![Page 1: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/1.jpg)
Applications of Parabolas:Highway Overpasses using Type 1
Vertical Curves
John CatlettMathematics Teacher
North Star High School
![Page 2: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/2.jpg)
What is a parabola?
• A parabola is the graph that results from an equation of the form:
• Parabolas are symmetric about a vertical line known as the Axis of Symmetry.
• The Axis of Symmetry runs through the maximum or minimum point of the parabola which is called the Vertex.
yAx2 Bx C
![Page 3: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/3.jpg)
• Examples of parabolas:
![Page 4: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/4.jpg)
Exploration of Parabolas
• How do the coefficients A, B and C affect the graph of the parabola?
• Let’s use Geogebra to find out:• http://isite.lps.org/dtravis2/jcat/geogebra/
parabolaexploration.html
![Page 5: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/5.jpg)
Parabolas and Vertical Curves• In road design, vertical curves are designed
using parabolas. There are four types:
![Page 6: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/6.jpg)
Focus: Type 1 Curves and Overpasses
![Page 7: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/7.jpg)
Design Factors• Parabolas used in the overpass design are
based on a number of factors:– The entrance and exit grades, g1 and g2
– The length of the vertical curve, L (NDOR uses 600 ft as the minimum L)
– The design speed (speed the road is designed for)– The sight stopping distance, S (based on design
speed and line of sight)– The elevation at the start of the curve, ElevBVC
![Page 8: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/8.jpg)
Equation of the parabola:yAx2 Bx C
Ag2 g1
2L, Bg1, C ElevBVC
![Page 9: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/9.jpg)
Finding the Parabola for Overpass Design
![Page 10: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/10.jpg)
• Lets find equations to fit some existing overpasses.
• http://isite.lps.org/dtravis2/jcat/geogebra/overpassexample1.html
• For our purposes today, we will use:– The NDOR minimum ( 600 ft ) for L– The distance above the roadway that runs below
the overpass at the start of the curve as the ElevBVC
(these calculations are more complex in real life)
![Page 11: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/11.jpg)
Now lets find the equation of the parabola using our formula,
yAx2 Bx C
Ag2 g1
2L, Bg1, C ElevBVC
![Page 12: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/12.jpg)
Example:The entrance grade, g1, is 3%, the exit grade, g2,
Is 2%, the ElevBVC is 8 feet and the horizontal length of the curve, L, is 900 feet.
Solution:
Equation: Check Solution:
http://isite.lps.org/dtravis2/jcat/geogebra/overpassequationchecker.html
A .02 .03
2900 0.00003 B0.03 C 8
y 0.00003x2 0.03x 8
![Page 13: Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d815503460f94a6591d/html5/thumbnails/13.jpg)
Final ThoughtsVideo: http://isite.lps.org/dtravis2/jcat/
Other things to consider?