Parabola

parabola

When you kick a soccer ball (or shoot an arrow, fire a missile or throw a stone) it arcs up into the air and comes down again ...

... following the path of a parabola!

(Except for how the air affects it.)

Try kicking the ball:

 

parabola

Definition

A parabola is a curve where any point is at an equal distance from:

  • a fixed point (the focus ), and
  • a fixed straight line (the directrix )

Get a piece of paper, draw a straight line on it, then make a big dot for the focus (not on the line!).

Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line.

Keep going until you have lots of little dots, then join the little dots and you will have a parabola!

 

parabola

Names

Here are the important names:

  • the directrix and focus (explained above)
  • the axis of symmetry (goes through the focus, at right angles to the directrix)
  • the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix.

 

parabola

Reflector

And a parabola has this amazing property:

Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.

And that explains why that dot is called the focus ...

... because that's where all the rays get focused!

parabolic dish

So the parabola can be used for:

  • satellite dishes,
  • radar dishes,
  • concentrating the sun's rays to make a hot spot,
  • the reflector on spotlights and torches,
  • etc

 

conic section parabola

We also get a parabola when we slice through a cone (the slice must be parallel to the side of the cone).

So the parabola is a conic section (a section of a cone).

Equations

Place the parabola on the cartesian coordinates (x-y graph) with:

  • its vertex at the origin "O" and
  • its axis of symmetry lying on the x-axis,

then the curve is defined by:

y2 = 4ax

parabola on coordinates

 

Example: Where is the focus in the equation y2=5x ?


Converting y2 = 5x to y2 = 4ax form, we get y2 = 4 (5/4) x,

so a = 5/4, and the focus of y2=5x is:

F = (a,0) = (5/4,0)

The equations of parabolas in different orientations are as follows:

parabola orientations
y2 = 4ax y2 = -4ax x2 = 4ay x2 = -4ay

Measurements for a Parabolic Dish

If you want to build a parabolic dish where the focus is 200 mm above the surface, what measurements do you need?

To make it easy to build, let's have it pointing upwards, and so we choose the x2 = 4ay equation.

And we want "a" to be 200, so the equation becomes:

x2 = 4ay = 4 × 200 × y = 800y

Rearranging so we can calculate heights:

y = x2/800

And here are some height measurements as you run along:

parabola orientations Distance Along ("x") Height ("y")
0 mm 0.0 mm
100 mm 12.5 mm
200 mm 50.0 mm
300 mm 112.5 mm
400 mm 200.0 mm
500 mm 312.5 mm
600 mm 450.0 mm
   

If you build one tell me, and I can include a picture of it!