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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

with

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

with

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 kph 0 N = 0 kg 50 kph 75 N = 8 kg + roll resistance (280 N) = 37 kg 100 kph 299 N = 30 kg + roll resistance (280 N) = 59 kg 150 kph 672 N = 69 kg + roll resistance (280 N) = 98 kg 200 kph 1194 N = 122 kg + roll resistance (280 N) = 151 kg 250 kph 1866 N = 190 kg + roll resistance (280 N) = 219 kg 300 kph 2688 N = 274 kg + roll resistance (280 N) = 303 kg 350 kph 3658 N = 373 kg + roll resistance (280 N) = 402 kg 400 kph 4778 N = 488 kg + roll resistance (280 N) = 517 kg

 0 mph 0 N = 0 kg 35 mph 94 N = 10 kg + roll resistance (280 N) = 38 kg 60 mph 278 N = 28 kg + roll resistance (280 N) = 57 kg 80 mph 495 N = 50 kg + roll resistance (280 N) = 79 kg 100 mph 773 N = 79 kg + roll resistance (280 N) = 107 kg 120 mph 1113 N = 113 kg + roll resistance (280 N) = 142 kg 140 mph 1515 N = 154 kg + roll resistance (280 N) = 183 kg 160 mph 1979 N = 202 kg + roll resistance (280 N) = 230 kg 180 mph 2505 N = 255 kg + roll resistance (280 N) = 284 kg 200 mph 3092 N = 315 kg + roll resistance (280 N) = 344 kg 220 mph 3742 N = 381 kg + roll resistance (280 N) = 410 kg 250 mph 4832 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!

 Speed Total resistance Required power 50 kph 355 N 5 kW = 7 hp 100 kph 579 N 6 kW = 22 hp 150 kph 952 N 40 kW = 54 hp 200 kph 1474 N 82 kW = 111 hp 250 kph 2146 N 149 kW = 202 hp 300 kph 2968 N 247 kW = 336 hp 350 kph 3938 N 383 kW = 520 hp 400 kph 5058 N 562 kW = 764 hp

 Speed Total resistance Required power 35 mph 374 N 6 kW = 8 hp 60 mph 558 N 15 kW = 20 hp 80 mph 775 N 28 kW = 38 hp 100 mph 1053 N 47 kW = 64 hp 120 mph 1393 N 75 kW = 102 hp 140 mph 1795 N 112 kW = 152 hp 160 mph 2259 N 162 kW = 220 hp 180 mph 2785 N 224 kW = 304 hp 200 mph 3372 N 301 kW = 409 hp 220 mph 4022 N 395 kW = 537 hp 250 mph 5112 N 571 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:

 Speed Power atthe wheels Power atthe engine 50 kph 7 hp 8 hp 100 kph 22 hp 25 hp 150 kph 54 hp 64 hp 200 kph 111 hp 130 hp 250 kph 202 hp 237 hp 260 kph 226 hp 264 hp 270 kph 250 hp 293 hp 280 kph 277 hp 324 hp 290 kph 306 hp 358 hp 300 kph 336 hp 393 hp 310 kph 368 hp 431 hp 350 kph 520 hp 609 hp 400 kph 764 hp 893 hp

 Speed Power atthe wheels Power atthe engine 35 mph 8 hp 9 hp 60 mph 20 hp 23 hp 80 mph 38 hp 44 hp 100 mph 64 hp 75 hp 120 mph 102 hp 119 hp 140 mph 152 hp 178 hp 160 mph 220 hp 257 hp 180 mph 304 hp 356 hp 200 mph 409 hp 479 hp 220 mph 537 hp 628 hp 250 mph 776 hp 908 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

with

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