that the power a car requires really does increase as the cube of speed,
figure A.13 shows the engine power versus the top speeds of a range of
cars. The line shows the relationship “power proportional to v3.”

### Electric cars: is range a problem?

People often say that the range of electric cars is not big enough. Electric
car advocates say “no problem, we can just put in more batteries” – and
that’s true, but we need to work out what effect the extra batteries have on
the energy consumption. The answer depends sensitively on what energy
density we assume the batteries deliver: for an energy density of 40 Wh/kg
(typical of lead-acid batteries), we’ll see that it’s hard to push the range
beyond 200 or 300 km; but for an energy density of 120 Wh/kg (typical of
various lithium-based batteries), a range of 500 km is easily achievable.

Let’s assume that the mass of the car and occupants is 740 kg, without
any batteries. In due course we’ll add 100 kg, 200 kg, 500 kg, or perhaps
1000 kg of batteries. Let’s assume a typical speed of 50 km/h (30 mph); a
drag-area of 0.8 m2; a rolling resistance of 0.01; a distance between stops
of 500 m; an engine efficiency of 85%; and that during stops and starts,
regenerative braking recovers half of the kinetic energy of the car. Charging
up the car from the mains is assumed to be 85% efficient. Figure A.14
shows the transport cost of the car versus its range, as we vary the amount
of battery on board. The upper curve shows the result for a battery whose
energy density is 40 Wh/kg (old-style lead-acid batteries). The range is
limited by a wall at about 500 km. To get close to this maximum range,
we have to take along comically large batteries: for a range of 400 km, for
example, 2000 kg of batteries are required, and the transport cost is above
25 kWh per 100 km. If we are content with a range of 180 km, however,
we can get by with 500 kg of batteries. Things get much better when we
switch to lighter lithium-ion batteries. At an energy density of 120 Wh/kg,
electric cars with 500 kg of batteries can easily deliver a range of 500 km.
The transport cost is predicted to be about 13 kWh per 100 km.

It thus seems to me that the range problem has been solved by the
advent of modern batteries. It would be nice to have even better batteries,
but an energy density of 120 Wh per kg is already good enough, as long
as we’re happy for the batteries in a car to weigh up to 500 kg. In practice
I imagine most people would be content to have a range of 300 km, which
can be delivered by 250 kg of batteries. If these batteries were divided
into ten 25 kg chunks, separately unpluggable, then a car user could keep
just four of the ten chunks on board when he’s doing regular commuting
(100 kg gives a range of 140 km); and collect an extra six chunks from
a battery-recharging station when he wants to make longer-range trips.
During long-range trips, he would exchange his batteries for a fresh set at
a battery-exchange station every 300 km or so.

Figure A.14. Theory of electric car range (horizontal axis) and transport cost (vertical axis) as a function of battery mass, for two battery technologies. A car with 500 kg of old batteries, with an energy density of 40 Wh per kg, has a range of 180 km. With the same weight of modern batteries, delivering 120 Wh per kg, an electric car can have a range of more than 500 km. Both cars would have an energy cost of about 13 kWh per 100 km. These numbers allow for a battery charging efficiency of 85%.
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