• The tube of air created in a time t has a volume Avt, where A is
the cross-sectional area of the tube, which is similar to the area of
the front view of the car. (For a streamlined car, A is usually a little
smaller than the frontal area Acar, and the ratio of the tube’s effective
cross-sectional area to the car area is called the drag coefficient cd.
Throughout the following equations, A means the effective area of
the car, cdAcar.) The tube has mass mair = ρAvt (where ρ is the
density of air) and swirls at speed v, so its kinetic energy is:

and the rate of generation of kinetic energy in swirling air is:

So the total rate of energy production by the car is:

 power going into brakes + power going into swirling air (A.2) = 1⁄2mcv3/d + 1⁄2ρAv3.

Both forms of energy dissipation scale as v3. So this cartoon predicts that
a driver who halves his speed v makes his power consumption 8 times
smaller. If he ends up driving the same total distance, his journey will
take twice as long, but the total energy consumed by his journey will be
four times smaller.

Which of the two forms of energy dissipation – brakes or air-swirling –
is the bigger? It depends on the ratio of

(mc/d)/(ρA) .

If this ratio is much bigger than 1, then more power is going into brakes; if
it is smaller, more power is going into swirling air. Rearranging this ratio,
it is bigger than 1 if

Now, Ad is the volume of the tube of air swept out from one stop sign
to the next. And ρAd is the mass of that tube of air. So we have a very
simple situation: energy dissipation is dominated by kinetic-energy-being-
dumped-into-the-brakes if the mass of the car is bigger than the mass of
the tube of air from one stop sign to the next; and energy dissipation is
dominated by making-air-swirl if the mass of the car is smaller (figure A.4).

Let’s work out the special distance d* between stop signs, below which
the dissipation is braking-dominated and above which it is air-swirling
dominated (also known as drag-dominated). If the frontal area of the car
is:

Acar = 2 m wide × 1.5 m high = 3 m2

I’m using this formula:

mass = density × volume

The symbol ρ (Greek letter ‘rho’) denotes the density.
Figure A.4. To know whether energy consumption is braking-dominated or air-swirling-dominated, we compare the mass of the car with the mass of the tube of air between stop-signs.
Figure A.5. Power consumed by a car is proportional to its cross-sectional area, during motorway driving, and to its mass, during town driving. Guess which gets better mileage – the VW on the left, or the spaceship?