Cars and Motorcycles - Physics for Game Programmers - page 217

Determining Wheel Radius
The acceleration and velocity expressions shown in Equations (8.11) and (8.13) are functions of
the wheel radius, but how can this quantity be determined? Fortunately, the wheel radius can
be calculated from information on the tire itself. Every tire will have a series of letters and numbers
that identify the tire. For example, the front tires of the Porsche Boxster S have the identification
225/40ZR-18. The first number, 225 for the Boxster, indicates the width of the tire in millimeters.
The number after the slash symbol, /, is the ratio of the tire thickness to the tire width expressed
as a percentage. The letters indicate the conditions for which the tire is designed. The last two
numbers represent the diameter of the wheel that fits the tire in inches. The tire radius is equal
to the wheel radius added to the tire thickness. Based on the tire designation, 225/40ZR-18, the
radius of the Boxster S front tire is 0.3186 m .

Gear Shifting
We learned in the last section that a lower gear (with a higher gear ratio) results in a greater
acceleration. So why not just stay in first gear all the time? Wouldn't that optimize the acceler-
ation of the car? This is the flip side to the question, “Why not connect the engine directly to the
wheels?” The answer is, “No, you shouldn't stay in first gear all the time.” The reason is that the
velocity of the car is a function of engine turnover rate and gear ratios, and there is a limit to
how fast the engine can turn over.
Every car engine has a characteristic known as a redline rpm value. The engine cannot
exceed this turnover rate for more than a brief period of time without causing damage to the
engine. On the Porsche Boxster S, the redline value is 7200 rpm . Using Equation (8.13) and the
data in Table 8-1, the theoretical maximum velocity for each gear at 7200 rpm can be computed,
and the results are shown in Table 8-2.

Table 8-2. Theoretical Maximum Velocity for Each Gear for the Boxster S
Gear Maximum Velocity ( m/s ) Maximum Velocity, ( km/hr )

First 18.3 65.8

Second 31.7 114.3

Third 45.9 165.4

Fourth 57.2 206.0

Fifth 68.5 246.4

Sixth 83.1 299.3

We can see from Table 8-2 that although the maximum acceleration occurs in first gear,
the maximum velocity the car can attain in first gear is 65.8 km/hr . At this point, you would
reach the redline rpm value and have to shift into second gear, which would provide the optimum
acceleration between 65.8 and 114.3 km/hr . At this point, the rpm level would reach the redline
value again, and the car would have to be shifted into third gear. Most transmissions are designed
so the shift point for optimum acceleration is at the redline value of the car.