Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/183



In laying out the curves on a railroad it will not do, on account of the high speed of trains, to pass abruptly from a straight stretch of track to a circular curve. In order to make the change of direction gradual, engineers make use of transition curves to connect the straight part of a track with a circular curve. Arcs of cubical parabolas are generally employed as transition curves.

The transition curve on a railway track has the shape of an arc of the cubical parabola $$y = \tfrac{1}{3}x^3$$. At what rate is a car on this track changing its direction (1 mi. = unit of length) when it is passing through (a) the point (3, 9)? (b) the point $$(2, \tfrac{8}{3})$$? (c) the point $$(1, \tfrac{1}{3})$$?




 * Solution.
 * $$\frac{dy}{dx} = x^2, \frac{d^2 y}{dx^2} = 2x$$.
 * Substituting in (40),
 * $$K = \frac{2x}{(1 + x^4)^{\frac{3}{2}}}$$.
 * (a) At (3, 9),
 * $$k = \frac{6}{(82)^{\frac{3}{2}}}$$ radians per mile = 28&prime; per mile.
 * (b) At $$(2, \frac{8}{3})$$,
 * $$K = \frac{4}{(17)^{\frac{3}{2}}}$$ radians per mile = 3&deg; 16&prime; per mile. Ans.
 * (c) At $$(1, \frac{1}{3})$$,
 * $$K = \frac{2}{(2)^{\frac{3}{2}}} = \frac{1}{\sqrt{2}}$$ radians per mile = 40&deg; 30&prime; per mile. Ans.
 * }
 * (c) At $$(1, \frac{1}{3})$$,
 * $$K = \frac{2}{(2)^{\frac{3}{2}}} = \frac{1}{\sqrt{2}}$$ radians per mile = 40&deg; 30&prime; per mile. Ans.
 * }
 * }

By analogy with the circle (see (38), &sect; 99), the radius of curvature of a curve at a point is defined as the reciprocal of the curvature of the curve at that point. Denoting the radius of curvature by R, we have