Car speed and energy consumption

The energy consumption of a car covering a given distance depends mainly on the speed of the vehicle. The faster a car drives, the more energy is required for propulsion. At higher speed the energy consumption of a car per unit distance increases proportional to the square of the speed. This is caused by the air drag which dominates over the tyre rolling resistance at high speeds.

Driving twice as fast therefore costs approximately four times as much fuel per unit of distance. The required power increases with the cube of the speed, so driving twice as fast costs theoretically eight times as much fuel per unit of time.

Fuel consumption by a traditional car causes air pollution by combustion. This air pollution produced over a given distance likewise increases quadratically with increasing speed. Neglecting details, reducing the speed from 130 to 100 km/hour entails theoretically that only a relative fraction of (100/130)2 = 0.59 = 59% of energy is expended, so that 41% less fuel is consumed over the same distance. This leads to a comparable reduction in air pollution from gasoline combustion.

Calculation

The energy  for propulsion supplied by a car's engine is spent on:
 accelerating the car (abbreviated to ac),
 overcoming the air resistance (aerodynamic drag, dr), and
 overcoming the rolling resistance (rr).
so that

The energy effects of wind, ascent and descent are not considered here, nor the repeated braking and accelerating when driving in a city.

Acceleration 
The kinetic energy  of a moving car with mass  and speed  can be calculated theoretically as

so for a car with a mass of 1,000 kg at a speed of 120 km/h (33.3 m/s)

(One kilowatt hour (kWh) equals 3.6 million joules (J).)

The required kinetic energy is supplied by the motor when the car accelerates starting from rest. It is lost when the driver brakes (unless the available kinetic energy is stored in, for example, a flywheel). Part of this energy can also be recovered from electric cars via recuperative braking and can be stored in a accumulator.

In practice, the real energy consumption of a car depends on many additional factors, including the traffic situation and personal driving style. In order to be able to make comparisons, standardized consumption tests have been devised, driving cycles, such as the New European Driving Cycle (NEDC) and the newer Worldwide harmonized Light vehicles Test Procedures (WLTP).

Without any resistance or friction, by Newton's First Law a car with speed  would continue at this speed forever.
However, air drag and rolling resistance cause energy losses which must be overcome to drive at a constant speed.

Air drag
Depending on its surface area , streamlining and speed , a car moves a lot of air, which causes energy loss by the aerodynamic drag force. We can imagine the moving air as a tube (trunk) of air behind the car, with a volume of the cross-sectional area of that tube times the length, which equals the distance traveled by the car. This length is then equal to the car speed times the elapsed time. The effective cross-section of the air tube is less than the front surface of the car , for example due to streamlining. This can be taken in account by introducing a drag coefficient  for the air resistance of about 1/3.
Suppose the surface area of the wind shield of a car is

then the effective volume  of that tube of swirling air behind the car, after a period of driving  at speed  equals

with  =  the distance traveled.
The mass of the displaced air over a journey of  meters is equal to the volume of the tube times the specific mass  of air, so
.

The energy loss due to air resistance is equal to the kinetic energy that the car imparts to the air.Using the value of  for the density of air  results in

Rolling resistance
The mathematical formula for the force due to the rolling resistance of the car tyres is:

with
 the roll coefficient which is around 0.01.
 the mass of the car, and
 the gravitational acceleration.

The corresponding energy loss caused by the rolling resistance is (the work done by the rolling resistance)

,

with  the distance covered, so the energy spent on overcoming the rolling resistance per unit distance is a constant.

Total energy consumption to overcome friction
Adding up the energy consumption required to overcome both air drag and tyre rolling resistance, we have

From the graphs it is seen that per unit distance the drag resistance dominates over the rolling resistance at higher car speeds.

The energy consumption over a distance and the air pollution are proportional to the speed squared
The energy consumption of a car at higher speeds is seen to be proportional to its speed  squared, and of course also to the distance  traveled.
So if the air resistance on the car dominates – and acceleration/braking and rolling resistance are of less significance for energy consumption, such as on the highway  – the energy consumption is proportional to the square of the car speed. If air pollution increases in line with petrol consumption, this means that traveling at 120 km/h over the same distance pollutes 15% less than at 130 km/hour, because (120/130)2 = 0.85 = 85% and 100% - 85% = 15%.

Because a car mainly produces heat and only uses 25% of the energy from the fuel for movement (an electric car uses about 90% for movement), the petrol must provide much more energy than the car uses for movement:

Car power is proportional to the cube of the speed 

Precise details of a car's energy consumption depend on the construction etc. of a car, but in general this quadratic relationship with speed is correct for speeds above 60 km/h, but refer to the accompanying figure. This can also be seen in the measured relationship between the power (energy per time, so energy times speed divided by distance)  of cars and the cube of their top speeds 
, so

Energy is proportional to the square of the speed and must be multiplied again by the speed to obtain the expended power. So the car power scales with the third power of the car speed. The consumption of energy per second increases eightfold  (23) if you drive twice faster.

See also
 Energy efficiency in transport
 Energy-efficient driving
 Fuel economy in automobiles
 Kinetic energy

General physics reference
 Fundamentals of Physics:

References

External links
  Previous book publication 2009. PDF free download.
 

 
Energy conservation
Fuels