Physics/Horsepower/Speed question

This question is for all of our physics buffs out there!

Having a decent biology background, but lacking a physics education, I have a few questions. Since I don’t know formulae, or laws or the equations, I’m just going on what I know until I can get some feedback from y’all!

Say you drove a 2000 BMW 328 ci. It has 197Hp. The top speed is rated at 142(governed, so for all intents and purposes, let’s say that’s it’s top speed).

If I wanted to go faster, by say, 10 miles an hour, how many more HP would it take?

From what I know, when it comes to electronics and power(watts), and sound, it takes approximately 3 times as many watts to achieve a 3db increase.

Since the only true difference here is the type of power, does the same hold true for mechanical power?

AcK!! Please help me here…this took up over an hour and a half discussion today, and I’m dying for an answer!

Thanks, guys :slight_smile:


You wimp…I would’ve used a 1970 Hemi Roadrunner Superbird ( ) with 425+ hp and top speed of around 240mph as an example :D, but whatever the case, you’re thinking along the right lines: as you go faster, an increase in speed (i.e. 190-200mph) takes a lot more power than the same increase at lower speed (i.e. 10-20mph). It’s probably something like “doubling [speed] requires squaring [horsepower]”…If nobody else says it first, I’ll look up and post the equation tomorrow.

Sorry, we were discussing different models of the BMW(I.E.-the 323, the 328 and the new 330). They differ by about 10-15 hp per automobile, and I said that even though there was more hp, it wouldn’t affect top speeds and quickness to a terribly noticeable degree.

Sorry I didn’t go for the musclecar analogy, but we were dealing with real-world situations.


Very tough question. I can’t claim to be a physics buff but I know a few things about cars. With cars everything is a trade-off. If all you want is a higher top speed you could just put bigger wheels on the thing (I know a guy who did this to his Camaro) or change the gear ratios in the tranny. Think of it this way, the very first time you ever rode a 15-speed bike, weren’t you amazed at how fast you could go in top gear on the straight and level compared to your old one speed. You didn’t become more powerful you just applied your power more effectively. It is possible, though, that these simple methods won’t help on a car as advanced as your BMW (i.e. even if you had a 50 speed bike you wouldn’t be able to go 152mph). This car may be using its available horsepower as effectively as possible.
However, if you still insist on achieving higher top speed through more horses, then you could modify the engine to produce its peak torque at a higher RPM and thus have more horspower (HP=torque X RPM) and thus more “high end power”. This would mean losing some “low end power” and thus being slower off the line (i.e. 0-60 time would suffer). You could also put a larger displacement engine in and have more torque (and more horsepower) at all RPM’s. Or you could put in a turbo charger and have more high end power with little or no drop in the low end. In order to get a feel for how a car’s performance will be effected by an increase in power you have to specify exactly how those horses are to be gained. You also have to specify the exact torque curve of the BMW’s current engine and the RPM’s the engine is revving at when the vehicle hits top speed. This info will help determine what the easiest method would be to achieve a top speed of 152.

On a more physics related note, I know that the total energy expended to travel through still air (i.e. no head wind, no tail wind) increases exponentially with speed. That is, if it takes X total power output to achieve speed Y then it takes more than 2X power output to achieve speed 2Y. In your exapmle you achieved 142mph with 197HP, I can assure you that it will take much more than 394HP to achieve 284mph in the same car. The exact amount of power you would need depends heavily on the aerodynamics of the car in question and I never understood aerodynamics to begin with. If anyone else out there could help us out I would be interested…

Thanks for responding Saruman…

We’re not talking about modifications to make it faster, we’re talking about what it takes in the theoretical world to GOfaster.

FWIW, I think you’re on to it at the bottom of your response.


One thing I do understand is aerodynamics having just spent four years studying aeronautics (the biggest mistake of my life).

Drag on a car will be similar to drag on any other subsonic vehicle.

(I’ve quoted this equation in another thread before)

So essentially, drag go’s up with the square of speed.

Cd (drag coefficient) depends on how streamlined the car is. A well streamlined car will have a long tapering back end (like the solar powered cars). This will stop ‘separation’ of the flow over the rear half which contributes to the pressure drag. The trade off is that the bigger the surface area the more skin friction drag (but this is small comparatively). What this amounts to is that cars that are long thin and pointy at the front are only like that to make them look fast. An interesting anecdote from one of my lecturers: After testing the Porsche 928s model in a wind tunnel, the engineers, just for a laugh, put it in backwards to show that the car is more streamlined when you’re driving in reverse. I think it was the 928, could have been the 924, they both look like an aerofoil going backwards.

As far as engined are concerned, I think I’m right in saying that you can extract more theoretical power if you run the engine hotter (Carnot efficiency?). This is certainly true of jet engines which they run as hot as possible. However, jet engines are much more efficient that IC engines, which may suffer from other factors that I don’t know about.

Well, if you’re talking theoretically, there should be no top speed: in an ideal situation, having a positive horsepower means that you are constantly adding energy, and so you should be constantly accelerating. It is only stuff like friction that keeps this from happening. Also, you can’t just increase the horsepower of a car and be done with it. Increasing the power usually means a bigger engine, and is often followed by adding a larger gas tank, getting stronger tires, etc., all of which increase the wright of the car. This more of an engineering problem than a physics one.

Oh, and as far as needing an exponential amount of power to get a linear increase in decibels: a decibel is defined in terms of the logarithm of the power. An increase from 20 db to 30 is actually much larger than an increase from 10 to 20, but our ears perceive them to be about the same, so the scale is set up to reflect this.

