takes a minute to accomplish, then you will be doing work at the rate of one foot pound per minute. If it takes one second to accomplish the task, then work will be done at the rate of 60 foot pounds per minute, and so on.

In order to apply these basic measurements to automobiles and their performance (whether you're speaking of torque, horsepower, Newton meters, watts, or any other terms), you need to address the three variables of force (torque, in this discussion), work (horsepower, in this discussion) and time (which none of us has enough of, in any discussion).

Why We Measure This Way

Awhile back, a gentleman by the name of Watt (the same gent who did all that neat stuff with steam engines) made some observations, and concluded that the average horse of the time could lift a 550 pound weight one foot in one second, thereby performing work at the rate of 550 foot pounds per second, or 33,000 foot pounds per minute, for an eight hour shift, more or less. He then published those observations, and stated that 33,000 foot pounds per minute of work was equivalent to the power of one horse, or, one horsepower.

Everybody else said OK.

For purposes of this discussion, we need to measure units of force (torque) from rotating objects such as crankshafts, so we'll use terms which define a twisting force, such as foot pounds of torque. A foot pound of torque is the twisting force necessary to support a one pound weight on a weightless horizontal bar, one foot from the fulcrum.

Now, it's important to understand that nobody on the planet ever actually directly measures horsepower from a running engine. What we actually measure (on a dynomometer) is torque, expressed in foot pounds (in the US), and then we calculate actual horsepower by converting the twisting force of torque into the work units of horsepower.

Visualize that one pound weight we mentioned, one foot from the fulcrum on its weightless bar. If we rotate that weight for one full revolution against a one pound resistance, we have moved it a total of 6.2832 feet (pi times a two foot circle), and, incidentally, we have done 6.2832 foot pounds of work.

OK. Remember Watt? He said that 33,000 foot pounds of work per minute was equivalent to one horsepower. If we divide the 6.2832 foot pounds of work we've done per revolution of the weight into 33,000 foot pounds, we come up with the fact that we have to rotate that weight at the rate of 5,252 revolutions per minute in order to do 33,000 foot pounds per minute of work, and thus do work at the equivalent rate of one horsepower. If we only move that weight at the rate of 2626 rpm, it's the equivalent of 1/2 horsepower (16,500 foot pounds per minute), and so on. Therefore, the following formula applies for calculating horsepower from a torque measurement:

Horsepower = ( Torque * RPM ) / 5252

OK, that seems reasonably meaningless standing there by itself. Now, what does all this mean in car land? Let's try to explain by example.

The Case For Torque

First of all, torque rules, from a driver's perspective. Any given car, in any given gear, will accelerate at a rate that exactly matches its torque curve, allowing for increased air and rolling resistance as speeds climb. Another way of saying this is that a car will accelerate hardest at its torque peak in any given gear, and will not accelerate as hard below that peak, or above it. Torque is the only thing that a driver feels, and horsepower is just sort of an esoteric measurement in that context. 300 foot pounds of torque will accelerate you just as hard at 2000 rpm as it would if you were making that torque at 4000 rpm in the same gear. Yet, if you start plugging figures into that formula, you can see that the horsepower would double at 4000 rpm, with the same level of acceleration. Therefore, horsepower isn't particularly meaningful from a driver's perspective, and the horsepower and torque numbers only get friendly at 5252 rpm, where they always come out the same.

In contrast to a torque curve (and the matching push back into your seat), horsepower rises rapidly with rpm, especially when torque values are also climbing. Horsepower will continue to climb, however, until well past the torque peak, and will continue to rise as engine speed climbs, until the torque curve really begins to plummet, faster than engine rpm is rising. However, as I said, horsepower has nothing to do with what a driver feels.

You don't believe all this?

Fine. Take your non turbo car (turbo lag muddles the results) to its torque peak in first gear, and punch it. Notice the belt in the back? Now take it to the power peak, and punch it. Notice that the belt in the back is a bit weaker? OK. Now that we're all on the same wavelength (and I hope you didn't get a ticket or anything), we can go on.

The Case For Horsepower

OK. If torque is so all-fired important, why do we care about horsepower?

Because (to quote a friend), "It's better to make torque at high rpm than at low rpm, because you can take advantage of gearing". 

