High School Mathematics Physics SMILE Meeting
08 October 2002
Notes Prepared by Porter Johnson

Don Kanner [Lane Tech HS, Physics]     Rocket Ship Physics
simulated the motion of a rocket ship in free space by blowing up a balloon and releasing it above the table.  The balloon expelled air and was propelled forward, in analogy to a rocket ship that expels burned fuel and is pushed forward.  Don reasoned that, when gas is expelled at a constant rate, the rocket ship will have an increasing acceleration, because its mass is continually decreasing.   The rate of change of acceleration with time, Da/Dt, which is commonly called the "bump" or "jerk", is non-zero in this case.  He asked us how to handle this case of changing acceleration. Porter Johnson commented that, while higher derivatives of position with respect to time can always be calculated, in Newtonian dynamics, nothing beyond the second derivative [acceleration] plays a fundamental role. For rocket dynamics in free space, it is sufficient to apply conservation of momentum, since the sum of the momenta of the rocket and of the expelled fuel does not change with time.  The forces between the rocket and fuel being expelled are equal and opposite, by Newton's third law, and thus the total momentum is conserved.  To explore the dynamics let m(t) be the mass of the rocket ship, which decreases with time.  At the beginning of a short time interval, the rocket has mass m and initial velocity v, whereas at the end of the time interval its mass is (m+Dm) and its velocity is (v + Dv) --- note that Dm, the increase of the rocket mass, is negative!! The expelled mass, - Dm, has speed (v - vex), where the relative speed of the expelled gas relative to the rocket is vex, the exhaust velocity. The requirement of momentum conservation is

m v = (m+Dm) (v + Dv) + Dm (v - vex)
Let us neglect products of small terms and make cancellations to obtain the fundamental relation:
® ®   m Dv = -Dm vex
If we divide this relation by the time interval Dt we have the relation
m a = m [Dv/Dt] = - [Dm/Dt]  vex
The right side of this expression, -[Dm/Dt] vex = Thrust is defined as the force acting on the rocket. It represents the momentum carried away by the expelled fuel per unit time.

If a rocket of mass m = 1000 kg is expelling gas at the rate of 10 kg/sec, and at an exhaust velocity of 500 meters/second, relative to the rocket, the thrust produced by the rocket has the constant value of 5000 Nt. The mass of the rocket at time t is m(t) = 1000 - 10 t in kg, so that the acceleration continually increases:

a(t) = Thrust / m(t) = 5000 / (1000 - 10 t) = 500 / (100 - t ) in m/sec2
Here is a table of values of a at various times, with the corresponding rocket mass.
Time (sec) Rocket Mass (kg) Acceleration (m/sec2) ** Speed (m/sec)
 0 1000 5 0
20 800 6.3 110
40 600 8.3 260
60 400 12.5 460
80 200 25.0  800 
90 100 50.0 1150
One may use the above fundamental relation ® ® to show that the initial mass of the rocket m0, the final mass m, and the final speed v of the rocket are given by the formula:
v = vex loge m0/ m
[** The numbers in the last column are obtained from this formula.]
Don, you've given us fuel for thought!

Bill Blunk [Joliet Central, Physics]     Molecular Shish Kabob
showed us the Matter Model Kit [ME-9825; $64.00], which he obtained from the 2002 Pasco Physics and Data Collection Catalog [http://www.pasco.com], which contains the following information:

The Matter Model allows students to better understand the structure of matter and the dynamic relationship between its atoms. The "atoms" of the Matter Model are brightly colored spheres specifically designed to facilitate the many uses of this kit in the physics and chemistry classroom. The bonds between the atoms are modeled with springs, a very popular model used in textbooks and other educational research material. The atoms can be configured into many patterns depending on the concepts to be investigated. Each atom can be opened and closed using two convenient snaps. The inside of each atom has a slot that allows students to place one of the included nuts. In this way, the mass of the atoms can be changed. In addition, the springs can be easily connected to or removed from the atom using the six quick connects. When we held the brass rod horizontally, the atoms were evenly spaced, being held in place by the tension in the springs. Then we shifted the rod to the vertical position, while holding the bottom atom fixed, and we noticed that the atoms were closer together at the bottom of the rod than at the top. This is a good model for the increase of pressure with depth inside a gas. The vibrational properties of this system were fascinating. Very good, Bill!

