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# How do horsepower relate to the real power of a car?

If you, as the proud owner of a powerful car, say that it has 380 hp, you'll get many Oohs and Aahs but if you say that it's got 550 Nm of torque you'll hear just a '...?!' - if at all.

No one really knows about torque although it is the important unit. That's why this article has been written. Have fun with it:

## Through force, work, distance and time to power

In order to move a weight over a distance you need a force. If the weight is glued to the ground and therefore doesn't move, no work will be done (at least not in a physical sense). So work is the movement of a weight across a distance (by applying a force). If you take the time into account now, you can calculate the power. Power is the duration of work, the time it takes to move a weight a certain distance. The more weight you move in a period of time the more power (ability to do work over time) you 'have'.

Let's play engine and stick a weight of 1N (= 0.102 kg) to the end of a stick that is 1m long and that you are trying to hold horizontally by grasping it at it's opposite end. The (twisting) force that is necessary is 1N × 1m = 1Nm.
Now imagine rotating that stick with the weight at the other end around 360° against a resistance of 1N. The work done will be 1N × 6.2832m = 6.2832Nm (6.2832m is the perimeter of the circle the weight at the end of the 1m-long stick moves along. Perimenter = Diameter × Pi = 2 × 1m × 3.14159... = 6.2832m).
The resistance of 1N is there to simulate the gravitational force you would have to deal with when moving the weight vertically. So if you'd lift a weight of 1N 6.2832m vertically you had done the same work of 6.2832Nm.

But what about the power to do so? Now time has to be involved, as you can see by looking at the definition of a horsepower: 1hp = 75kp × m/s. 1kp (kilopond) is equivalent to 9.80665N, so 1hp = 735.5N × m/s which means that when lifting a weight of 735.5N (75 kg) one meter every second that's the power of one horsepower. Lifting twice the weight in the same time will double the power (2 hp) as will lifting the same weight in half the time and so on. More work (in the eyes of a physicist!) would not be done though, because the formula 'force × distance' does not care about time.

In order to return to cars and engines, let's take the BMW 850CSi for example. The engine of this car will produce 380 hp at 5300 rpm. So you could lift a weight of 380 × 75 kg = 28500 kg one meter every second!

The unit of horsepower is N × m/s. In our example with the weight and the stick the unit of the work was N × m (Nm). If we multiply this by the rpm (unit 1/s) we get the unit of power. If you're looking somewhat sceptical now, that's a common reaction. So let's calculate at which rpm 1 Nm of work equals 1 hp:

1 hp = 75 kp × m/s = 735.5 N × m/s

Now we set this equal to our torque multipiled by the still unknown rpm and get

735.5 N × m/s = 6.2832 Nm × n 1/s

Now we can compute n:

n = 117.058

An engine which has 1 Nm of torque at 117.058 revolutions per second (that is 7023.5 revolutions per minute) then has the power of 1 hp. With that information it is easy to get the following formula to calculate the power:

 hp = Nm of torque × rpm 7023.5

So no one measures the power of an engine directly. A dynamometer only measures torque which will be, related to the engine speed, converted into horsepower by the previous formula! That means that horsepower is more or less an imaginary unit and not something you could see, feel or measure. That may startle one or the other but that's really the way it is!

## Torque and acceleration

How fast a car accelerates depends exclusively on torque. It's the Newton-meters you feel in your back. Any car accelerates in any gear just as hard as the torque curve dictates. At the torque peak the car will accelerate most, at higher and lower rpm less. © BMW

That does mean that it makes absolutely no difference where the torque peak lies. It won't affect the amount of acceleration if it is at 2000 rpm instead of 4000 rpm, although the power would double at 4000 rpm (see the formula).

Simple math proves this. The engine's torque is fed through the gearbox and the differential to the wheels so that a certain amount of Newton-Meters arrives there. Taking the CSi for example at 4000 rpm in first gear this would be 550 Nm × 4.254 × 2.93 / 2 = 3428 Nm that arrive at every rear wheel (loss in the drivetrain left out here). The first factor represents the first gear in the gearbox, the second the rear differential. And because the torque is fed to two wheels there is the division by two.

Calculating the force that moves the car forward is easy now: divide by the radius of a wheel because it is the lever arm. The rear tyre of an 8 series has almost exactly two meters circumference and thus a radius of 2m / 2π = 0.318m. Now 3428Nm / 0.318m = 10780N which means that every rear wheel moves the car forward with 10780 Newton which totals to 10780 × 2 = 21560N (which equals 2200kg!).

