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Air- and roll resistance

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.

Roll resistance is calculated as follows:

FRoll = Cr × m × g


Cr being some coefficient which is about 0.015 in our case,
m being the vehicle's mass, including passengers (about 1900 kg for the 8 series),
and g being the earth's gravitational acceleration (9.81 m/sec2).

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,
Cd being the drag coefficient,
D being the density of air (1.29 kg/m3) and
v being the velocity in m/sec.

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.

0 kph0 N =0 kg
50 kph75 N =8 kg+ roll resistance (280 N) =37 kg
100 kph299 N =30 kg+ roll resistance (280 N) =59 kg
150 kph672 N =69 kg+ roll resistance (280 N) =98 kg
200 kph1194 N =122 kg+ roll resistance (280 N) =151 kg
250 kph1866 N =190 kg+ roll resistance (280 N) =219 kg
300 kph2688 N =274 kg+ roll resistance (280 N) =303 kg
350 kph3658 N =373 kg+ roll resistance (280 N) =402 kg
400 kph4778 N =488 kg+ roll resistance (280 N) =517 kg

0 mph0 N =0 kg
35 mph94 N =10 kg+ roll resistance (280 N) =38 kg
60 mph278 N =28 kg+ roll resistance (280 N) =57 kg
80 mph495 N =50 kg+ roll resistance (280 N) =79 kg
100 mph773 N =79 kg+ roll resistance (280 N) =107 kg
120 mph1113 N =113 kg+ roll resistance (280 N) =142 kg
140 mph1515 N =154 kg+ roll resistance (280 N) =183 kg
160 mph1979 N =202 kg+ roll resistance (280 N) =230 kg
180 mph2505 N =255 kg+ roll resistance (280 N) =284 kg
200 mph3092 N =315 kg+ roll resistance (280 N) =344 kg
220 mph3742 N =381 kg+ roll resistance (280 N) =410 kg
250 mph4832 N =493 kg+ roll resistance (280 N) =521 kg

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:

P =(FRoll + FAir) × v
=(Cr × m × g + A/2 × Cd × D × v2) × v
=Cr × m × g × v + A/2 × Cd × D × v3

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!

SpeedTotal resistanceRequired power
50 kph355 N5 kW =7 hp
100 kph579 N6 kW =22 hp
150 kph952 N40 kW =54 hp
200 kph1474 N82 kW =111 hp
250 kph2146 N149 kW =202 hp
300 kph2968 N247 kW =336 hp
350 kph3938 N383 kW =520 hp
400 kph5058 N562 kW =764 hp

SpeedTotal resistanceRequired power
35 mph374 N6 kW =8 hp
60 mph558 N15 kW =20 hp
80 mph775 N28 kW =38 hp
100 mph1053 N47 kW =64 hp
120 mph1393 N75 kW =102 hp
140 mph1795 N112 kW =152 hp
160 mph2259 N162 kW =220 hp
180 mph2785 N224 kW =304 hp
200 mph3372 N301 kW =409 hp
220 mph4022 N395 kW =537 hp
250 mph5112 N571 kW =776 hp

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:

SpeedPower at
the wheels
Power at
the engine
50 kph7 hp8 hp
100 kph22 hp25 hp
150 kph54 hp64 hp
200 kph111 hp130 hp
250 kph202 hp237 hp
260 kph226 hp264 hp
270 kph250 hp293 hp
280 kph277 hp324 hp
290 kph306 hp358 hp
300 kph336 hp393 hp
310 kph368 hp431 hp
350 kph520 hp609 hp
400 kph764 hp893 hp

SpeedPower at
the wheels
Power at
the engine
35 mph8 hp9 hp
60 mph20 hp23 hp
80 mph38 hp44 hp
100 mph64 hp75 hp
120 mph102 hp119 hp
140 mph152 hp178 hp
160 mph220 hp257 hp
180 mph304 hp356 hp
200 mph409 hp479 hp
220 mph537 hp628 hp
250 mph776 hp908 hp

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
Cd: being the drag coefficient (0.29 for the 8 series)
D: Density of the air (1.29 kg/m3)
Cr: roll resistance coefficient (about 0.015)
m: Mass of the car in kilograms (about 1900 kg for an 8 series)
g: The earth's gravitational acceleration (9.81 m/sec2)
v: Velocity in m/sec (= kph / 3.6 or mph / 2.2374)
1.17: Factor to compensate the energy loss in the drivetrain (1.15 for front wheel drive)
PEngine: Engine power in W (divide by 736 to get hp)

With all the values that are constant for the 8 series you get:

PEngine = (0.387 kg/m × v3 + 280 N × v) × 1.17