ACTION Instructions: Start juggling three balls (perhaps earth balls or globes against an incline Galileo plane held by three or more volunteers willing to orbit a ball on the parachute. If Appendix A Script is used.)

Welcome to: **JUG:007A JUGGLEOLOGY**

In our first course, JUG:123(A), we spoke of

John A. Wheeler's demonstration* that a tossed

object's spacetime geodesic flight path is a

free fall parabola in space.

Now I may toss a ball and expect it to

follow a geodesic straight line

path in the spacetime of planet earth.

That geodesic straight line is **the parabola** in each toss.

ACTION Instructions: Have some folk hold up measuring equipment to get 16" x 16" dimensions.

Let's say I toss 16" high by 16" wide over and over again.

FIRST TRANSPARENCY • The 16 by 16 inch UNIT TOSS

Call this a "UNIT TOSS".

Do these parabolas - - **geodesics** - - amaze you?

They **astonish** me!! I hardley believe this is happening.

Now let's attempt to find an equation for this Unit Toss.

Let's simplify our task first by flipping this

parabol upside down putting the vertex at the

origin with coordinates (0, 0).

SECOND TRANSPARENCY • A Simple Parabola *y = (1/4) x²*

The catching point is now located up high with coordinates (8, 16).

Algebra students know how to find the value

of *a* in *y = a x^2*.

We use the following method.

When *y = 16* and *x = 8* on this graph we have:

*16 = a 8^2*.

* a = 16/64 = 1/4 *!

Then *y = (1/4) x^2*.

However, we need our graph to represent the UNIT TOSS

in the upright position - vertex at the top.

A reflection of this curve in the x axis

will do that inversion for us.

THIRD TRANSPARENCY An Inverted Parabola *y = -(1/4) x²*

A flip like that moves the point (8, 16) to the

point (8, -16) and likewise all other (x, y)

coordinates become (x, -y).

An equation change from *y = +(1/4) x^2* to

*y = -(1/4) x^2*.

parrots this downward flip.

Now let us push the new curve up from

its depth of sixteen inches.

The point (8, -16) moves to (8, 0) and the

vertex goes from (0, 0) to (0, 16)

Here is the picture.

FOURTH TRANSPARENCY • Centered UNIT TOSS *y = -(1/4) x² + 16*

What is the new equation resulting from the upward push?

Yes you are right to proclaim

* y = -(1/4) x^2 + 16*

for the points of this curve.

I am still not satisfied.

I want the Unit Toss to start at the origin.

A translation of eight inches to the right will do that:

Let * (x - 8) * replace * x * to commence

firing, that is, tossing from the origin.

Good, but tossing from the origin (0, 0) will change the equation:

Consider writing the equation * y = -(1/4) x^2 + 16*

with a *(x-8)* replacement for *x*.

* y = -(1/4) (x - 8)^2 + 16* first,

then simplify to get:

* y = -(1/4) x^2 + 4 x*

FIFTH TRANSPARENCY • UNIT TOSS *y = -(1/4) x² + 4 x*

All the while - unknown to you - my student, Jennie, has

been videotaping from the side.

At about 30 frames per second averaged over several

UNIT TOSSES she found the time per toss = .576 seconds.

Jennifer said; " Mr. Thayer, the Unit Toss has a .576 sec.

time interval."

Galileo demonstrated that the horizontal

motion of a toss was linear.

That means *x = r t* is appropriate

where *r = 16"/.576 sec*.

*x = 27.8 t* and a substitution of this

time formula for *x* into:

* y = -(1/4) x^2 + 4 x* makes y related to time in:

* y = -193 t^2 + 111 t*.

SIXTH TRANSPARENCY • UNIT TOSS Time Study *y = -193 t² + 111 t * And *x = 27.8 t*

Consider this:

Light from earth travels around half way to

the moon in a short time of 0.576 sec.

And - Did you notice that an "accereration do to gravity

constant" ** fell** out of our developement!

That is true. Check it out!

Realize that any student can find it without Newtonian physics!

The UNIT TOSS is interesting but lets have

some physical activity right now!

ACTION GET TWO BALLS AND SAY:

Hay, Let's try a new rhythm - how about TOSS - - TOSS

to celebrate the UNIT JUGGLE!

SEVENTH TRANSPARENCY • Music and guidelines for a RHYTHM RECESS

We can work on coordination and rhythm with two objects.

First some instructions.

Find the BALL #2 page and make a second paper ball.

Hold one paper ball in each hand.

Throw the right-hand ball in an arc toward your left hand.

Say TOSS as this ball peaks.

And, just as it peaks, throw the second ball

in an arc underneath it toward the right hand.

Catch the first ball in your left hand

and the second in your right.

Aim to get your peaks out in front of opposite ears.

Say TOSS - TOSS in rhythm.

Ok, go ahead and stand up, everyone, and toss two balls.

Hands out.

Eyes on the peaks.

When one ball peeks, toss the other ball.

Same height, 16 to 20 inches each - and - no handoffs.

PAUSE Move around the audience helping the TOSS - TOSS pattern for each person.

ACTION
Music turned off and short soft toot on whistle.

Ok, RECESS is OVER. Please take a seat and

save the paper balls for next time.

*

•Next JUG:009A UFTOJ Claude E Shannon

•Back to the modern course list

For more information and ideas about working with parabola projects you may link to a page written by one of the these math artists:

William V. Thayer, PedLog contact us!