You could have at least sandwiched it in a riddle...
Imagine the first length of the rail as a straight line - the chord of a circle.
Draw two radius lines perpendicular to the expanded rail - one from the end, one from the middle of the rail, making one half of the expanded rail and the two drawn radii a section of a circle.
The radius line that comes from the middle of the expanded rail bisects the chord (line where the unexpanded rail was) and is perpendicular to it.
Thus the full radius, the middle radius (this side is 1m shorter), along with half of the original length of the rail (50m) form a right triangle.
Using the Pythagorean formula, we can then find the length of the radius:
$$50^2 + (r-1)^2 = r^2 $$
$$50^2 + r^2 - 2r + 1 = r^2$$
$$50^2 - 2r + 1 = 0 $$
$$ 2501/2 = r $$
$$ r = 1250.5 $$
We can then use trig to find the included angle:
$$sin(\theta) = 50/r = 50/1250.5$$
$$\theta = 0.0399946679 rad$$
The length of half of the expanded track, then, is:
$$r*\theta = 1250.5*0.04.... = 50.0133323m$$
Thus, the total length of the track is 100.026665 m