• Technical data
The two main forces that slow a vehicle down are air and roll resistance. This article explains the math behind it, shows how much power is necessary to achieve a certain speed, and how much more power is needed to go even faster.
Air- and roll resistance
Roll resistance is calculated as follows:
FRoll = Cr × m × g
Cr being some coefficient which is about 0.015 in our case,
So the roll resistance of an 8 series is 0.015 × 1900 kg × 9.81 m/sec2 = 280 N
That number, as you can see, does not depend on the speed but on the car's weight only. So it will become more and more insignificant the faster the car will be. Nevertheless the engine always has to overcome a force of 280 N = 28.5 kg to keep the 8 moving.
Next comes the formula for air resistance:
FAir = A/2 × Cd × D × v2
A being the frontal area of the car in m2,
Now speed comes into the equation. As many values become constants if the formula is applied to one car only, we will now simplify this: The BMW 8 series has a frontal area of 2.07 m2. This is compensated by the very low drag coefficient (Cd) of 0.29 which leaves us with an air resistance area of 2.07 m2 × 0.29 = 0.6 m2. Here you can see how Cd influences the 'size' (from the point of view of the airflow) of the car. The lower Cd the easier it is for the air to pass the obstacle. The car becomes more streamlined. The 850CSi's Cd is 0.31 but it has a different frontal area (unknown to me - lowered chassis, different mirrors).
Now half of the air resistance area has to be multiplied by the density of our atmosphere: (0.6 m2 × 1.29 kg/m3) / 2 = 0.387 kg/m.
The force of the air resistance can now be calculates very easily: FAir = 0.387 kg/m × v2. Because speed is squared in the equation, extreme forces can be expected at high velocities.
That's an interesting table but it doesn't help much. What is missing is the power in Watts that is necessary to achieve those speeds. It is calculated the following way:
Here you can see that the needed power rises to the power of three which means eight times as much power for twice the speed and 27 times the power for triple speed!
From 250 kph / 160 mph on the required power rises very quickly. Now it becomes clear why Bugatti needs 1000 hp in its 16.4 Veyron in order to pass 400 kph / 250 mph as planned. But remember, those values in the tables here are valid only for the BMW 8 series or cars with identical aerodynamics.
Still we are not finished because the calculated horsepower must be at the wheels, not at the engine! That means the engine power has to be even higher in order to compensate the energy loss of gearbox and drivetrain. This loss is about 17% with rear wheel driven and 15% with front wheel driven cars. So for the RWD 8 series you end up with the following values:
The factors for drivetrain loss are guessed - somewhat. It seemed to be a reasonable average when looking up this data on the internet and although it seems to be a bit on the high side, the power and top speed of the Alpina B12 5.7 coupé as well as my personal experience seem to confirm the choice.
It goes without saying that the transmission must be carefully chosen/developed so that the top speed will be achieved at the engine's power peak. The 380 hp of a stock 850CSi will never get you near 300 kph because the engine develops them at 5300 rpm and 250 kph. Beyond that power drohp off again and reduces top speed. So if you keep the standard gearbox you will need some engine tuning to get a higher top speed. Which brings us to the next point.
Because of the power of three in our equation, the engine has to undergo extensive surgery to provide a noticeable change in top speed. To be only ten percent faster requires a third more engine power (1.13 = 1.33), and with the maximum of 10% power increase that common tuning chips for normally aspirated engines provide, only a 3% higher top speed is possible (cubic root of 1.1). With previously possible 290 kph it will bring you very close to the magic 300, but weaker cars will get from, let's say 160 kph before to only 165 kph afterwards which isn't even worth mentionning.
But now again the generic formula which calculates the reqired engine power at a given speed:
PEngine = ((A/2 × Cd × D × v3) + (Cr × m × g × v)) × 1.17
A: being the frontal area in m2
With all the values that are constant for the 8 series you get:
PEngine = (0.387 kg/m × v3 + 280 N × v) × 1.17