# The Physics, Astronomy and Mathematics of the Solar System

## Discovering Conic Sections in the Motion of Heavenly Bodies

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## Classroom activities

Each of the class activities will correspond to the subject headings above for a total of six lessons. In outline form the lessons are as follows:

- 1) Discovering Conics (Introduction)
- 2) The circle
- 3) The ellipse
- 4) The parabola
- 5) The hyperbola
- 6) Unified conics

### First lesson

The first lesson will include a hands on activity where students will be given styrofoam cones along with a piece of piano wire for slicing the cones. The students will be instructed to slice the cones at different angles. The students will describe in writing the various shapes that are formed.

There will also be a demonstration with a sundial. In a darkened room a high intensity lamp will be moved about the gnomon of a sundial. Students will write their observations.

### Third lesson

For the lesson on the ellipse the class will begin with discussion of Kepler's laws. The students will then complete the following worksheet:

*
Worksheet
*

Work with a partner. Place the piece of paper with the vertical line and horizontal line on the cardboard in the landscape position (so that the horizontal line is longer than the vertical line). Push the pushpin into the center of the paper where the two lines intersect. Take the string loop and place it around the pushpin. On the other end of the loop position your pencil and take the slack out of the string. Trace all the way around.

- 1. What shape have you drawn?
- 2. If the horizontal line is the
xaxis and the vertical line is theyaxis, write the standard equation for this shape.

Now, take the two pushpins and place them on the horizontal line equidistant from the center such that there will still be some slack if the string loop is placed around the pushpins. One partner should secure the pins as the other takes the string loop and places it around the pushpins. Take the pencil, removing slack from the string loop, and trace out a figure.

- 3. What shape have you drawn?

The points where the pushpins are placed are each called the focus; in the plural, foci. Label them F
_{
1
}
and F
_{
2
}
respectively. Now, pick any three points on the figure you have drawn and label each P
_{
1
}
, P
_{
2
}
, and P
_{
3
}
. Using your ruler, measure the distance, as best as you can, from each focus to each point on the figure to complete the table below.

Distance in cm | F
_{
1
}
| F
_{
2
}
| Sum F
_{
1
}
+ F
_{
2
}

P
_{
1
}

P
_{
2
}

P
_{
3
}

Using a new sheet of paper create two more ellipses using different points on the horizontal (
*
x
*
) axis. Be sure that the foci points are equidistant from the center. Complete the tables below for each.

Distance in cm | F
_{
1
}
| F
_{
2
}
| Sum F
_{
1
}
+ F
_{
2
}

P
_{
1
}

P
_{
2
}

P
_{
3
}

Distance in cm | F
_{
1
}
| F
_{
2
}
| Sum F
_{
1
}
+ F
_{
2
}

P
_{
1
}

P
_{
2
}

P
_{
3
}

- 4. The sum listed in the last column of each table is also known as the focal radii. Looking at the sums in the last column, what conjecture are you able to make about the focal radii? (remember, there may be some measurement error).

Now, using the three ellipses you have already drawn, measure the distance along the horizontal (longer) axis to each vertex of the ellipse and write it in the table below in the column labeled major axis. From the tables above also write the focal radii (Sum F
_{
1
}
+ F
_{
2
}
) for each of the ellipses.

(table available in print form)

- 5. Using the data from the table above, what conjecture are you able to make about the relationship between the focal radii and the major axis?

Now, measure along the vertical (shorter) axis from vertex to vertex and write the result below.

(table available in print form)

Assume your ellipses were drawn on the co-ordinate plane with the vertical line as the
*
y
*
axis and the horizontal line as the
*
x
*
axis in order to answer the following questions.

- 6. What is the relationship between the
xco-ordinate of the vertex and the measure of the major axis?- 7. What is the relationship of the
yco-ordinate of the vertex and the measure of the minor axis?- 8. What conjecture can you make about the measure from each of the foci to the vertex at the minor axis?

Use the above data and your conjectures to make generalizations about ellipses in order to answer the following questions. It may also be useful to recall the distance formula.

- 9. If the length of the major axis of an ellipse centered at the origin is 10 units, what are the
xco-ordinates of the vertices?- 10. If the length of the minor axis of the same ellipse is 6 units, what are the
yco-ordinates of the vertices?- 11. What is the distance from each of the foci to the vertex at the minor axis?
- 12. What will be the
xco-ordinates of the foci for this ellipse?- 13. For the ellipses we made today we know that one half the measure of the major axis (semi-major axis) will equal the absolute value of the
xco-ordinate of the vertex of that axis. What is the quotient of the absolute value of thexco-ordinate of the vertex of the major axis divided by the length of the semi-major axis (i.e., what is the quotient of a number divided by itself)?- 14. What is the quotient of the absolute value of the
yco-ordinate of the vertex and the measure of the semi-minor axis?

### Fourth lesson

Using a garden sprayer a demonstration of a parabolic trajectory will be shown. Students will then answer a series of questions on what factors will alter the trajectory, such as pressure, sprayer height, sprayer angle.

### Sixth lesson

The final lesson will include an interactive website where students will answer the following questions: What conic section do you get when 0
*
e
*
1? When e > 1? When
*
e
*
= 1? How does the shape of the conic section change as
*
e
*
gets closer to 0? How does the shape of the conic section change as
*
e
*
gets closer to 1? Explain why the eccentricity is equal to the distance from the vertex
*
E
*
to the focus
*
F,
*
divided by the distance from
*
E
*
to point
*
A
*
on the directrix. Drag point
*
P
*
on the ellipse. What happens to the distances between
*
P
*
and the focus and between
*
P
*
and the directrix? What happens to the ratio of these distances? Why is it correct to say that the eccentricity of a given ellipse is constant? Does the eccentricity remain constant as the distances between
*
P
*
and the focus and
*
P
*
and the directrix change? Describe the changes that occur in the graph and in the eccentricity as point
*
E
*
moves closer to the focus and as it moves closer to the directrix. What do you understand better about conic sections as a result of working this problem?
^{
4
}