When a car moves along a flat road the engine has to work to overcome two main resistances – air resistance and rolling resistance (the drag in tyres, wheel bearings etc). The top speed of the car is determined by the amount of engine power available and the size of these retarding forces. The maths to work out these equations for an actual vehicle are very simple. In order to calculate the top speed we need to work out the size of the retarding forces.
Defined as the force needed to just start a car rolling on flat ground this force is mainly a function of vehicle weight. You can measure it yourself fairly easily with a pair of bathroom scales or a spring balance. Just hold the scales vertical against the rear bumper and push until the car starts to move. You might find that once the car is rolling the force needed to keep it just moving falls slightly. This lower force is the number you are after. Copyright David Baker and Puma Race Engines
For most cars the force in pounds can be estimated as follows:
Rolling resistance (lbs) = vehicle weight (lbs) x 0.012 to 0.015 (I usually take 0.013 as a good average)
Obviously if the tyres are flat or a wheel bearing is half seized this force can alter a fair bit but we will see later that it is air resistance that is the main obstacle to top speed so even a large error in the rolling resistance calculation won’t matter much. Rolling resistance is taken to be a constant i.e. not varying with vehicle speed although this is really somewhat of a simplification. For an average car weighing 2500 lbs this force is therefore in the region of 33lbs.
This is a function of the frontal area (FA) of the car and its coefficient of drag (Cd). Often car magazine tests show these numbers and all manufacturers will have the data if they can be persuaded to release them. Most modern cars have drag coefficients between 0.3 and 0.4 with a few really streamlined ones as low as 0.28 or so. The Cd is a measure of how “slippery” a shape is as the air goes round it. Copyright David Baker and Puma Race Engines
Frontal areas tend to lie between 19 and 23 square feet for European cars (we can exclude 4 wheel drive yank tanks and similar from this exercise because who cares how fast they go anyway?)
The drag in pounds goes up with the square of speed and can be calculated from the following formula:
Air resistance (lbs) = FA x Cd x 0.00256 x speed squared (speed in mph)
Average family cars have a top speed of 120 mph or so these days so let’s have a look at the size of this force at that speed. We’ll assume the car has a frontal area of 21 square feet and a Cd of 0.35
Air resistance (lbs) = 21 x 0.35 0.00256 x 120 x 120 = 271 lbs
As you can see this is a much larger force than the rolling resistance. In fact rolling resistance only makes a major difference to vehicle dynamics at very low speeds (under 60 mph or so) and means that heavy cars use more power and therefore have poor fuel consumption at low speeds. At higher speeds the air resistance becomes paramount and so even heavy cars can show good fuel consumption if they are well streamlined.
The final step is to relate the drag figures above to the power required to overcome them. If we add rolling resistance and air resistance together we get total drag in pounds. Power required is then calculated as:
Power (bhp) = Total drag x mph / 375
We could if required split the power into the amounts needed to overcome each drag separately. The equations would then become:
Power to overcome rolling resistance = weight x 0.013 x mph / 375
Power to overcome air drag = FA x Cd x 0.00256 x mph cubed / 375
Hopefully something of major importance should be clear from the above. We already know that it is air resistance that is the major element in this equation and we can see that we need to incorporate mph cubed in the power equation for air drag. As a simplification therefore we can say that power required is closely related to mph cubed – i.e. to double the speed of a vehicle we need 8 times the engine power. Alternatively we can express this as top speed is a function of the cube root of engine power. This means that engine modifications will have a much greater impact on acceleration (which is directly related to power) than top speed. Also that is why an old engine which is down on power might accelerate slowly but still have close to its original top speed. So next time your mate tells you in the pub that he put a K&N air filter in his car and the top speed went up by 10 mph you can explain exactly why that isn’t going to be very likely.
Let’s say we want to increase the top speed of a car by 10% – how much extra power do we need? Increase in power required is related to increase in speed cubed – i.e. to 1.10 cubed = 1.33. So we need about 33% extra power to achieve 10% increase in top speed. Copyright David Baker and Puma Race Engines
Alternatively let’s say we tune an engine and achieve 10% extra power – how much will top speed go up by?. Speed is proportional to the cube root of power – i.e. to the cube root of 1.10 = 1.03. So speed will only increase by about 3%.
What this all means for you hopefuls who bolt on go faster goodies like chips, exhausts and the like. You will see hardly any increase in top speed. To get significant increases in top speed requires serious engine surgery.
The power calculated above is power delivered to the wheels and NOT flywheel power – i.e. we need to allow for transmission losses to get back to engine power required. Transmission losses will be the subject of another article but for brevity we can take the following as good assumptions. Front wheel drive cars will lose 15% of the engine power as transmission and tyre losses and rear wheel drive cars will lose 17%. This assumes manual gearboxes and I could care less how fast autos or 4 wheel drive cars go !
So divide by 0.85 or 0.83 as appropriate to convert from wheel bhp to flywheel bhp.
