Chapter 2.5, Problem 44E

(Modeling) Stopping Distance on a Curve Refer to Exercise 43. When an automobile travels along a circular curve, objects like trees and buildings situated on the inside of the curve can obstruct the driver's vision. These obstructions prevent the driver from seeing sufficiently far down the highway to ensure a safe stopping distance. In the figure, the minimum distance d that should be cleared on the inside of the highway is modeled by the equation

$d=R\left(1-\cos \frac{\theta }{2}\right).$

(Source: Mannering. F. and W. Kilareski. Principles of Highway Engineering and

Traffic Analysis, Second Edition. John Wiley and Sons.)

(a) It can be shown that if θ is measured in degrees, then $\theta \approx \frac{57.35}{R},$ where S is the safe stopping distance for the given speed limit. Compute d to the nearest foot for a 55 mph speed limit if S = 336 ft and R = 600 ft.

(b) Compute d to the nearest foot for a 65 mph speed limit given S = 485 ft and R = 600 ft.

(c) How does the speed limit affect the amount of land that should be cleared on the inside of the curve?

Reference Exercise:

(Modeling) Highway Curves A basic highway curve connecting two straight sections of road may be circular. In the figure, the points P and S mark the beginning and end of the curve. Let be the point of intersection where the two straight sections of highway leading into the curve would meet if extended. The radius of the curve is R, and the central angle θ denotes how many degrees the curve turns. (Source: Mannering, F. and W. Kilareski,

Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.)

(a) If R = 965 ft and θ = 37°, find the distance d between P and Q.

(b) Find an expression in terms of R and θ for the distance between points M and N.