Game Development Reference
In-Depth Information
The maximum velocity can be found using the standard equation for finding the roots of a
quadratic equation.
2
−±
c
c
−
4
c c
v
=
2
2
1
3
(8.25)
max
2
c
1
Let's use Equation (8.25) to compute the maximum velocity of the Porsche Boxster S when
the car is in sixth gear. The drag coefficient of the car is 0.31, the empty mass is 1323
kg
, and the
radius of the front wheels is 0.3186
m
. We'll assume that the coefficient of rolling friction is
0.015, the frontal area of the car is 1.94
m
2
, and the weight of the driver is 70
kg
. The air density
value will be taken to be 1.2
kg/m
3
. The car is assumed to be driving over flat, level ground, so
the slope angle,
q
, is equal to zero.
The maximum velocity will occur in the higher
rpm
range, so the engine torque will be
approximated by Equation (8.17c). This means that the
b
coefficient is equal to -0.032 and the
d
coefficient has a value of 457.2. In sixth gear, the maximum velocity of the car is likely to be
drag-limited, so we'll use Equation (8.25) to compute the maximum velocity.
1
D
CA
r
1 0.31*1.2 *1.94
c
=−
=−
=−
2.59
e
−
4
(8.26a)
1
2
m
2
1393
22
2
2
60
gGb
60 * 0.84 *3.44 * 0.032
c
=
k
= −
= −
0.018
(8.26b)
2
2
2
2
p
mr
2
p
*1393* 0.3186
w
gGd
0.84 *3.44 * 457.2
c
=
k
−
m
g
=
−
0.015*9.8
=
2.83
(8.26c)
3
r
mr
1393* 0.3186
w
0.018
±
0.018
2
+
4 * 2.59
e
−
4 * 2.83
m
s
km
hr
(8.26d)
v
=
=
75.4
=
271.5
max
−
2 * 2.59
e
−
4
The other solution to the
v
max
equation gives a negative value and can be ignored. The
manufacturer's published value for the top speed of the Boxster S is 266
km/hr
. Considering the
assumptions and simplifications that went into our model, it did a pretty good job of computing
the top speed of the vehicle.
The value of 271.5
km/hr
is the drag-limited maximum velocity of the Boxster S. Let's compare
it to the redline-limited value calculated using Equation (8.23).
2
p
r
Ω
2
p
* 0.3186 * 7200
m
s
km
hr
v
=
w li e
=
=
83.1
(8.27)
=
299.3
60
gG
60 * 0.84 *3.44
k
So the Boxster S could go 299
km/hr
in sixth gear if there were no aerodynamic drag and no
rolling friction. To get a feeling for the relative magnitudes for some of the force terms, let's
compare the magnitude of aerodynamic drag and rolling friction on the Boxster S when it has
reached a speed of 271.5
km/hr
. The drag force can be calculated from Equation (8.3).
1
0.31*1.2 * 75.4 *1.94
N
F
=
2
=
2051
(8.28)
2