Parabola

What does a fountain have in common with a baseball? What possible connection can there be between an MG headlight and the Golden Gate Bridge?

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An important mathematical curve called the parabola. A spout of water takes the shape of a parabola for the very same reason that when we throw a ball in the air its path follows this same curve.The parabola also has some interesting physical properties that make it the ideal shape for the reflector in a car headlight and for a suspension bridge.

What is a parabola? Mathematically, it is the set of all points that are equidistant from a point and a line. The line is called the directrix and the point is called the focus. Each point on the parabola is as far from the directrix as it is from the focus, so distance 1 (labelled d1) is always equal to distance 2 (labelled d2).

Using this definition, we can draw a parabola using a triangle, t-square, tack and a piece of string. The string is the same length as the side of the triangle. The tack is the focus of the parabola and the t-square is the directrix. Putting the point of our pen against the string, and sliding the triangle along the t-square--always keeping the pen against the triangle, our pen will draw a parabola. Why? For any position of the pen as it moves, the length marked "c" from the pen to the focus,.is the same length as from the pen to the t-square vertically down the triangle.

Using the definition of a parabola, it is possible to derive an equation and then graph the parabola on a coordinate system. An example of an equation for a parabola, and the graph of the equation, are shown below:

Another way a parabola is defined mathematically is as one of four curves called the Conics. These curves can all be cut from a cone. If we cut a cone at an angle, we will get a parabola. The beam of light from a flashlight is in the shape of a cone, and held at different angles, will form first a circle, then an ellipse, and next the curve of a parabola.

The parabola has an unusual and useful mathematical property: parallel lines coming in to the parabola will reflect off the curve and pass through the focus. A line drawn tangent to the curve shows how this happens. The line coming in makes an angle with the tangent. This angle is called the angle of incidence, and it will always be equal to the angle of reflection.

The metal surface of a radar receiving telescope uses this property of the parabola to gather in sound or light waves and focus them at a point. That is why this point of the parabola is called the focus. Both a car headlight and a flash-light use this principle in reverse. The metal reflecting surface inside the light is in the shape of a parabola, and the light bulb is placed at the focus. Light rays are reflected in a beam of light. An object thrown in the air ... even droplets of water propelled by a hose or fountain ... follow a path forced on them by the vertical pull of gravity.

When you hit a baseball, or throw a ball in the air for a friend to catch, it goes up and then gravity pulls it down in a smooth arc called a parabola. You can find some information about how to throw a "curve ball" at the following link, on the Exploratorium website. The Exploratorium is a fascinating hand-on science museum in San Francisco:

http://www.explorato rium.edu/baseball/curve.html

You can find out more about the mathematics and science of baseball at another link on the Exploratorium website:

http://www.explorato rium.edu/baseball/index.html

There is another curve that looks very much like the parabola, but has different mathematica properties. This curve is called the Hyperbola, and is explored in greater detail at the following link:

Go to The Hyperbola