Valley (sag) vertical curve design for night driving: If N is the algebraic difference of grades (as a fraction) and S is the headlight sight distance in metres, what is the standard expression for the required curve length L when L ≥ S (IRC night visibility condition)?

Introduction / Context:
Valley (sag) curves are governed by comfort (rate of change of acceleration) and, at night, by headlight sight distance. For night visibility, the dipped headlight beam must illuminate the road surface over the stopping sight distance S. The standard relationship links curve length to grade difference and headlight geometry.

Given Data / Assumptions:

• N = algebraic difference of grades (in fraction, e.g., 0.02 for 2%).
• S = headlight sight distance in metres.
• Standard headlight height 1.0–1.2 m approximated as 1.0–1.2 m with an upward divergence of about 1 degree → represented by constants 1.5 and 0.035 in the design equation.
• Case considered: L ≥ S (long curve condition).

Concept / Approach:
For night visibility control of valley curves, the headlight beam striking the road surface must not be intercepted by the pavement before the sight distance S. This geometric condition yields the empirical relation used in practice.

Step-by-Step Solution:

Verification / Alternative check:
For short curves (L < S), the alternative form is L = 2 * S − (1.5 + 0.035 * S) / N, which provides consistent lengths around the transition between cases, confirming internal logic.

Why Other Options Are Wrong:
Option (b) is dimensionally inconsistent; (c) ignores headlight parameters; (d) belongs to crest (summit) curves using eye/object heights; (e) is the short-curve case, not applicable when L ≥ S.

Common Pitfalls:
Using N in percent instead of fraction; applying the summit-curve formula to valley curves; mixing the L ≥ S and L < S equations.

Final Answer:
L = (N * S^2) / (1.5 + 0.035 * S)