For an extreme example of this, I'll leave car land for a moment, and describe a waterwheel I got to watch awhile ago. This was a pretty massive wheel (built a long time ago down in Sturbridge, MA), rotating lazily on a shaft which was connected to the works inside a flour mill. Working some things out from what the people in the mill said, I was able to determine that the wheel typically generated about 2600(!) foot pounds of torque. I had clocked its speed, and determined that it was rotating at about 12 rpm. If we hooked that wheel to, say, the drive wheels of a car, that car would go from zero to twelve rpm in a flash, and the waterwheel would hardly notice.

On the other hand, twelve rpm of the drive wheels is around one mile per hour for the average car, and, in order to go faster, we'd need to gear it up. If you remember your junior high school physics and the topic of simple machines, you'll remember that to gear something up or down gives you linear increases in speed with linear decreases in force, or vice versa. To get to 60 miles per hour would require gearing the output from the wheel up by 60 times, enough so that it would be effectively making a little over 43 foot pounds of torque at the output (one sixtieth of the wheel's direct torque), which is not only a relatively small amount, it's less than what the average car would need in order to actually get to 60.

Applying the conversion formula gives us the facts on this. Twelve times twenty six hundred, over five thousand two hundred fifty two gives us: 6 horsepower.

OOPS. Now we see the rest of the story. While it's clearly true that the water wheel can exert a bunch of force, its power (ability to do work over time) is severely limited.

At The Drag Strip

OK. Back to car land, and some examples of how horsepower makes a major difference in how fast a car can accelerate, in spite of what torque on your backside tells you.

A very good example would be to compare the "LT1" Corvette (built from 1992 through 1996) with the last of the "L98" Vettes, built in 1991. I'm sorry to mention the "C" word in this august publication, but there just isn't a better example to use. Figures as follows:

The cars are geared identically, with identical tires, and car weights are within a few pounds, so it's a good comparison.

First, each car will push you back in the seat (the fun factor) with the same authority - at least at or near peak torque in each gear. As mentioned, torque is what pushes you back into the seat, so with identical torque production and everything else pretty much equal, each car will tend to feel about as fast as the other to the driver. The LT1 will actually be significantly faster than the L98, though, even if it won't push you into the seat any harder. If we mess about with the formula, we can begin to discover exactly why the LT1 is faster. Here's another slice at that formula:

Torque = ( Horsepower * 5252 ) / RPM

If we plug some numbers in, we can see that the L98 is making 328 foot pounds of torque at its power peak (250 hp @ 4000), and we can infer that it cannot be making any more than 263 pound feet of torque at 5000 rpm, or it would be making more than 250 hp at that engine speed, and would be so rated. In actuality, the L98 is probably making no more than around 210 pound feet or so at 5000 rpm, and anybody who owns one would shift it at around 46-4700 rpm, because more torque is available at the drive wheels in the next gear at that point.

On the other hand, the LT1 is fairly happy making 315 pound feet at 5000 rpm, and is happy right up to its mid 5s red line.

So, in a drag race, the cars would launch more or less together. The L98 might have a slight advantage due to its peak torque occurring a little earlier in the rev range, but that is debatable, since the LT1 has a wider, flatter curve (again pretty much by definition, looking at the figures). From somewhere in the mid range and up, however, the LT1 would begin to pull away. Where the L98 has to shift to second (and give up some torque multiplication for speed, a la the waterwheel), the LT1 still has around another 1000 rpm to go in first, and thus begins to widen its lead, more and more as the speeds climb. As long as the revs are high, the LT1, by definition, has an advantage.

There are numerous examples of this phenomenon. The Integra GS-R, for instance, is faster than the garden variety Integra, not because it pulls particularly harder (it doesn't), but because it pulls longer. It doesn't feel particularly faster, but it is.

A final example of this requires your imagination. Figure that we can tweak that Corvette LT1 engine so that it still makes peak torque of 340 foot pounds at 3600 rpm, but, instead of the curve dropping off to 315 pound feet at 5000, we extend the torque curve so much that it doesn't fall off to 315 pound feet until 15000 rpm. OK, so we'd need to have virtually all the moving parts made out of unobtanium, and some sort of turbocharging on demand that would make enough high-rpm boost to keep the curve from falling, but hey, bear with me.

