Vertical curves – length from allowable rate of change of grade If r is the permitted rate of change of grade (per unit length) and g% is the algebraic change in grade, what length L of the vertical curve should be provided?

Introduction / Context:
Vertical curve length is often governed by comfort (rate of change of slope) and sight distance. Designers limit the rate at which grade changes to ensure driver comfort and smooth profile.

Given Data / Assumptions:

• r = allowable rate of change of grade per unit length (e.g., % per 100 m, or % per m depending on agency convention).
• g% = algebraic difference between the two approach grades (e.g., +g1 and -g2).

Concept / Approach:
For a parabolic vertical curve, the rate of change of grade is uniform and equals g / L in consistent units. Limiting this to r gives the design equation.

Step-by-Step Solution:
Uniform rate condition: (change in grade) / length = r.Thus, g / L = r.Rearrange to L = g / r (use consistent units for r and g).

Verification / Alternative check:
Check units: if r is in % per 100 m, insert the 100 m factor explicitly to maintain consistency.

Why Other Options Are Wrong:

• r / g and other combinations invert or mis-scale the relation.
• Formulas with 100 in numerator/denominator are unit-specific and not generally valid without context; the general relation is L = g / r.

Common Pitfalls:
Mixing units (e.g., % per 30 m) without converting before applying L = g / r.

Final Answer:
L = g / r