Transportation - Energy and the Environment - page 200

in the earth's gravitational field. Finally, when the vehicle accelerates to a higher speed, power is
needed to increase the kinetic energy of the vehicle.
The aerodynamic drag force acting on a vehicle is conveniently given as the product

is the mass density of air (kg/m 3 ), A is the frontal area (m 2 ) of the ve-
hicle (about 80% of the height times the width), V is the vehicle speed (m/s), and C D is the drag
coefficient. The latter is empirically determined by testing vehicle models in a wind tunnel, and
its value, generally in the range of 0.25-0.5, depends very much on the vehicle shape and external
smoothness. Given the fact that a passenger vehicle must enclose the passengers in a safe structure
employing good visibility and without excess material, it is difficult to reduce C D below about
0.25. The corresponding power P air needed to overcome the air resistance is the product of the
force times the vehicle velocity:

C D AV 2 ρ / 2, where ρ

V 3
2

) ρ P air = ( C D A (8.10)

The power required to overcome air resistance thus increases as the cube of the vehicle speed. At
high vehicle speeds, air resistance is the major factor in determining the requirement for engine
power.
It might seem that wheels should present little or no resistance to forward motion. While this
resistance is small, it cannot be reduced to zero. The source of this resistance lies in the deflection
of the tire where it comes into contact with the ground. This deflection is necessary to support the
weight of the vehicle and to provide a contact area between the tire and the road that is needed to
prevent the tire slipping along the road surface. The deflection is such that, when the wheel rotates,
the road surface exerts a retarding torque on the wheel, which must be overcome by the engine
drive system, and a corresponding retarding force on the wheel axle. This force is usually specified
as C R mg , and the corresponding power P roll becomes

P roll = C R mgV (8.11)

where m is the vehicular mass, g is the acceleration of gravity ( mg is the total vehicle gravity
force supported by the tires), and C R is a small dimensionless constant whose value depends upon
the tire construction and pressure. Stiff, highly pressurized tires will have smaller C R , but will
be more prone to slip and transmit road unevenness to the vehicle. The values of C R lie in the
range 0.01-0.02. Because the rolling power grows only as the first power of the vehicle speed, it
is generally smaller than P air at high speeds but can become larger at low speeds.
When climbing a hill of rise angle θ , the force of gravity acting on the vehicle, mg , has a

component mg sin θ opposing the forward motion. The power required to maintain a steady climb

rate is

P hill = mgV sin θ (8.12)

Finally, the instantaneous power required to accelerate the vehicle, P acc , is simply the time
rate of increase of vehicular kinetic energy, m

( + )

V 2

/

1 2, or

m

V 2 d

dt

( 1 + )

V dV
dt

P acc = = m ( 1 + ) (8.13)

2