Cars and Motorcycles - Physics for Game Programmers - page 212

As was the case with projectiles, aerodynamic drag acts in the opposite direction to the
velocity of the vehicle. We'll discuss aerodynamic drag in more detail a little later in this chapter.
The final force in the basic force diagram is due to rolling friction, which was introduced in
Chapter 7. This force acts on all four wheels and resists the rolling motion of the car. The total
rolling friction force, F R , is equal to the total normal force, F N , multiplied by the coefficient
of rolling friction for the vehicle, m r .

FF = m = m g cos q (8.4)

R r N r

The net force on the car parallel to the direction the car is driving, F Total , is equal to the
sum of the forces due to engine torque, gravity, aerodynamic drag, and rolling friction.

T 1

2

F =−

w

m mg cos q − mg sin q − C r v A

(8.5)

Total r D

r 2

w

For the car shown in Figure 8-1, the sign on the parallel gravity force term, mg sin q , in
Equation (8.5) is negative to indicate that it is pulling the car backwards. The acceleration of the
car at any given time is equal to the net force on the vehicle divided by the mass of the vehicle, m .

T 1 CvA r

2

a =−

w

m g cos q − g sin q −

D

(8.6)

r

rm 2 m

w

The velocity of the vehicle at any given time can be found by integrating Equation (8.6).
Before we can integrate Equation (8.6), however, we need to evaluate the torque that is applied
to the wheels.

Engine Torque and Power
When the engine runs, it generates a torque that is used to drive the car forwards or backwards.
As we shall see later in this chapter, the torque generated by the engine is typically not the same
as the torque applied to the wheels. The engine torque is a function of the rate at which the
engine is turning over.

T =Ω () (8.7)

e e e

Ω e , in Equation (8.7) is usually expressed in terms of revolutions
per minute, or rpm . If the engine torque is plotted as a function of engine turnover rate, the
result is what is known as a torque curve . These curves are usually available for a given car from
the manufacturer or from other sources. A typical torque curve is shown in Figure 8-2. One
characteristic of engine torque is that it does not always increase with increasing engine turn-
over rate. The torque in the curve shown in Figure 8-2 increases with increasing rpm until it
reaches a peak value of 309 N-m at about 4600 rpm . As the engine turnover rate increases
beyond 4600 rpm , the torque delivered by the engine decreases.

The engine turnover rate,