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How It’s Used

The science that studies objects traveling through the air is known as ballistics.  Armies have been studying ballistics for centuries.  In more advanced ballistics, you have to consider wind and air resistance on the object.

9.1.3 Vertical Curves 

The problems of hitting or missing targets with parabolic curves are very similar to a type of problem faced by civil engineers when designing highways or railroads.  Roadways built in hilly terrain require horizontal curves , which turn right or left, and vertical curves, which curve up or down.  Parabolic curves are used as vertical curves on roads over hills and at the bottom of valleys.

When the xy coordinate system is used to plot vertical curves, x is the distance from the location where the vertical curve starts and y is the height or elevation of the particular point on the curve.

Example 9.4:  A team of civil engineers determines that the most economical route of a planned roadway will have a vertical curve with the equation:

y = 0.00005x² - 0.06x + 100  where   x = the horizontal distance, in feet, and

                                                              y = the vertical distance, or elevation, in feet.

Another team of engineers is designing an overpass 800 ft. from the point where the vertical curve starts (x = 800 ft.). To meet state regulations, the overpass must have an 18 ft. clearance above the highway.  The engineers designing the overpass must therefore make sure that the bottom is 18 ft. above the road at that point.  To keep costs down, the engineers don't want to make the overpass taller than it needs to be.  What is the minimum elevation (y = ?) of the bottom of the overpass?

Solution:  To determine the minimum elevation of the bottom of the overpass, we need to find the elevation (y-value) of the roadway at the location below the overpass (x = 800).

at x = 800 ft. 

y = 0.00005(800 ft.)² - 0.06 (800 ft.) + 100

  = 84 ft.

For the bottom of the overpass to be at least 18 ft. above this, the elevation of the bottom of the overpass must equal: 

84 + 18  = 102 ft. 

The minimum elevation of the bottom of the overpass is 102 ft.

Note - The calculations involved in plotting vertical curves can be made much easier by using computers or programmable calculators.  Computers usually recognize exponents with a symbol known as a caret (^).  For example, with a spreadsheet program, the equation: 

 y = -0.2x² + 4  wherex = the contents of cell A1 

would have the spreadsheet equation of:  

=-0.2(A1^2)+4 

Most spreadsheet programs can even plot the curve for you.

Other Uses for Parabolas

Parabolas are also found in many other places.  Satellite dishes use a special property of parabolas to collect electromagnetic waves.  A dish with a parabolic cross-section will focus all parallel waves coming to it to a single point.  This allows us to amplify signals coming from thousands or even millions of miles away.

Quadratic equations, the equations that describe parabolas, occur often in math.  In the examples here we’ve looked at how to find y when you know the value of x.  The standard form of a quadratic equation is shown below:

y = ax² + bx + c 

For most quadratic equations, you need to find the value of x when you know the value of y.  There are several techniques to solve quadratic equations that are explained in the book.