Acton's Railroad Problem

Q: In Numerical Methods that Work [1] there is a famous "railroad track problem": 1 foot of extra track is inserted into 1 mile of  railroad track, and it bows up in a circular arc. Find the maximum height it  achieves off the ground.

How can I obtain a numerical solution to this problem and is it analytically solvable?

A: From the figure, with , we obtain the following system of equations.

Here is a one-line numerical solution.

Hence the maximum height it achieves off the ground is 44.5 feet.

Generalizing the equations by writing as the length of straight track and as the length of curved track, where , and eliminating the nontrigonometric variables, we find

Even this simple equation is not analytically solvable.

However, as Ronald Bruck (bruck@pacificnet.net) points out, it is easy to determine series representations for the  answer. Since and explicitly noting that depends on , we obtain the series expansion for .

Using series inversion, we obtain .

We can use this expansion to determine and . Since , the series for is

Since , the series for is

Note that the order is important here--we need to determine before we determine . We truncate these series expansions as follows.

Then, for the original problem,

Since , we find

This result is in good agreement with FindRoot.

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