To determine the length of a one-way sag vertical curve joining a -3% grade with a +5% grade, we can use the formula:
Length = (V^2) / (a * g)
Where:
- V is the design speed in feet per second
- a is the algebraic difference between the initial and final grades
- g is the acceleration due to gravity (32.17 ft/s^2)
Given:
- Design speed (V) = 40 mph
First, let's convert the design speed from mph to feet per second:
V = 40 mph * (1.47 ft/s per mph) = 58.8 ft/s
Next, calculate the algebraic difference between the initial and final grades:
a = (-3% + 5%) / 100 = 0.02 (since -3% is negative, and +5% is positive)
Now, substitute the values into the formula:
Length = (58.8^2) / (0.02 * 32.17)
Calculating this, we get:
Length ≈ 12,139 ft
Therefore, the length of the one-way sag vertical curve is approximately 12,139 feet.