High School Mathematics-PHysics SMILE Meeting
07 October 2003
Notes Prepared by Porter Johnson
Porter Johnson touted the book The Golden Ratio and Fibonacci Numbers {Springer Verlag 1997: ISBN 981-02-3264-D).  In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed.  In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relation to the golden ratio are presented.
Roy Coleman passed out information on weather and rain, that came from articles by Tom Skilling, which appeared in recent issues of the Chicago Tribune. They contained the following information:
Properties of Typical Clouds:
Cloud Type Water vapor density Vertical height Horizontal spread Mass
Cirrus 0.1 g/m3 1 km 25 km ´ 25 km 8 ´108 kg
Cumulus 0.2 g/m3 1 km 1 km ´ 1 km 2 ´105 kg
Cumulus Congestus 0.8 g/m3 5 km 2 km ´ 2 km 2 ´10 8 kg
Cumulonimbus 1.0 g/m3 10 km 6 km ´ 6 km 4 ´109 kg
Useful data for developing physics problems! Thanks, Roy!

Roy Coleman [Morgan Park HS, Physics]     Estimating the Speed of Falling Rain Drops
First, go out and buy a car in which the rear window slopes at a modest angle (say, q » 30°) to the horizontal.  Then, drive your car during a rainstorm, and find the minimum speed v0 at which falling raindrops do not strike that rear window.  If there is no wind, the speed of the falling raindrops vf  should satisfy the relation vf / v0 = tan qSimple, non?  If the wind is blowing, repeat the observation while going in the opposite direction as well, and take the average.  Be sure not to crash into anything while doing this experiment!
Now, there's a good reason to get rid of the old clunker and get a new car.  Nice work, Roy!

Bill Blunk [Joliet Central HS, physics]       Paper Match Rocket and More
constructed a launching pad using a piece of cardboard paper, and a paper clip bent to support a light object leaning against it [a paper match stick -- wooden matches are too heavy to work].  First he tried to launch the paper match just by putting it on the launching pad and lighting it.  The launch fizzled, because the match just sat still and burned.   Why didn't it go flying away?  After some discussion it was decided that there was no net impulse given to the paper match in this process, since the exhaust gases from the match were sent in all directions.  One must find a way to direct the flow of exhaust gases to provide a net impulse.  So, Bill wrapped the head of the paper match several times with a small piece of aluminum foil, pressed it tightly at the top so that exhaust gases would come only out the bottom, and placed the paper match head-up onto the launch pad.  Bill then lit another match and held it under the wrapped head of the match on the launch pad.  There were audible hissing sounds from inside the foil, as the match head ignited and (anaerobic) combustion began.  A split second later, the paper match jetted off the launch pad, and bounced off the ceiling.  Details on construction and operation were given in last week's summary:  mp092303.htmlBill, you really set things on fire, intellectually speaking!

Bill also presented an extension of last week's lesson on balancing an egg [mp092303.html] at the Autumnal equinox.  In particular, he pretended to "balance" a golf ball on a horizontal board.  Of course, one would not expect the ball to move, because it is round -- unlike an egg placed on end. and it did not move.  Then, Bill slowly tilted the board up on one end, making an angle of about 30° to the horizontal -- and the golf ball still did not move!  Amazing!  After extensive cross-examination by the group, Bill finally admitted that the experiment was a hoax.  Namely, the golf ball was spherical in shape, but its center of mass lay significantly below the geometrical center.  Bill had made his annual pilgrimage to Amazing Toys in Great Falls MT.  This item can also be ordered through their website http://www.amazingtoys.netVery slick, Bill!

Siegerschnecke -- which means Snail Race auf Deutsch.  Bill called attention to a very important race between trained snails which was held in Cremonia (Alpine Italy) last Summer.  By holding a piece of lettuce and crawling in front of the snail, the winning snail trainer (female, age 11) had coaxed the snail to travel 1 meter in 450 seconds, corresponding to an average speed of about 2 mm/sec. As prize for this victory, she and her pet snail received a lettuce bowl.  This speed is significant, in that it is greater than a typical drift velocity of electrons in a conducting wire, even at relatively high currents. And, think of how proud the winning team must be in this annual event, described in the (Deutsch) website (with pictures) given here: http://www.toponline.ch/area-1.rub-39.art-39031.tce
Fascinating topics and spectacular stuff, Bill!

Imara Abdullah [Douglas Academy,  science]        Posters
provided us with poster paper, colored markers, and tape, and asked each of us to prepare a poster to illustrate some concept or process in mathematics or science.  We came up with the following displays:

Name Display description  Concept or process illustrated
Porter Johnson blank sheet of paper Vacuum, empty space, cosmic void
Bill Colson Flow chart 3 ® 1 3 conditions for triangle congruence
Roy Coleman I'm a p r2 (big wheel)  Area and circumference of circle
Elizabeth Roombos Rock hurled off cliff Projectile motion
Marilynn Stone Click-clack apparatus Momentum conservation
Monica Seelman 45°-45°-90° triangle Pythagorean Theorem
Earl Zwicker Sequential images of ball
on inclined plane
Galileo experiments in mechanics
Imara Abdullah Walking dog around block Perimeter
John Bozovsky Kneeling carpenter drilling into wall Niels Bohr (kneeling and boring)
Larry Alofs Rectangle at new IIT student center   Golden rectangle  -- or not?
Jane Shields Colored strips on paper Northern lights
Babatunde Taiwo
Rocks thrown simultaneously
up and down
Do they hit the ground
at the same time?
Walter McDonald Headlight beam image Illumination: Inverse square law
Rich Goberville Projectile shot from cannon Action-reaction Forces
Bill Shanks Plumb bob demons Universal gravitation
Fred Farnell Light charged balls on strings Coulomb's Law: Electrostatics
Leticia Rodriguez See-through skeleton Systems in human body
John Bozovsky Truck accident How the Mercedes bends
Imara showed us how to display individual posters, using two sheets of transparent Plexiglas™ sheets, held by two binder clips.  Nifty, eh!

