It is given that the safe stopping sight distance is 128 m. What should be the length of the vertical curve if it is formed due to meeting of two gradients +3% and –5%?
- A302 m
- B298 mCorrect
- C307 m
- D295 m
Solution
Concept
The length of a summit curve is designed to provide adequate sight distance for driver safety. Standard IRC specifications use a driver eye height of 1.2 m and object height of 0.15 m, leading to the simplified expression L = N × S² / 4.4 for L > S.
Formula
N = g1 - g2 L = (N × S²) / 4.4 (for L > S)
Step-by-step solution
1. Calculate the total deviation angle: N = +3% - (-5%) = +8% = 0.08 2. Assume L > S and use the IRC standard summit curve formula: L = (N × S²) / 4.4 3. Substitute values: L = (0.08 × 128²) / 4.4 = (0.08 × 16384) / 4.4 = 297.89 m ≈ 298 m 4. Check assumption: Since L = 298 m > S = 128 m, the assumption L > S is valid.
Explanation
The deviation angle N is the algebraic difference between the two gradients, giving N = 0.03 - (-0.05) = 0.08. Assuming the length of the summit curve L is greater than the stopping sight distance S (128 m), the formula is L = (N × S²) / 4.4. Substituting the values yields L = (0.08 × 128²) / 4.4 = 297.89 m, which rounds to 298 m and satisfies the assumption L > S.
Why the other options are wrong
A: Result from minor calculation errors or using wrong height assumptions. C: Result from using L < S formula incorrectly. D: Incorrect rounding or algebraic error in computing N × S² / 4.4.
Shortcut
Remember 128² ≈ 16400, then L ≈ 0.08 × 16400 / 4.4 ≈ 1312 / 4.4 ≈ 298 m.
Common mistake
Adding the gradients algebraically without accounting for signs (e.g., taking N = 3 - 5 = -2% instead of 3 - (-5) = 8%).
Correct answer: B) 298 m