Safe length of a valley (sag) curve for night driving conditions: Which expressions for length L (metres) in terms of sight distance S (metres) and algebraic difference of grades N (in fraction) are used?

Introduction / Context:
Valley (sag) curves must be designed to ensure safe sight distance at night under headlight illumination. Unlike summit curves (governed by line of sight over the crest), sag curves are governed by the geometry of the light beam, the driver’s eye height, and the upward inclination of the headlight beam.

Given Data / Assumptions:

• S = required sight distance (m) for design speed (typically stopping sight distance).
• N = |g1 − g2| = algebraic difference of grades in fraction.
• Headlight beam geometry is represented by the constant 1.5 + 0.035 S (metres), combining typical headlight height and upward divergence.

Concept / Approach:
For sag curves, two design cases are used depending on whether the curve length L is longer than or shorter than S. The formulas relate the curve’s rate of change of grade to the illuminated length on the pavement so that the required sight distance is available at night.

Step-by-Step Solution:
Case 1 (L ≥ S): ensure full sight distance on the curve by using L = (S^2 * N) / (1.5 + 0.035 S).Case 2 (L < S): ensure adequate headlight reach beyond the curve by using L = 2S − (1.5 + 0.035 S)/N.Select the option that includes both cases.

Verification / Alternative check:
For typical highway speeds, plugging design SSD values into the formulas yields lengths consistent with standard sag curve charts and ensures comfortable rate of change of acceleration (‘jerk’) while meeting night visibility needs.

Why Other Options Are Wrong:

• Single expressions ignore the two-case nature (L ≥ S vs. L < S).
• Expressions like L = S * N do not capture headlight geometry and are dimensionally or conceptually incorrect.

Common Pitfalls:

• Confusing sag-curve night visibility formulas with crest-curve sight distance formulas.
• Using N in percent instead of fraction; always convert percent to fraction before use.

Final Answer:
Both (a) and (b)