Without friction: Nsin(theta) = mv^2/r (horizontal, centripetal) Ncos(theta) = mg (vertical, equilibrium) Dividing: tan(theta) = v^2/(rg) => v = sqrt(rg*tan(theta)) (only one safe speed)

With friction (maximum speed — friction acts inward and downward along the bank): Radial: Nsin(theta) + fcos(theta) = mv^2/r Vertical: Ncos(theta) = mg + fsin(theta) With f = muN: v_max = sqrt(rg(tan(theta) + mu)/(1 - mu*tan(theta)))

With friction (minimum speed — friction acts outward and upward): v_min = sqrt(rg*(tan(theta) - mu)/(1 + mu*tan(theta)))

If tan(theta) <= mu, then v_min = 0 (the car can be stationary without sliding).