We were all on our feet and involved!  Beautiful Activity, Imara.

Fred Schaal [Lane Tech HS, mathematics]        Lubbock*** [TI-83] Helper
used his calculator and plotter to extend his lesson of 29 February 2000 [mp022900.htm] concerning tangent lines and asymptotic curves.  This lesson had, in turn, been an extension of a lesson by Bill Colson on 11 November 1999 [ph110999.htm] on producing intricate geometrical patterns. These comments by Porter Johnson are excerpted from the first lesson:

The formula for line segment lying in first quadrant and intercepting the x-axis at x = a and intercepting the y-axis at y = b - a, where 0 < a < b, is

x/a + y/(b-a) = 1
The asymptotic curve, which represents the envelope of all the straight lines, can also be written as
y = (Ö b - Ö x )2     or      Ö x + Ö y = Ö b      or     (x - y)2 + b2 = 2 b (x + y) 

Note: some browsers will incorrectly render the square root sign (Ö).

 The curve is a parabola with the line of symmetry [axis] lying along the line x = y. Bill used the graphing calculator to extend the problem to an extended region:   -a < x < a and -a < y < a.  What did it look like? Amazingly, the original asymptotic curve (caustic) was symmetrically extended to all four quadrants:

 Ö |x| + Ö| y| = Öb

How come?

Fred's trusty programmer, Bill Colson, developed and displayed the graphs for xp, where the real, non-integer variable p increases from 1 to 2.  The curves all passed through x = 1, but became steeper below that point for x < 1 as p was increased.  For x > 1, it was the other way around.  Fred showed the graphs for negative x as well.  Surprisingly, for p = 1.1, 1.3, 1.5, 1.7, and 1.9 the curves were negative, whereas for p = 1.2, 1.4, 1.6, and 1.8 they were positive.  How come?

Comments by Porter Johnson:  For negative x, there is a real, positive value of  p whenever p is a rational fraction of the form a / b, where the integer b is even.  Furthermore, there is a real, negative value of  xwhen p=a / b is rational, when b is odd. The gremlins and lubbockians*** of calculator programming were definitely on your side for this one, Fred.

Q: How many Texas Tech football supporters does it take to change a light bulb?
A: Don't be silly; there's no electricity in Lubbock*** [home of Texas Instruments™]!
Bigger plots are definitely better!  Good, Fred!

Ann Brandon [Joliet West HS, physics]        Inertia and Seat Belts
put a "bear crash doll" onto a physics collision cart, and released the cart at the top of a plank inclined at a moderate angle (about 30°) to the horizontal table.  The cart rolled down the plank, off its edge, and onto the table. At that point the bear-doll flipped out of the cart, and landed on its head. This was a perfect illustration of Newton's Second Law:  A force is needed to change the velocity of the bear.  A force -- supplied when the bear-doll struck the table -- is required to change the direction of motion of the bear-doll; otherwise it would just keep going and going and going.  It also illustrates the need to wear seat belts in an automobile.  Next she strapped the bear into the cart with a rudimentary (rubber band) seat belt, and again released the cart from the top of the plank.  This time, the bear whacked its head against the "dashboard", and experienced severe whiplash.  Again, Newton's Laws are at workAlso, the need to wear shoulder straps, as well as seat belts, was shown by Ann, using an additional rubber band.

The collision of the two Boston Red Sox outfielders [Johnny Damon and Damian Jackson] http://sports.espn.go.com/mlb/playoffs2003/news/story?id=1632209 chasing after a pop fly during the baseball playoff games was discussed. One of the players suffered a concussion, and was unconscious for a few minutes, although no brain damage apparently occurred as a result of the collision.  A great danger in a concussion is brain swelling, which must be treated quickly to prevent permanent brain damage.
It was also mentioned in discussion that Princess Diana and all other occupants of the car died as a result of the automobile crash  in Paris -- except for the front "shotgun" passenger, who was wearing a seat belt.  For details see the website Animation of Princess Diana Collision:  http://www.knottlab.com/aboutus.aspx?page=hi-profile%20cases&id=1 and  Princess Diana:  Cause of Death:  http://www.dianaspeaks.info/AutopsySummary.html

Ann, you showed us the physics and importance of buckling up with these elegantly simple experiments!

Porter Johnson [IIT, physics]        Physics of Baseball  
asked the following questions concerning baseball:

He suggested that certain insights are contained in the perfect gas law, PV = n RT, where P is the gas pressure, V  is the gas volume, n is the number of moles of gas, R is the gas constant, and T is the absolute temperature. The density of the gas r is given by the number of moles per unit volume n/V, multiplied by the molecular weight M.  Thus, we may write the Perfect Gas Law in the form P = r R T / M.  The relevant quantity  for air resistance is the density r, which may be expressed in terms of M, T, P as
r =  (M P) / (R T)
The density of air is higher at low temperature {cool days}, at high air pressure {sea level), and in a gas of higher molecular weight (more O2 and N2; less H2O). Under these conditions, there is more air resistance, so that well-hit balls will not travel quite so far.  In addition,  pitches will curve more, move around more,  and "drop" more, making them harder to hit.  Pitchers prefer such conditions for these reasons, which are explained by simple physics.

Go Cubbies!

Notes taken by Porter Johnson