If you drag raced a stock LT1 with this car, they would launch together, but, somewhere around the 60 foot point, the stocker would begin to fade, and would have to grab second gear shortly thereafter. Not long after that, you'd see in your mirror that the stocker has grabbed third, and not too long after that, it would get fourth, but you'd wouldn't be able to see that due to the distance between you as you crossed the finish line, still in first gear, and pulling like crazy.

I've got a computer simulation that models an LT1 Vette in a quarter mile pass, and it predicts a 13.38 second elapsed time, at 104.5 mph. That's pretty close (actually a bit conservative) to what a stock LT1 can do at 100% air density at a high traction drag strip, being power shifted. However, our modified car, while belting the driver in the back no harder than the stocker (at peak torque) does an 11.96, at 135.1 mph, all in first gear, of course. It doesn't pull any harder, but it sure as heck pulls longer. Of course, if you do the numbers, you'll find that it's also making 900 hp, at 15,000 rpm.

At The Bonneville Salt Flats

Looking at top speed, horsepower wins again, in the sense that making more torque at high rpm means you can use a stiffer gear for any given car speed, and thus have more effective torque at the drive wheels.

Finally, operating at the power peak means you are doing the absolute best you can at any given car speed, measuring torque at the drive wheels. I know I said that acceleration follows the torque curve in any given gear, but if you factor in gearing vs. car speed, the power peak is IT. A BMW example will illustrate this.

At the 4250 rpm torque peak, a 3 liter E36 M3 is doing about 57 mph in third gear, and, as mentioned previously, it will pull the hardest in that gear at that speed when you floor it, discounting wind and rolling resistance. In point of fact (and ignoring drive train power losses), the rear wheels are getting 1177 foot pounds of torque thrown at them at 57 mph (225 foot pounds, times the third gear ratio of 1.66:1, times the final drive ratio of 3.15:1), so the car will bang you back very nicely at that point, thank you very much.

However, if you were to re-gear the car so that it is at its power peak at 57 mph, you'd have to change the final drive ratio to approximately 4.45:1. With that final drive ratio installed, you'd be at 6000 rpm in third gear, where the engine is making 240 hp. Going back to our trusty formula, you can ascertain that the engine is down to 210 foot pounds of torque at that point (240 times 5252, divided by 6000), but if you do the arithmetic (210 foot pounds, times 1.66, times the 4.45), you can see that you are now getting 1551 foot pounds of torque at the rear wheels, making for a nearly 32% more satisfying belt in the back.

Any other rpm (other than the power peak) at a given car speed will net you a lower torque value at the drive wheels. This would be true of any car on the planet, so, theoretical "best" top speed will always occur when a given vehicle is operating at its power peak.

"Modernizing" The 18th Century

OK. For the final-final point (Really. I Promise.)  What if we ditched that water wheel, and bolted a 3 liter E36 M3 engine in its place? Now, no 3 liter BMW is going to be making over 2600 foot pounds of torque (except possibly for a single, glorious instant, running on nitromethane), but, assuming we needed 12 rpm for an input to the mill, we could run the BMW engine at 6000 rpm (where it's making 210 foot pounds of torque), and gear it down to a 12 rpm output, using a 500:1 gearset. Result? We'd have 105,000 foot pounds of torque to play with. We could probably twist the whole flour mill around the input shaft, if we needed to.

The Only Thing You Really Need to Know

Repeat after me: "It's better to make torque at high rpm than at low rpm, because you can take advantage of *gearing*." For any given level of torque, making it at a higher rpm means you increase horsepower - and now we all know just exactly what that means, don't we?

Thanks for your time.

March 1999 BIMMER

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Under the Hood: Do You Believe in Fate? by Greg Scott

March 1999 New Members by Barry Tarr

Keys In Hand: Start Your Engines by Fred Beck

The Best Years of Our Lives by Yale Rachlin

Horsepower and Torque: A Primer by Bruce Augenstein

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July 2002 Board Meeting Minutes August 2002 Boston Bimmer

March 2001 Board Meeting Minutes May 2001 Bimmer

1998 Income and Expense Report May 1999 BIMMER

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