# Reality check

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Let us check ourselves by calculating very approximately how much fuel is used to move the two car side mirrors over the distance of 100 km, and compare it with the typical car fuel consumption. If it will turn out that the amount of fuel needed to push the mirrors through the air is very small, then probably our hope to explain the difference between old and modern mirrors by the desire to reduce their drag is unfounded. Crude estimates of the type we are going to do are very common among good engineers, and are dubbed “back of envelope calculation”, for the manner in which those calculations are often done.

Knowing that the heating value of petrol is about 45000 kJ/kg, the petrol density is about 0.75 kg/L, and the engine efficiency is about 0.25, we obtain that 1 L of petrol provides 1L×0.25×0.75 kg/L×45000 kJ/kg = 8437 kJ of useful energy.

Now, the work of the drag of two mirrors moved by the distance of 100 km is 100 km. From our video observations we know that in the car frame of reference the velocity in the mirror wake is small, that is the fluid particles have the velocity close to the velocity of the mirror itself. Therefore, in the frame of reference linked to the ground, the velocity in the wake will be close to the velocity of the car, U=V.

Let as first assume that the car is moving at 10 m/s=36 km/h=22.37 mph. Given that the air density at normal conditions is 1.225 kg/m3,  the energy required for two mirrors to travel 100 km is E=2×(ρWHU2/2)×100 km. We do not know the dimensions of the wake of a car mirror, but we can guess it to be close to the dimensions of the mirror itself, which we estimate as 6 cm by 15 cm, obtaining the wake area of roughly 100 cm2=0.01 m2. Hence, E = 2 x 1.225 kg/m3×0.01 m2×(100 m2/s2 /2)×100 km=1.225 N×100 km=122.5 kJ. Accordingly, the fuel consumption is increased by these mirrors by 1.225/8437 L/km=122.5/8437 L/(100 km) =0.015 L per 100 km. Fuel consumption of cars  varies, but roughly we can take it to be 10 L/100 km. Then, mirrors are responsible for (0.015/10)×100%=0.15% of the fuel consumption. Reducing this might or might not worth the trouble for a car manufacturer. However, the drag force is proportional to the square of U. Hence, if the car moves not at 22 mph, but at 110 mph, as it does on a motorway, the same calculations would give that the mirrors were responsible for 3.6% of the fuel consumption. Will the manufacturers care about that much? Yes, they will. Our idea has passed the first reality check.

Now it is time to do some work. There will be two pieces: first, to do a back-of-envelope estimate of how much the drag can be reduced by streamlining the mirror. Second, to try to get a better streamlined mirror than the one in our videos.

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