JUG7

"Out Juggling In My Class" by William V. Thayer

Act 7. JUG:007 Juggleometry

copyright © 1996 Wm. V. Thayer, All Rights Reserved

Instructions: Start juggling the pins with pictures of Galileo and formulas on them. Flaunt the pins and face.

Galileo is the main source for the next course.

JUGGLEOMETRY

This is the trigonometry course of my MAJOR with my UNIT JUGGLE at the center.

A unit juggle is a sequence of 16 inch high by 16 inch wide UNIT TOSSES from hand to hand.

Hold up the green 16 inch cut out then turn it slowly to male a parabola turn into a straight line.

Let's look at the parabolic path of a ball's motion due to gravity.

The horizontal position is closely modeled by our old friend d = r t.

In this case x = r t where r is the horizontal rate and t is the time in flight.

To find the horizontal rate, r, one needs to know a total flight time, F.

I tried using a STOP WATCH.

I put the stop watch in my mouth.

Then used my tongue to start and stop it.

That was a real tongue twister.

Then I thought of using video tape to get the height of the toss.

Consider how long it takes to drop a ball 16 inches using s = 193 t².

This transparency will help you follow my steps.

Set 16 = 193 t², then t² = 16/193 = .0829, and taking the

square root of both sides gives t = .288 sec.

Since up equals down flight time we take twice .288 making total flight time

F = .576 sec.

The horizontal rate is found by: r = 16/.576 = 27.8 in/sec.

Then the horizontal model of motion is: x = 27.8 t inches.

Now let us consider the vertical direction of motion.

Again we use Galileo's model s = - 193 t² with a shift to start from the origin.

That shift makes our basic model for vertical motion: y - k = -193 (t - h)²

for peak height k at time h.

A vertical translation of k = 16 inches and a time translation of h = .288 sec is

needed to have the peak of the flight at the right height and time.

With substitution and simplifying we end up with y = -193 t² + 111 t inches.

Now we have a unit toss given by horizontal position x = 27.8 t inches and vertical position given

by y = -193 t² + 111 t inches.

If t = .1 seconds then x = 2.78 inches and y = 9.17 inches.

I looked into research on the time interval it takes a person to catch an object and

found it was plus or minus 15 milliseconds.

That corresponds to a variation of plus or minus 1.6 inches on the length of the ball's path.

Now we see that PATTERN HEIGHT IS THE JUGGLER'S CLOCK.

Now we do the GEOMETRY of the unit toss.

Present a graph.

Correct x-y coordinates help us make computer game programs simulate flying objects.

Let's model the path with y = a x² + b x + c.

This transparency may help you follow the discussion.

Substitution of (0,0) for x and y respectively gives c = 0.

Likewise, x intercept = 16, when y = 0, and vertex (8,16) yield two equations for unknowns a and b.

By solving this system of two equations and substitution we get:

y = -(1/4) x² + 4 x.

Hold up a graph and point to

The FOCUS OF THE UNIT TOSS is one inch below the VERTEX.

Put a ruler on the UNIT TOSS as a tangent line and then a secant line.

Direct measurements such as angle, slope, even area under a curve can easily be made from a large graph.

Balance UNIT JUGGLE on a finger or something.

For example, we counted graph grid squares to find area equal to 171 square inches.

Hold a 4" ball up to fit the curve at the vertex.

Jennifer, my student, found that the tangent circle at the vertex has a 4 inch diameter.

Use a flex tape and say:

Jeff found the length of the curve to be 37.2 inches.

Table of various height tosses.

Mike looked at several different juggling heights then announced a theorem.

Every toss must be aimed at twice the peak height on the axis of symmetry. CHANT C.

Workbook exercises give more information related to the unit toss and other juggling heights.

The set of parallel profile curves of a ball in flight are not parabolas.

May all this be too much too fast? - -

Music for their RHYTHM RECESS juggling.

Should we take a Rhythm Recess. We can work on coordination and rhythm with two objects.

Act 8.

Back to the course catalog.

For more information please contact: William V. Thayer, Mathematics Department, St. Louis Community College at Meramec, 11333 Big Bend Blvd., St. Louis, MO 63122-5799, Telephone 314 984 7866 office
email: thayer@stl-online.net, home page: http://www.stlcc.cc.mo.us/mc/users/thayer

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