Question

Given \( G_1 \) and \( G_2 \) are the slopes of the approach and departure grades of a vertical curve, respectively.
Given \( |G_1| < |G_2| \) and \( |G_1| \neq |G_2| \neq 0 \), Statement 1: \( +G_1 \) followed by \( +G_2 \) results in a sag vertical curve.
Statement 2: \( -G_1 \) followed by \( -G_2 \) results in a sag vertical curve.
Statement 3: \( +G_1 \) followed by \( -G_2 \) results in a crest vertical curve.
Which option amongst the following is true?

• A. Statement 1 and Statement 3 are correct; Statement 2 is wrong
• B. Statement 1 and Statement 2 are correct; Statement 3 is wrong
• C. Statement 1 is correct; Statement 2 and Statement 3 are wrong
• D. Statement 2 is correct; Statement 1 and Statement 3 are wrong

Hint:

Remember: A sag curve has both grades positive but with the second grade smaller, while a crest curve has the first grade positive and the second negative.

Answer 1

The Correct Option is A

Step 1: Understanding vertical curves.
In road design, vertical curves are used to provide smooth transitions between different road grades. There are two types of vertical curves: - Sag curve: A concave curve where the grade changes from a negative to a less negative value or from a positive to a less positive value. - Crest curve: A convex curve where the grade changes from a positive to a less positive value or from a negative to a less negative value.

Step 2: Evaluating the statements.
- Statement 1: \( +G_1 \) followed by \( +G_2 \) results in a sag vertical curve.
This is true. In a sag vertical curve, both grades are positive but the second grade \( G_2 \) is smaller than the first \( G_1 \), making the curve concave.
- Statement 2: \( -G_1 \) followed by \( -G_2 \) results in a sag vertical curve.
This is false. If both \( G_1 \) and \( G_2 \) are negative, it means the slope is descending throughout, and there's no change to create a curve. This would be a straight, descending path.
- Statement 3: \( +G_1 \) followed by \( -G_2 \) results in a crest vertical curve.
This is true. If \( G_1 \) is positive and \( G_2 \) is negative, the curve transitions from an upward slope to a downward slope, making it a crest curve.
Thus, the correct answer is (A): Statement 1 and Statement 3 are correct; Statement 2 is wrong.
\[ \boxed{\text{The correct option is (A).}} \]