Other Forces Affecting Deceleration

In general, a number of forces affect deceleration. Aerodynamic drag helps to slow the car regardless of how much traction is available. Gravity may help or hurt, depending on the slope. And, as with cornering, humps and dips affect the normal force, and consequently, the tires' grip level.

We define $\ensuremath{\hat{\mathbf{p}}}$ as the unit vector tangent to the track in the direction of travel. If the track has a slope such that the gravitational force has a component in this direction, then it will work against the frictional forces that are slowing the car. We can write the total force slowing the car as

$$\ensuremath{\mathbf{F}_{\mathrm{b}}} = -\ensuremath{\mathbf... ...ath{\hat{\mathbf{p}}} - \mu F_n \ensuremath{\hat{\mathbf{p}}}$$ (25)

Using the expressions for the forces found when calculating cornering speed (equations 16, 17, 18, 19) and expressing the drag force as $F_d = v^2\beta$ we arrive at the expression for acceleration under braking.

$$a_b = g(\ensuremath{\hat{\mathbf{z}}}\cdot\ensuremath{\ha... ...suremath{\hat{\mathbf{r}}}\cdot\ensuremath{\hat{\mathbf{n}}}))$$ (26)

Since this expression depends on the car's position on the track as well as its speed, we can no longer calculate an expression for $v(x)$ as in equation 24. Instead, we will use equation 26 to predict the car's speed a short distance ahead, and then repeat using the speed and position from the previous iteration. In effect, we run a short braking simulation to see if braking is necessary.