Don Kanner [Lane Tech HS, Physics] Rocket
Ship Physics
Don 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 nonzero 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  v_{ex}),
where the relative speed of the expelled gas relative to the rocket is v_{ex},
the exhaust velocity. The requirement
of momentum conservation is
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:
Time (sec)  Rocket Mass (kg)  Acceleration (m/sec^{2})  ** 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 
Bill Blunk [Joliet Central, Physics]
Molecular Shish
Kabob
Bill showed us the Matter Model Kit [ME9825; $64.00],
which he
obtained from the 2002 Pasco Physics and Data Collection Catalog
[http://www.pasco.com], which
contains the following information:
Maria Vinci
[Evergreen
Park HS, Mathematics] Tiling and Tessellation
Maria passed around the book The Graphical Work by the
Dutch
graphical artist M C Escher (18981972) [Taschen GmBH
1989; ISBN 3828858641], 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 Escher:
http://www.mathacademy.com/pr/minitext/escher/index.asp.
PJ comment: The preparation of periodic microcrystalline samples
of protein
structures, such as DNA, is a crucial component in Xray
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 Xray scattering of microcrystals 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?
Walter 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:
L: Length 
R: Resolution 

#1  3  2 
#2  7  1 
#3  20  0.5 
Monica Seelman [St James Elem] Shoelaces,
Bows, Knots, and Topology
Monica 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:
Fred
Farnell [Lane Tech HS, Physics] A Slow Train
Fred used traction feed computer paper to lay out a 27 meter
"track" on the floor of his
classroom. He released a slowmoving, batteryoperated 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 stopwatches to record the time required for the
train to
travel to their locations. A distancetime 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 speedtime and accelerationtime graphs were constructed from the
distancetime 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
Larry brought in a batteryoperated 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 # 80104. 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 v^{2}/R,
so that
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