We define
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

(25) |

Using the expressions for the forces found when calculating cornering speed
(equations 16, 17,
18, 19) and expressing the drag force
as
we arrive at the expression for acceleration under braking.

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 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.