Quick answer: The car likely can go 10 mph faster with no increase in horsepower. If it weren’t for this governor thingie in the way.

The governor (it used to be a mechanical device) is one of the parameters programmed in the onboard engine management system. Essentially, it starts to throttle back the engine power as the vehicle speed approaches the programmed limit, which in this case is 142mph. More horsepower won’t get you past this artificial barrier.

So, if you could hack into the car’s firmware, you could find this top speed parameter and set it higher. Some people make a living doing this, believe it or not. Heard of performance chips? Any respectable seller of aftermarket BMW parts sells these performance chips, and they alter lots of other engine performance parameters as well. Shift points, air/fuel ratios, spark timing, and such. Generally you get an increace in hp, but that’s not really my point.

'Nother quick answer: you only need to raise the hp at the top end of the rpm range.

Back to the problem. Let’s assume that the top speed was limited at something like 185mph. But the real-life, good day with primo fuel, low humidity, new air filter and everything - top speed was 142 miles per hour. At this point you’ve reached an equilibrium - i.e., the forces pushing the car forward (engine power)are exactly equal to the forces acting against the car, mostly wind resistance.
A few real pieces of data and some bold assumptions can get us going, but the answer is still elusive. One trouble is wind resistance doesn’t act linearly, but it is a squared function IIRC. And another little twist involves the hp/rpm curve, and the rpm/speed curve.

For any given gear, the vehicle speed increases linearly with engine rpm; e.g. 2nd gear at 4000 rpm = 40 mph and 2nd gear at 6000 rpm = 60mph (my old Z-car would do this). Hp though, is generally linear only in the early and mid rpm range, and starts to flatten out some before finally starting to DROP at the higher end of the rpm scale. So really, you get the hp peak well before your engine rpm is maxxed out, and hp falls with increasing rpm.

So then, drive her up to 142mph and check the rpm reading. (in my old Z-car, this would be 7100rpm, - in theory anyway) Then look up the engine hp on the hp/rpm chart at 7100 rpm. Probably a good bit less than the 197hp spec.

Now is when I run out of real answers. I’d try plotting curves of hp/rpm and speed/rpm on the same chart and see what happens at 7100rpm. Then I’d extrapolate the speed/rpm curve another 10mph (7600rpm, in theory, in the Z)and follow the hp curve another 500rpm. How much did the hp drop over this interval?

I would boldly assume this is the hp deficit you seek. A first aproximation anyway.

BTW, I was going to be the first response to this post, but by time I got done, several others had already posted. Sorry if lots of this is repetitive.

Suspicious Mind is correct, but I think the punchline could stand some amplification.

Power = Force * Velocity

The dominant force here is aerodynamic drag. Sure, there is internal friction in the engine, rolling resistance in the tires, and other stuff. Pretend for a moment that those are negligible, and deal just with the aerodynamics.

That equation SM quoted is important:

Drag = 1/2 * air density * frontal area * Cd * U^2.

1/2, the air density, and the frontal area of the car will all be the same, regardless of speed. I think that, for much of the range of speeds (really, Reynolds numbers) that we’re talking about for a car, the drag coefficient (Cd) is pretty much constant. That means that all of that boils down to:

Drag = Constant * U^2

The Force mentioned in the first equation is nothing more (given the assumption above that nothing else matters :)) than the drag.

Power = Constant * U^3

Under those simplifying assumptions, then, power required goes as velocity cubed.

If 142 mph is your top speed (if it were limited by power, not by the governor), than a 10 mph increase in top speed would require roughly a 23% increase in power.

great answer brad_d, you get an A :slight_smile:

This is not “what can Sam do to make his BMW faster”(I’ll just call Dinan for that):), it is “In a perfect controlled environment, how many HP’s would it take to squeeze 10 extra miles out of a car engine that it totally, tapped out”. Once again, let’s assume that 142 is the top speed, and there is no governor in the system.

So, the BMW 3 series has a Cd of .30 and a frontal area of approximately 25.89 ft2.

Simple, I’m assuming the equation you and Brad speak of is F=1/2CdApv^2 =

F=1/2(.30)(25.89^2)(.080lb-mass/ft^3)142^2= What?

Oy…I think I fried something! :slight_smile: what does this tell me? I can’t make much sense of it, let alone do the math.

What is lb-mass/ft^3?

Now I know why I didn’t do physics…


a 23% increase in power would translate into 197HP X 1.23= 242HP. An increase of 45HP would be required to gain a measly 10mph more on the top speed. To double the top speed to 284mph using brad_d’s estimate would require 8 times the power (a wopping 1576HP). IIRC from reading Car and Driver, formula one race cars can go 250mph or more on something like 1400HP (they have far less drag, but this seems like a ballpark figure).

Well, even thought the answer was there and I was too engrossed trying to figure this stuff out to see it, I hope I was nearing that conclusion with all the numbers I put up.

And thanks for all of your responses. You’ve proved my point that you have to make quite a substantial difference in hp to get more speed out of near-identical cars.

Just out of curiosity, someone mentioned that this was also an exponential function. Does that mean that to go from 10-20mph more to 20-30 mph more you’d have to, like square it or something?


It takes twice as many watts to achieve a 3 dB increase. But the dB (little d, capital B) scale is logarithmic, unlike units of horsepower or speed.

A 3 dB increase in power is twice as much, and a 10 dB increase is ten times as much. There’s no reason that you couldn’t use the dB scale for engine power. Increasing your engine power by 26% would be a 1 dB increase.

The reason the dB scale is used for sound is that out ears hear this way. A 1 dB change is about the smallest difference in sound level that a human can differentiate.