### Brain Teasers

# Rising Train Tracks

Henry is a new civil engineer, and is charged with planning the laying of a new type of ultra-rigid rails that expand only one inch per mile of length in temperature extremes. Henry's plan calls for the tracks (each of which is one mile in length) to be laid end-to-end with no gap between them, pinned to the ground at every other juncture, as shown below:

v_______ _______vv_______ _______v

Henry's boss sees the plan and says, "You fool! What about when the tracks expand?"

Henry replies, "The expansion will only be an inch over the whole mile length. If the tracks buckle, it will only be by an inch or two at most. The trains will certainly have no trouble bumping over these buckles."

Henry's boss says, "You're fired." Why was the boss so critical?

v_______ _______vv_______ _______v

Henry's boss sees the plan and says, "You fool! What about when the tracks expand?"

Henry replies, "The expansion will only be an inch over the whole mile length. If the tracks buckle, it will only be by an inch or two at most. The trains will certainly have no trouble bumping over these buckles."

Henry's boss says, "You're fired." Why was the boss so critical?

### Hint

Would Pythagoras have the same plan?### Answer

The rise in the middle of the tracks would have been much greater than Henry surmised. Using the Pythagorean Theorem, you will have a triangle with a base of one mile (or 63,360 inches), and a hypotenuse of one mile plus one inch (or 63,361 inches). To find the height, take the square root of (the square of the base minus the square of the hypotenuse). SQRT((63,361)^2 - (63,360)^2) = 355.98 inches, or over 29 feet!Hide Hint Show Hint Hide Answer Show Answer

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## Comments

Wrong! The Pythagorean theorem requires a right triangle - if the mile of track is buckled in the middle, the Hypotenuse is half of 1 mi.+ 1 in. and the base, half of 1 mile with a congruent triangle running along the other half mile of track (Draw the diagram.) The heighth of the track is a mere 14.8 feet - still enough to get the schmuck fired!

Each length of track is one mile, and they are only pinned at one end. So you will have two abutting right triangle, each with a base of one mile, or you will have an isosceles triangle with a base of two miles and two sides of (1 mile + 1 inch) each.

I think unklemyke approximated the situation much better than the Answer did. If I recall correctly from a class on the behavior of non-rigid bodies, the buckling of such an object (idealized conditions assumed) would be half a period of a sinusoidal wave... (somewhat looks like an arc) From the problem, the perimeter of the half-wave would be 1mi plus 1in, and the half-period of the wave would be 1mi. I think it is possible to calculate the amplitude from this information. Can't do that though. . Just here to say unklemyke's approximation is much better.

This problem is wrong...there should be a rating for that! lol

If I made an error, I am happy to fix it. How is it wrong, beta7? I tried answer the bending question by making the rails "ultra-rigid". I removed the calculus of a sinusoidal semi-period by allowing the track to rise on one side instead of bending at the middle. Please let me know where the errors lie, and I will change the problem. Thanks!

The main problem that I was seeing is that nothing indicated how long the individual tracks were. The implication is that they are 1 mile in length, but that is not explicitly stated.

My mistake, GT. Thanks for pointing out the oversight. I will put in a correction.

Great teaser! That is certainly a larger height than you would at first imagine.

Being from Canada, the only issue I had was having to look up the number of inches in a mile.

Being from Canada, the only issue I had was having to look up the number of inches in a mile.

I don't have a problem with the math, just the physics. The two ends are quite unlikely to stay butted together for a rise of 29 feet over a mile-length section of track in each direction. Since the track is only pinned on one side, what would likely happen is that the one piece of track would offset to the left and one to the right (or possibly it rises high enough that both collapse to the same side.)

Not pinning the track on both ends is a flaw worthy of being fired.

Not pinning the track on both ends is a flaw worthy of being fired.

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