As you can see, the connection of torque and thrust couldn't be more direct. More torque means more thrust - independent of rpm value and independent of power at that rpm!

Horsepower are therefore completely unimportant for acceleration, aren't they...?

The power rises until the torque drops faster than rpm rise, so power does somehow tell you how fast a car accelerates. Why?

The secret is the moment you have to shift up, the moment you have more torque available in the next higher gear and therefore would be able to accelerate harder. So the higher the rpm before you have to shift the longer you can stay in a gear and don't have to sacrifice engine power for speed (because of the higher gear ratio).

Here is a table which shows on a BMW 850CSi when and why you have to shift. The table shows the torque (in relation to rpm) that is available directly at the engine and, much more important, the torque available behind the gearbox which is passed to the wheels (the final drive hasn't been taken into account here because it's a constant factor). In addition you can see how the engine speed changes when shifting.

 rpm Torqueengine 1st gear  (4.254 : 1) 2nd gear  (2.534 : 1) 3rd gear  (1.682 : 1) 4th gear  (1.235 : 1) 5th gear  (1.000 : 1) 6th gear  (0.831 : 1) Torque rpm2nd gear Torque rpm3rd gear Torque rpm4th gear Torque rpm5th Torque rpm6th gear Torque 1500 420 Nm 1768 Nm => 893 1064 Nm => 995 706 Nm => 1101 518 Nm => 1214 420 Nm => 1246 349 Nm 2000 440 Nm 1871 Nm => 1191 1114 Nm => 1327 740 Nm => 1468 543 Nm => 1619 440 Nm => 1661 365 Nm 2500 470 Nm 1999 Nm => 1489 1190 Nm => 1659 790 Nm => 1835 580 Nm => 2024 470 Nm => 2077 390 Nm 3000 500 Nm 2127 Nm => 1787 1267 Nm => 1991 841 Nm => 2202 617 Nm => 2429 500 Nm => 2493 415 Nm 3500 530 Nm 2254 Nm => 2084 1343 Nm => 2323 891 Nm => 2570 654 Nm => 2834 530 Nm => 2908 440 Nm 4000 550 Nm 2339 Nm => 2382 1393 Nm => 2655 925 Nm => 2937 679 Nm => 3238 550 Nm => 3324 457 Nm 4500 540 Nm 2297 Nm => 2680 1368 Nm => 2987 908 Nm => 3304 666 Nm => 3643 540 Nm => 3739 448 Nm 5000 530 Nm 2254 Nm => 2978 1343 Nm => 3318 891 Nm => 3671 654 Nm => 4048 530 Nm => 4155 440 Nm 5500 470 Nm 1999 Nm => 3276 1190 Nm => 3650 790 Nm => 3671 580 Nm => 4453 470 Nm => 4570 390 Nm 6000 400 Nm 1701 Nm => 3574 1013 Nm => 3982 672 Nm => 4405 494 Nm => 4858 400 Nm => 4986 332 Nm

As you can see, the torque at 6000 rpm behind the gearbox in 1st and 2nd gear is much higher than the torque in the next higher gear at any rpm. That means that in order to accelerate best you not only can rev the engine to over 6000 rpm in the first two gears, you have to do it (the engine of the 850CSi allows speeds of up to 6400 shortly).

In 1st gear, at 6000 rpm, you have 1701 Nm torque behind the gearbox. If you then shift into second, the engine speed will drop to about 3600 rpm and the torque to about 1350 Nm.
In 4th gear you have 494 Nm of torque at 6000 rpm (because of the ratio of 1.235 instead of 4.254 like in 1st). Shifting into fifth would result in 530 Nm of torque at 4858 rpm. So there is absolutely no point in revving the engine up to 6000 in fourth gear.

## rpm and torque

As the table shows, much power is 'wasted' because of the ratio in high gears. That's why it is most important to stay in any gear as long as possible, and to do that you need the torque peak to be at high rpms.

Another example: Take two 850CSi and replace one engine with one who has the same torque but whose torque peak is at 2000 rpm istead of 4000. When racing each other from a dead stop, the modified CSi will be faster off the line because its 550 Nm are available at 2000 rpm already, whrere the normal one has only 440 Nm.
After its peak the amount of torque rapidly falls and around 3000 rpm the driver of the modified CSi will have to shift into a higher gear because there he has more torque available. So engine power is sacrificed for speed. And now it's the stock CSi that blasts away because his opponent suddenly has just more than half the torque than in first gear, whereas he can go on in 1st until 6400 rpm. When the stock CSi shifts into second gear, the modified one will almost have to shift into third.

