So I was driving in my car thinking to myself "I wonder how far I would go (before starting to roll backwards) if I just took my foot off the brakes" I tried to figure it out myself but could not. SO I am asking the smartest people in the world, you guys!
Parameters: Write a generalized equation representing the distance a car travels based on f(x) and any other physical constants/variables needed. f(x) is a function that represents the cross section of the road you are traveling on. Thus the slope at any given point represents the angle of the road. You may treat the car as a box sliding with friction.
This is a simple situation of force balancing. Enumerate all the forces on the car:
The sum of these forces must be zero. Drag and rolling resistance act in the opposite direction of motion. The gravity term depends on orientation. Without losing generality, we can write the equation of motion as: $$0 = F_A - F_D - F_R + F_g \\ C_R\frac{dx}{dt} + C_D \left(\frac{dx}{dt}\right)^2 - mg\cos \theta = F_A.$$
If we have no acceleration (equiv. no braking), then $F_A = 0$. Of course, $\theta$ is measured as the angle with respect to level ground. This angle may change relative to the $x$-position, so it may be more appropriate to consider the gravity term as $mg\cos \theta(x)$.