(Modeling) Length of a Sag Curve When a highway goes downhill and...

Question

(Modeling) Length of a Sag Curve When a highway goes downhill and then uphill, it has a sag curve. Sag curves are designed so that at night, headlights shine sufficiently far down the road to allow a safe stopping distance. See the figure. S and L are in feet.

The minimum length L of a sag curve is determined by the height h of the car's headlights above the pavement, the downhill grade θ₁ < 0°, the uphill grade θ₂ > 0°, and the safe stopping distance S for a given speed limit. In addition, L is dependent on the vertical alignment of the headlights. Headlights are usually pointed upward at a slight angle α above the horizontal of the car. Using these quantities, for a 55 mph speed limit, L can be modeled by the formula

(θ₂  -  θ₁)S²
L = ————————— ,
         200(h + S tan α)

where S < L. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.)

Compute length L, to the nearest foot, if h = 1.9 ft, α = 0.9°, θ₁ = -3°, θ₂ = 4°, and S = 336 ft.

Accepted Answer

Welcome back. Everyone in this problem. A sag curve in a highway is characterized by its downhill and then uphill direction. It must be properly designed to provide the drivers with the necessary safe stopping distance to avoid collision with any obstruction on the road. The minimum length L of a Saga curve depends on the height of the car's headlights edge the downhill grade theta one which is negative the uphill grid, theta two which is positive and the safe stopping distance. S for a particular speed limit. Moreover, it depends on the angle of inclination of the headlights alpha with respect to the horizontal. And here we can see a diagram of our S A curve and a car driving in the S curve. Now, the following formula can be used for a 55 mile per hour speed limit. And here's our formula L equals theta two, theta one multiplied by S squared, all divided by 200 multiplied by the expression H plus S than alpha where S is less than L note that the units for LH and S should be feet and all the angles should be in degrees for H equals 1.7 ft. Alpha equals 0.7 degrees. Theta one equals negative two degrees, theta two equals five degrees and S equals 312 ft. Calculate the length L to the nearest foot. Now, here we have a scenario. Oops, before I forgot before I forgot for the answer choices. A is 591 ft. B is 618 ft. C is 693 ft and D is 727 ft. I mean, how could I forget that guys? I'm sorry. Now, here we have a diagram and basically what we're saying is that we'd like to find the minimum length of the Saga curve. OK. And we are given a formula for the length of the SAG curve. Now for this formula. And in this scenario, if we notice here, the SAG curve has theta two, theta one S and alpha as its variables. And for the information that's given we have S we have data too or we, we have H we have alpha, sorry, we have theta one and we have theta two. So we can plug these values into our formula to find the value of L. So let's go ahead and do that, ensure that your calculator is set in degree mode. That's very, very important because that's what the formula is working with. So if we substitute these values into our formula, then that tells us that L the minimum length of the S curve is going to be equal to theta two which is five degrees minus theta one, which is negative two degrees. OK. Remember to show that we're subtracting a negative. OK. And all of that is being multiplied by S squared. So that's 312 ft all squared. OK. And we're dividing all of that right by 200 multiplied by H which is 1.7 ft plus S than alpha. So that's 312 ft multiplied by the tangent of 0.7 degrees. So this is exactly what we're going to put in our calculator minus the units here for feet and, and degrees and so on. So I want you to put this into your calculator and let's see what you get. Well, when you do that, a value that it should read on your calculator is 618.1144691 ft. OK. That's, that's, that's the one that you're supposed to get. But now here we want to calculate L to the nearest foot. So if we round up to the nearest foot, it will be approximately 618 ft. And we know our answer is correct because remember if we look back in our formula, we know that S should be less than L. And here on our diagram, we can also see it S OK, which is the safe stopping distance would naturally have to be less than the minimum length of the S curve. And here we realize that 312 ft is less than 16 618 ft. Therefore, we're sure we're correct. When we say that b 618 ft is a, is the right answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Sag Curve Design

Trigonometric Functions in Road Design

Mathematical Modeling of Road Safety

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