Maria Vinci [Evergreen Park HS, Mathematics]     Tiling and Tessellation
passed around the book The Graphical Work by the Dutch graphical artist M C Escher (1898-1972) [Taschen GmBH 1989; ISBN 3-8288-5864-1], which contained various patterns, tilings, and tessellations. [For more details on the life of Maurits Cornelis Escher and his works see the website M C Escher by Cordon Art BV [http://www.mcescher.com/].  Maria showed various tessellated figures that students made in her classes, using images of an elephant or a human face in making periodic tilings.  Although Escher was primarily a graphical artist, he understood mathematics rather well, and his work has had a profound influence on mathematicians; for details see the website Mathematical Art of M C Escherhttp://www.mathacademy.com/pr/minitext/escher/index.asp PJ comment: The preparation of periodic micro-crystalline samples of protein structures, such as DNA, is a crucial component in X-ray scattering to determine the atomic structure of these materials.  For example, the double helical structure was deduced by Watson and Crick upon the basis of analysis of X-ray scattering of micro-crystals of DNA.  Thus, tessellations are also important throughout modern science.  We get the picture, Maria!

Walter McDonald [VA Hospital; Bowen HS]    Fractals:  How Long is the Coastline of Florida?
explained that the length of certain intricate curves is indeterminate, because the lengths depend upon the scale of resolution.  For example, a tourist brochure may advertise that the coast of the State of Florida is 6000 km [4000 miles] in length, but even this estimate is imprecise, since it would be impossible to follow all the nooks and crannies that separate the land from the sea.  As the scale of resolution of the measurement decreases, the length increases.  He showed some "self similar curves", for which the structure has the same form when viewed at various scales --- including one on which we measured the following lengths with various resolutions:

 #1     3 2
 #2    7 1
#3    20 0.5
He calculated the fractal dimension D from the conditions (R2/R1)D = (L1/L2and (R3/R2)D = (L2/L3).  For this case, D is around 1.3 - 1.4. Walter gave us the following web-based references on fractals: You helped us to see fractals almost everywhere we look, Walter! Good job!

Monica Seelman [St James Elem]     Shoelaces, Bows, Knots, and Topology
taught us how to tie double and triple knots that can be untied by pulling the cord at one end, and she tried to figure out a pattern for such knots. She passed out a piece of cardboard rolled and taped into a cylindrical shape, which had two holes punched in it at one end, to simulate a shoe.  Also, she gave us black and white shoelaces that had been cut in half and tied together, so that each lace has a black half and a white half.  She gave us these methods for making double and triple knots:

She tried without success to tie quadruple knots that could be undone with just one string pull. Can anybody else figure out how to do it? Monica and Earl Zwicker showed various knots that could be untied by merely pulling an end of the string. Comment by PJ: Knot theory is a branch of topology, and knots that can be undone by pulling are not considered "knots" in the strict topological sense. It was a very clear presentation about a very knotty problem, Monica!

Fred Farnell [Lane Tech HS, Physics]     A Slow Train
used traction feed computer paper to lay out a 27 meter "track" on the floor of his classroom.  He released a slow-moving, battery-operated toy train engine [He got it at Radio Shack; it requires 4 batteries for operation.], which students kept on the paper track by pushing it occasionally with a stick.  Students were located along the track with stop-watches to record the time required for the train to travel to their locations.  A distance-time graph was constructed from the data, which was a fairly straight line of slope 0.5 meters/sec.  [A smaller, faster toy made the 27 meter trek in about 13 seconds.]  The speed-time and acceleration-time graphs were constructed from the distance-time graph by taking slopes.  He signaled the students to begin timing by lowering a rod that he held over his head --- this method of initiation is similar to the music conductor's downbeat, which signals the orchestra to begin playing a piece.  A fresh approach, Fred.  We knew that bigger is better, and sometimes slower is better, as well.

Larry Alofs [Kenwood Academy, Physics]     Flying Bat Toy
brought in a battery-operated Flying Bat Toy, which he obtained at the Kane County flea market.  The toy was manufactured in China and distributed by MGN Company as Item # 8-0104.  He attached the bat toy to a cord that was connected to a pivot on the ceiling, turned on the flapping wings, and released the bat.  The bat soon executed uniform circular motion of radius R about 1 meter, in a horizontal plane. He estimated the speed v of the bat [about 2 m/s] by timing its revolution, and estimated the angle q between the wire and the vertical [about 30°].  He then applied Newton's laws to the motion of this conical pendulum, so that T cos q = m g, and T sin q = m v2/R, so that

v2 = g R tan q .
This relation is roughly satisfied.  Very clever, Larry! --- but just where are we going to get some bats to put into our belfries??

We ran out of time before Bill Shanks, Ann Brandon, and Fred Schaal could give their presentations.  They will have "first shot" at our next meeting, Tuesday 22 October.  See you there!

Notes taken by Porter Johnson