In which case meeting of two grades results in a vertical curve classified as a summit curve?
- A+3% meets +5%
- B–2% meets +3%
- C+3% meets +2%Correct
- D–3% meets –2%
Solution
Concept
A summit curve is formed when a positive gradient meets a milder positive gradient, a positive gradient meets a level line/negative gradient, or a flatter negative gradient meets a steeper negative gradient (i.e., when g₁ > g₂).
Formula
N = g₁ - g₂ If N > 0, Summit Curve If N < 0, Valley/Sag Curve
Step-by-step solution
1. Identify g₁ and g₂ for each option and calculate the algebraic difference N = g₁ - g₂: 2. Option (A): N = +3% - (+5%) = -2% (Valley curve) 3. Option (B): N = -2% - (+3%) = -5% (Valley curve) 4. Option (C): N = +3% - (+2%) = +1% > 0 (Summit curve) 5. Option (D): N = -3% - (-2%) = -1% (Valley curve)
Explanation
A summit curve (or crest curve) is formed when the deviation angle N = g₁ - g₂ is positive (convexity upwards), meaning g₁ > g₂. For option (C), g₁ = +3% and g₂ = +2%, giving N = +3% - (+2%) = +1% > 0, which results in a summit curve.
Why the other options are wrong
A: g₁ (+3%) is less than g₂ (+5%), giving N = -2%, which forms a valley curve. B: g₁ (-2%) is less than g₂ (+3%), giving N = -5%, which forms a valley curve. D: g₁ (-3%) is less than g₂ (-2%), giving N = -1%, which forms a valley curve.
Shortcut
A summit curve forms whenever the first gradient g₁ is algebraically greater than the second gradient g₂ (g₁ > g₂). Only option (C) satisfies +3 > +2.
Common mistake
Confusing signs when calculating N, or assuming any two positive gradients form a valley curve.
Correct answer: C) +3% meets +2%