STATEMENT OF PROBLEM

Curvature Preliminaries

How would you measure the sharpness of a turn? In accordance with our notion that a small circle has greater curvature than a large one, the curvature of a circle is defined to be 1/radius. For noncircular curves, we can find the curvature at a point by finding a circle that is tangent to the curve and whose second derivative (remember, the second derivative gives us concavity) matches that of the circle. This circle is the circle closest approximation to the curve at that point. We then say that the curve has the same curvature as that circle at that point.

Your Problem

In many cases, a vehicle's speed is restricted by the curvature of the path it travels. If the curvature of the path is 1/r, with r in feet, then the maximum speed of our vehicle, in mph, is approximated by 30(1-1/(r+1)^(1/4)). The graph below shows that curvature has a substantial effect on the maximum speed of the vehicle.

Input := 

speed[r_] = 30(1-1/(r+1)^(1/4));
Plot[speed[r],{r, 0, 12},
	PlotLabel -> "speed vs. radius of curvature",
 	AxesLabel -> {"radius [ft]", "speed [mph]"}];

Find the minimum time needed to safely travel 1000 feet down a straight road, make a 90-degree turn of radius 30 feet, and then continue straight for another 1000 feet. Assume that the maximum safe acceleration and deceleration as 2 mph per second.