Did it look as if the modified car would win at first, everyone will see that it is better to have the torque peak at high rpm as soon as the modified car falls back rapidly. The BMW Formula-1 engine has about the same torque as the engine of the E46 M3. The M3 has its torque peak at 5000 rpm whereas the Formula-1 engine has its at about 16000. Let's guess who can have the accelerator pedal floored for a longer time...

Or another little example:
Maximum torque at high rpm = good = petrol engine
Maximum torque at low rpm = bad = diesel engine
What good is all the torque if you almost have to shift into 6th gear at 65 mph?

Now, can you apply this in real life? Well, more or less because there are as good as no normally aspirated diesel engines available today (only turbo diesels). This might have led you to think that the upper statement is wrong. But comparing normally aspirated engines and engines using forced induction (by turbo- or supercharger) is like comparing apples and oranges. So petrol engine = good, diesel engine = bad refers of course to comparisons within those two concepts.

If you say 'Diesel' nowadays, you always mean turbodiesel, without fully realizing it. That makes diesel engines look better but compared to an equally turbocharged petrol engine they still do not stand a chance. Yes, of course is a diesel more economical but this site is about emotions, not about common sense.

So what's so special about a turbo engine? It's the torque curve. As was said before, it is torque that is responsible for acceleration. The more torque the quicker the car accelerates. It has been stated why it's better to have the torque peak at high revs, too. But why shouldn't we have high torque at low revs as well?

Because the characteristics of normally aspirated engines don't allow us to. Torque rises up to a certain point and then falls off again. So the goal of engineers designing normally aspirated engines is to move that point to the highest possible revs. There are no such problems with a turbocharged engine. The turbocharger forces the air into the cylinders which the pistons would have to suck into instead while moving downwards. The engine management now is controlling the amount of air flowing into the cylinders, depending on rev count. So the engine has maximum torque almost everytime.

In order to visualize both concepts, here are power- and torque diagrams of two representative cars:  © sport auto Lamborghini Diablo GT Porsche 996 Turbo

The Lambo shows the characteristics of a normally aspirated engine very clearly, a torque peak, whereas the Porsche has a typical torque plateau (560 Nm from 2700 to 4600 rpm). While the Diablo reaches its maximum accerleration only around 5500 rpm the Porsche does it from 2700 rpm until 4600 rpm! But it also shows a turbo-lag. This is the rev range before the turbocharger starts working. Around 1000 revs the Porsche power plant is pathetic (200 Nm) but at 2000 rpm there is already two and a half times as much torque (500 Nm). Today's turbos engage quite softly but earlier there was a radical increase of torque at that point.

Let's imagine we would bolt turbos to the Lamborghini engine which would then build up maximum torque from 2000 rpm on as well so that we had 630 Nm from 2000 until 5500 rpm. That would give us brutal acceleration but - and that's interesting - wouldn't result in a higher horsepower figure! Check it...

That's why, despite their low power, turbo diesels accelerate surprisingly well. Because of the turbocharger they build up torque early and keep it up over a wide rev range. Horsepower, as you can see, are not very descriptive.

But you cannot even rely on the torque figure all the time. Let's get back to the M3 and Formula-1 example. Both engines have roughly the same torque, the M3 at 5000, the F1 at 16000 rpm. If you put the F1 engine into the M3, the acceleration would be about the same but you could reach higher speeds with the new engine. You could go three times as fast: about 800 kph (500 mph) - theoretically of course, as air resistance would slow you down much much earlier. Okay, that wouldn't be very intelligent, so let's put a different transmission into our car so that at maximum revs it reaches the same speed as with the original M3 engine. But that does mean that the gear ratio is much shorter (about a third) than before and that suddenly three times as much torque will arrive at the wheels and our car accelerates three times faster.

In addition the gearbox has to be specially designed for the engine characteristics. What good is it when you shift into the next higher gear at the limiter and revs drop to regions with not enough torque? So a gearbox that fits the engine characteristics well is very important, too.

# Conclusion

In order to know how a car accelerates, you have to have a look at the torque curve, the gearbox and take the weight of the car into account. Then - and only then can you expect a correct result. Knowing the horsepower figure is quite useless in such a case.