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A steel rail track of length 1 km was laid at an ambient temperature of 20 deg C with no gaps for thermal expansion. When the temperature rose to 25 deg C, the track buckled and formed an isosceles triangle shape. Given the coefficient of linear expansion of steel is 14 * 10⁻⁶ per K, find the height of the buckle in metres (to the nearest integer).

1. 65 m
2. 54 m
3. 87 m
4. 43 m

Correct answer: 54 m

Solution

When the rail buckles into an isosceles triangle, the two slant sides together have length equal to the new (expanded) rail length, while the base remains the original track length (1 km). Using the Pythagorean theorem on half the triangle gives h = sqrt((L'/2)² - (L/2)²). With L = 1000 m, delta_T = 5 K, alpha = 14e-6 /K, delta_L = L*alpha*delta_T = 1000*14e-6*5 = 0.07 m. L' = 1000.07 m. h = sqrt((500.035)² - 500²) = sqrt(500.035² - 500²). Using difference of squares: (500.035)² - 500² = (500.035 - 500)(500.035 + 500) = 0.035 * 1000.035 = 35.001. So h = sqrt(35) ≈ 5.92 m. That gives ~6 m. But the standard answer for this classic problem with a 1-km rail and 5 K rise is approximately 54 m for alpha = 1.2e-5 (12e-6). Let me recheck with alpha = 12e-6: delta_L = 1000*12e-6*5 = 0.06 m. h = sqrt((500.03)² - 500²) = sqrt(0.03 * 1000.03) = sqrt(30.0009) ≈ 5.48 m ≈ 5 m. Still ~5 m. Wait - the classic JEE answer is often quoted as 54 m for a different setup. With the given alpha = 14e-6: delta_L = 0.07 m, h ≈ sqrt(35.001) ≈ 5.92 ≈ 6 m. Nearest integer = 6 m.

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