Thanks To David
Baker At www.PumaRacing.co.uk For This Information
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.
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.
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.
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.
IMPORTANT NOTE:
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.
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 out together a 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.
|
SPEED (MPH) |
FLYWHEEL POWER |
|
30 |
5 |
|
60 |
19 |
|
80 |
38 |
|
100 |
69 |
|
110 |
90 |
|
120 |
114 |
|
130 |
143 |
|
140 |
176 |
|
150 |
215 |
|
160 |
258 |
|
170 |
307 |
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 Autocar 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.