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The centripetal acceleration can be calculated from the car's speed and trajectory as $a = v^2 c(x)$ where $x$ is the distance along the track and $c(x)$ is the curvature of its path at that distance. If the car is driving on the racing line, the curvature can be obtained as shown in section 2. The curvature as a function of distance for the example track (figure 1) is shown in figure 6.

Figure 6: Curvature of the racing line for the example track. Curvature peaks near the turns. Curvature is positive for left turns.
Image curvature

Banking and elevation changes can affect the maximum safe speed for a corner. If a corner is at the crest of a hill, the car will get light and lose some traction. In general, gravity, normal, and frictional forces must sum to the centripetal force. Forces normal to the road can be ignored, so we have

...ath{\hat{\mathbf{q}}} + \ensuremath{\mathbf{F}_{\mathrm{\mu}}}
\end{displaymath} (15)

where $\ensuremath{\hat{\mathbf{q}}}$ is the unit vector parallel to the road and away from the center of curvature. The frictional force is $-\mu F_n \ensuremath{\hat{\mathbf{q}}}$ where $\mu$ is the coefficient of static friction and the normal force given by
F_n = \ensuremath{\mathbf{F}_{\mathrm{g}}}\cdot\ensuremath{...
\end{displaymath} (16)

where $F_a$ is aerodynamic downforce. We substitute the following expressions for the forces
$\displaystyle \ensuremath{\mathbf{F}_{\mathrm{c}}}$ $\textstyle =$ $\displaystyle -mv^2c\ensuremath{\hat{\mathbf{r}}}$ (17)
$\displaystyle \ensuremath{\mathbf{F}_{\mathrm{g}}}$ $\textstyle =$ $\displaystyle -mg\ensuremath{\hat{\mathbf{z}}}$ (18)
$\displaystyle F_a$ $\textstyle =$ $\displaystyle \alpha v^2$ (19)

and solve for $v$.
v_\mathrm{max} = \left(\frac{g(\ensuremath{\hat{\mathbf{z}}...
...cdot\ensuremath{\hat{\mathbf{n}}}) - \mu\alpha/m}\right)^{1/2}
\end{displaymath} (20)

The fastest way around the track is to stay as close as possible to \ensuremath {v_\mathrm {max}} without going over. Figure 7 shows part the \ensuremath {v_\mathrm {max}} curve for the example track.

Figure 7: Maximum speed on the example track's racing line for the a car capable of 1g lateral acceleration.
Image v-max

next up previous
Next: Braking Up: Speed Control Previous: Speed Control
Sam Varner 2012-01-18