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ACTION Instructions: Start juggling the pins with pictures of Galileo and formulas on them. Flaunt the pins and face of Galileo.
Galileo is the main source for the next course.
JUG:007 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.
ACTION Hold up the green 16 inch cut out then turn it slowly to make 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.
FIRST TRANSPARENCY • The horizontal model of motion for the UNIT TOSS
Consider how long it takes to drop a ball 16 inches using s = 193 t².
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 inches/.576 seconds = 27.8 in/sec.
Then the horizontal model of motion is: x = 27.8 t inches.
SECOND TRANSPARENCY • Now lets consider the vertical direction of motion for the UNIT TOSS.
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.
THIRD TRANSPARENCY • Summary of the UNIT TOSS x and y coordinates based on time t.
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 = 0.1 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 plus or minus 1.6 inch variation on
the length of the ball's path.
So we see that PATTERN HEIGHT IS THE JUGGLER'S CLOCK.
FOURTH TRANSPARENCY • Now we do the GEOMETRY of the UNIT TOSS.
Now we do the GEOMETRY of the unit toss.
ACTION Present a graph.
Let's model the path with y = a x² + b x + c.
Substitution of (0,0) for x and y gives c = 0.
Likewise, x intercept = 16, where 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.
ACTION Hold up a graph and point to
The FOCUS OF THE UNIT TOSS is one inch below the VERTEX.
ACTION 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.
ACTION Balance UNIT JUGGLE on a finger or something.
For example, we counted graph grid squares to find area equal to 170 2/3 square inches.
ACTION Hold a 4" ball up to fit the curve at the
Jennifer, my student, found that the tangent circle at the vertex has a 4 inch diameter.
ACTION Use a flex tape and say:
Jeff found the length of the curve to be 37.2 inches.
ACTION 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 Mike's Juggling Theorem
The set of parallel profile curves of a ball in flight are not parabolas.
Some workbook exercises may give more information related to the
UNIT TOSS and other juggling heights 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!
FIFTH TRANSPARENCY • Music and guidelines for their RHYTHM RECESS.
We can work on coordination and rhythm with two objects.
First some instructions.
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.
Correct x-y coordinates help us make computer game programs simulate
flying objects.
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.
Ok, RECESS is OVER. Please take a seat and save the paper balls for next time.
•Next JUG:009 UFTOJ Claude E Shannon
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