| |
 |
 |
Exercise
1:
Go to Newtonian
Gravitation and the Laws of Kepler and read about how
conic sections are demonstrated in the motions of the heavenly
bodies (planets, comets, etc.). Find the following:
| 1) |
Two
examples of bodies with almost circular orbits (stating
one focus of each) |
| 2) |
Two
examples of bodies with elliptical orbits (stating
one focus of each) |
| 3) |
One
examples of a body with a parabolic orbit |
| 4) |
One
example of a body with hyperbolic motion |
| For
the last two, you may need to investigate this site
further. (Try going home
to see what is there.) |
|
|
Exercise
2:
Parabolas aren't
just very important to the study of astronomy because they
describe the paths of some comets! Did you see the movie Contact?
Visit The
Circle of Curvature, the Big Dish, and the Little Green Men
and give a brief mathematical description of big dish at Arecibo
and write a paragraph about why they are using the dish. Now
go read about Liquid
Mirror Telescopes. Write a paragraph on the advantages
of this technology and how it works. |
|
Exercise
3:
Exercises 1 and
2 had you investigate how parabolas are used in space exploration.
To see a demonstration of how the parabolic dishes collect
their information, go try this Focus
of a Parabola applet. Do you think a telescope would be
more effective with the dish being wider or narrower? |
|
Exercise
4:
This JAVA applet,
Drawing
Parabolas, will allow you to change the values of a, the
distance from the vertex to the focus, and see what happens
with the graph. Read the instructions and answer the question
asked on the site. |
|
Exercise
5:
Visit this parabola
JAVA applet and experiment with the values of p. (Note
that they are using the letter "p" where our text
uses an "a".)
Let
p = 50, then p = 25, then p = 2, then p = -2,
then p = -25
and, finally p = -50.
What
conclusions can you draw based on what you've seen? What
happens when you let p = 0? Now let p = 0 in your standard
equation for the parabola. What happened?
(answer)
*********
NOTE!! CHECK a = p!!!!!!
|
|
Exercise
6:
Did you know that,
if you were in a room with an ellipsoidal ceiling and you
were positioned at one focus with your friend at the other,
you could whisper across a crowed room and, not only would
your friend here you perfectly, no one else would be able
to? To appreciate the power of the ellipse, visit Reflective
Properties of Ellipses. When you play with this applet,
picture the whisper shooting from one focus to the other.
Also, be sure to move the two foci together to see what happens. |
|
Exercise
7:
Now that you've
seen how particles (or whispers) can "bounce around"
an ellipse, go to Reflective
Properties of Hyperbolas and see what happens with the
hyperbolic shape. |
|
Exercise
8:
Go to Key Curriculum
Press' site, The
Conic Sections as the locus of Perpendicular Bisectors.
It will take a few minutes to load. Play with the JAVA applet
by dragging the red dots around with your mouse. Specifically,
notice what is happening with the eccentricity while you have
an ellipse. Answer the site's first three questions. |
|
Exercise
9:
To investigate
parametric curves a little more quickly, let's visit this
special parametric
curve calculator. When you first enter the site, these
equations will appear:
x =
t * cos(t)
y = t * sin(t)
(Note:
You will have to scroll down to see the graph!)
Go
ahead and plot these and make a sketch (by hand) of the
graph you see. Now, change the equations to:
x =
5* cos(t)
y = 5 * sin(t)
What
happened to the graph? (Be sure to pay close attention to
the "tick" marks on the x and y axies!) Now try:
x =
5 * cos(t)
y = 15 * sin(t)
What
happened this time? Arrive at a conclusion about what was
happening with the first set of parametric equations you
graphed as opposed to selecting specific values (like 5
and 15)... What caused the first set to form the spiral?
(answer)
|
|
Exercise
10: (new)
Go to the search
engine HotBot
and find one real-world application of conic sections. Do
a brief write-up of your findings. |
|
Exercise
11: (new)
Parametric equations
allow us to graph a whole new world of things. To see some
great examples, go back to the Famous
Curves Index. Write down the formulas for:
The
Astroid,
The Epicycloid,
The Hypocycloid and
The Involute of a Circle.
|