Table for an Average Car
The maths above is so simple it should only take a few minutes to put together a spreadsheet to work out the power required at any speed for your own car if you have the weight, Cd and fA. To give an idea of power levels required for an average car I have put together the table below. It assumes a car weighing 2500 lbs with driver, 21 square feet FA, 0.35 Cd and front wheel drive so transmission losses are 15% of the flywheel power. The other thing that can be estimated from this power requirement is the fuel consumption. A well designed petrol engine running on a lean cruise fuel/air mixture should have a specific fuel consumption of between 0.50 and 0.55 lbs weight of fuel per horsepower generated per hour. At very low throttle openings the consumption will be a tad worse because the cylinders aren’t filling completely which hurts efficiency. This means that small engines will always produce better economy than large ones, partly because at a given road speed they are operating closer to full throttle and partly because their smaller (and/or fewer number of) cylinders will waste less power, and fuel, in overcoming internal friction. An imperial gallon of fuel weighs about 7.5 lbs (those puny American gallons are about 20% smaller). From the speed and flywheel power requirement we can now calculate the expected fuel use per hour and therefore the mpg. In practise the estimated mpg for very low speeds will be optimistic because most engines aren’t working very efficiently at such small throttle openings. The same will apply to the mpg for very high speeds because most engines are getting close enough to full throttle to require a richer full power fuel mixture up there. The table is therefore only a rough guide of course and obviously the real world steady speed economy is different for every car. It hopefully makes the point though that the best way of improving your mpg is just to go slower! Copyright David Baker and Puma Race Engines
|SPEED (MPH)||FLYWHEEL POWER|
In the Real World
People are forever claiming how fast their cars go based on speedo readings. Most speedos read way fast – the law allows a 10% error and most manufacturers set speedos somewhat fast so that you won’t get get done for speeding and then sue them. I have tested a number of cars and the average error is about 5% to 7% fast – i.e. when the speedo shows 100 mph you are really doing about 93 to 95 mph. Many magazines do a speedo accuracy test as part of their report – have a look at Auto car tests or similar. Trying to calculate engine power based on speedo readings is a waste of time unless you know the speedo error. Small errors in measured top speed lead to much larger errors in calculated horsepower due to the cube law above. It is easy enough to find out what this speedo error is – any rolling road should have a calibrated roller which you can use to test speedo mph against true mph – ask the guy to do this for you next time you have your car set up.
Another way is to time the car against the motorway marker posts which are 100 metres apart. I find this is a perfectly accurate way of checking things if you hold a steady speed for half a mile or so (8 posts). You can do the maths yourself though.
Also bear in mind that if the wheels and tyres are non standard sizes then the speedo will not read the same either. Even tyre wear makes a difference with new tyres reading about 2% slower on the speedo than worn out tyres due to the change in diameter due to the tread depth. I had to recalibrate my speedo after fitting new tyres to get complete accuracy again.
The other factor that always comes into play when anyone tries to find out the top speed of their car is a psychological one – you tend to ignore tests that show a low speed and only take any notice of ones where the car goes well. This tends to mean that the average person remembers the speed when the slope was slightly downhill and there was a tailwind – any other occasion gets forgotten as being a bad test. This isn’t helped by the fact that a level road tends to appear slightly uphill to most people so when they pick a stretch of motorway to have a thrash on it is often a downhill stretch.
To actually achieve the theoretical top speed that a car with a given drag and a given engine bhp should be capable of does of course require that the car be geared correctly. In other words the engine needs to be at the rpm at which it produces maximum power at the theoretical top speed. However, having said that, it is all much less critical than people tend to realize. Because top speed is fairly insensitive to engine power as shown above, there will be only be a small decrease in speed for a relatively large drop in engine power. Most engines have a broad spread of power around peak rpm – for 500 rpm either side of peak the power falls very little. So provided that the gearing is close enough to fall into this 1000 rpm band at the theoretical top speed the car will usually achieve it or very close to it.
So for a car producing peak power at 6,000 rpm you will reach almost the same top speed anywhere in the 5,500 to 6,500 rpm area. In other words there is something like a 15% spread of possible gearing in top gear that will do the job. That is also why so many cars reach very similar top speeds in both 4th and 5th gear – it’s just you get there a bit faster in 4th. For very ‘peaky’ highly tuned engines where power rises and then falls very abruptly, the choice of gearing becomes more critical. This is only of real concern to race type engines though.
As an aside the whole business of gearbox ratios for road cars is very much over-rated. There is a current fad for expensive 6 gear conversions which make next to no difference to the overall performance of the car. I might write an article on it one day but for now I will content myself with advising anyone considering such a mod to forget it. Spend the money on serious engine work instead. To go into the maths of why such conversions don’t show much benefit is too time consuming for now.
It takes a lot of power to increase the top speed of a car significantly. The cube rule above is a good guide. You can’t just change the gearing as so many people seem to think. If you look through a few magazine tests you will see that most cars show top speeds that follow very closely the figures shown in the table above. One more thing to bear in mind is the way magazines test cars. Often they do not have long enough test straights to reach the absolute top speed and when banked tracks are used these also tend to scrub off a few mph. Most of this applies only to very fast cars – i.e. 150 mph plus machines. You have to read between the lines to see whether the tester thinks that the top speed shown is a fair representation of what the car in question could do.