Summary
Highlights
The video introduces the concept of sight distance, defining it as the roadway length visible to a driver for design and operation purposes, specifying stopping sight distance and passing sight distance. A problem is presented: designing the length of a vertical summit curve with tangent grades of 2.2% and -1.8%, and a required stopping sight distance of 100 meters.
The first assumption is that S is less than L. This means that heights h1 and h2 (driver's eye height and object height) will lie on the curve. Standard values for h1 (1.14 meters) and h2 (0.15 meters) are introduced, noting that these are older standards. The arithmetic sum of the grades (A) is calculated as 0.04 (in decimal).
The method for finding the curve length (L) is explained using the square property of the parabola. The stopping sight distance (S) is expressed as the sum of two horizontal distances, x and y, which are then related to L, h1, h2, and A. The derived equation is solved using a calculator or software (Mathcad in this instance), yielding L = 94.47 meters.
The calculated L (94.47 meters) is compared to the given S (100 meters). Since S (100m) is greater than L (94.47m), the initial assumption (S < L) is found to be incorrect. This means h1 and h2 do not lie on the curve, but rather on the tangent grades.
A derived formula for the case where S is less than L is presented: L = (A * S^2) / (200 * (sqrt(h1) + sqrt(h2))^2) or (A * S^2) / (2 * (sqrt(h1) + sqrt(h2))^2) if A is in decimal. Using this formula gives the same result of L = 94.47 meters, confirming the manual calculation.
The video then addresses the correct scenario where S is greater than L. In this case, h1 and h2 are on the tangent lines. The main equation becomes S = L/2 + x + y, where x and y are related to h1, h2, and the grades (g1 and g2) by basic trigonometry (rise over run).
To find the minimum required length, the equation for S is differentiated with respect to the grades (g1 or g2). The grades are treated as variables, while the arithmetic sum 'A' remains constant. The derivation involves several steps of calculus and algebra to establish relationships between g1, g2, h1, h2, and A.
Substituting the derived expressions for g1 and g2 back into the main equation for S > L, and using the given values for S, h1, h2, and A, the length L is calculated to be approximately 94.15 meters. This value is consistent with the condition that S (100m) is greater than L (94.15m), making this the correct answer.
A simplified formula for the case where S is greater than L is presented: L = 2S - (200 * (sqrt(h1) + sqrt(h2))^2) / A (or with 2 in the denominator instead of 200 if A is in decimal). This formula also yields L = 94.148 meters, matching the manual approach. The video concludes by emphasizing the importance of understanding the derivation for educational purposes.