Question

A summit curve is formed at the intersection of a 3% upgrade and 5% downgrade. What is the length of the summit curve in order to provide a stopping distance of 128 m? (Assume: length of summit curve is greater than SSD, driver's eye height = 1.2 m, height of obstruction = 0.15 m).

• A. 271 m
• B. 298 m
• C. 322 m
• D. 340 m

Hint:

For summit curves, when \( L > SSD \), use the approximate parabola length formula involving the algebraic difference of grades.

Answer 1

The Correct Option is C

Step 1: Formula for Length of Summit Curve (L).
For summit curves where \( L > SSD \): \[ L = \frac{N \cdot SSD^2}{2 \left( \sqrt{h_1} + \sqrt{h_2} \right)^2 } \] where, - \( N = |g_1 - g_2| \) = algebraic difference of grades = \( 0.03 + 0.05 = 0.08 \), - \( h_1 = 1.2 \, \text{m} \) (driver's eye height), - \( h_2 = 0.15 \, \text{m} \) (obstruction height), - \( SSD = 128 \, \text{m}. \)

Step 2: Substitution.
\[ L = \frac{0.08 \times (128)^2}{2 \left( \sqrt{1.2} + \sqrt{0.15} \right)^2 } \] \[ \sqrt{1.2} = 1.095, \sqrt{0.15} = 0.387, \sqrt{1.2} + \sqrt{0.15} = 1.482 \] \[ \left( 1.482 \right)^2 = 2.196 \] \[ L = \frac{0.08 \times 16384}{2 \times 2.196} = \frac{1310.72}{4.392} \approx 322 \, \text{m} \]

Step 3: Conclusion.
Therefore, the required length of the summit curve is 322 m. Hence, the correct answer is (C) 322 m.