High School Mathematics-Physics SMILE Meeting
1997-2006 Academic Years
Kinematics

28 October 1997: Ann Brandon [Joliet West HS]
(Formulas for Uniformly accelerated motion)


vf = v0 + a t
d = 1/2 a t2 + v0 t
v = [vf + v0]/2 = d/t
vf2 = v02 + 2 a d

there are 4 formulas because there are 5 variables [ vf, v0, d, a, t] and each has one variable missing. End of story.

25 November 1997: Arlyn van Ek [Illiana Christian HS]
did a variation on the boat crossing the river using a sheet of moving paper and a powered car. The setup is based on a problem in the standard textbook by Merrill, concerning a "paper river". One must add velocities as vectors to determine the direction of travel of the powered car on the moving river. If the car is aimed straight across, it is deflected downstream by a certain angle by the motion of the paper. Interestingly enough, to make the car go straight across, it must have a net upstream motion relative to the paper, but it is not correct to aim it upstream by the same angle as it went downstream when going straight across in the rest frame of the paper.

20 January 1998 Carol Zimmerman Lane Tech High School
She brought in a cartoon of FOR BETTER OR FOR WORSE. A guy is standing on an overpass 30 meters high. He has to drop a package into the truck as it goes under the overpass. The truck is 100 meters away. If it starts from a velocity of 0 and accelerates at a rate of 4.9 meters per second-squared. How many seconds should he wait, after the truck starts, to drop the package?

She showed the math involved and the truck took 6.4 seconds to get to the bridge, and the rock took 2.5 seconds to fall, thus the guy should wait for about 3.9 seconds before dropping the package, and breaking the truck window [or worse].

28 September 1999: Bill Blunk (Joliet Central HS)
showed us an effective way to use paradigms to teach physics. As an example, using the distance (d) an object travels in free fall under gravitational acceleration (g) for a time (t), we know that d = gt2/2. Let's suppose that g = 10, t = 2 and d = 20. Bill challenges his students: If you can give me the proper units for g, t and d, I'll give you a ten! When a student gets a correct answer, he gives them ten cents! Neat!

26 October 1999: Ann Brandon (Joliet West HS)
(handout) did Physics Off the Cliff (Conceptual Physics) with us. How long would it take a steel ball bearing (slingshot amenities, K-Mart) to fall from the top of a table 0.79 m high and strike the floor? Using h = gt2/2, knowing h and g, one calculates t = 0.40 s. We rolled the ball off the end of a horizontal table, observing that it took 1.55 s to roll 1.0 m off the end, so it had an initial horizontal velocity of 1.0 m/1.55 s = 0.63 m/s. Moving horizontally with that velocity while falling for 0.40 s to the floor, it would strike the floor at (0.63 m/s)(0.40s) = 0.25m from below the table edge. On target, observed! A beautiful way to show the independence of horizontal and vertical motion!  For edification concerning the Cartoon Laws of Physics, see the website http://www.cc.gatech.edu/classes/cs8113f_97_spring/cartoon.html.

26 September 2000 Marilynn Stone (Lane Tech HS)
made an inclined plane by placing a book on the table and leaning a grooved plastic (about 1 ft) ruler against it - using tape to fasten its lower end to the table. She placed a steel ball (a little over 1 cm diameter) in the groove at the top end of the ruler and released it. It rolled down the groove, onto the table, traveled horizontally across the table, rolled off the edge, and fell to the floor. Marilynn took the horizontal distance from the bottom of the ruler to the edge of the table to be 0.5 m. By measuring the time it took the ball to roll off the edge of the table (0.54 s) we could calculate the horizontal speed:

 vx = d/t = (0.5 m)/(0.54 s) = 0.93 m/s.

The height (y) of the table was measured to be 0.92 m, and y = gt2/2. So putting in values for y and g = 9.8 m/s2, we found the time of fall to be t = 0.43 s. When the ball fell off the table and was accelerated downward by gravity, it continued to move with the same horizontal speed as it fell, so it moved a distance

x = vxt = (0.93 m/s)(0.43 s) = 0.4 m 

from the edge of the table by the time it hit the floor. Marilynn placed a cup at that position, released the ball as before, and sure enough! - the ball fell into the cup! (Though it did bounce right out, due to elastic forces!) Just to convince us, she did it again.

"Students are always surprised to see that it actually works, as predicted by the physics," 

Marilynn told us. Great!

10 October 2000 Betty Roombos (Lane Tech HS)
explained how she shows her students to do vector problems. We are given two displacement vectors:

6 km  at 30o E of S
5 km at 20o N of W
Find the resultant and its direction. Betty would have them draw the vectors using pencil, protractor and paper, and constructing and measuring the resultant. But then they would do it analytically, by resolution of the vectors into their compass direction components, addition of those components, and then using algebra and trig to determine the resultant vector. With the aid of us with calculators, Betty carried the numbers through at the board, and it worked out well!

25 September 2001 Fred Farnell (Lane Tech HS, Physics) Follow The Bouncing Ball
Fred led us through an exercise that addresses accuracy, error, and variation in the process of measurement.  The motivation for his presentation was his past experience.  As an example, one group would measure a density of a given material to be 0.60 g/cm3, whereas another group would measure it to be 0.62 g/cm3.  Are these measurements different, or are they really equivalent?   How do we learn to appreciate the issue?

Our exercise involved dropping a ball on the floor or table from a height of 1 meter, and measuring the time between the first bounce and the sixth bounce. A series of stop watches were passed out, and we recorded these measurements, obtained by watching the bounces, hearing them without seeing them, and seeing them without hearing them:

Ball Dropped on Floor
Bounces Seen and Heard
Times in Seconds
Ball Dropped on Floor
Bounces Heard; Not Seen
Times in Seconds
Ball Dropped on Table
Bounces Seen; Not Heard
Times in Seconds
Ball Dropped on Floor
Move Hand with Bounces
Times in Seconds
3.05 3.28 3.76 3.75
3.44 3.73 3.80 3.75
3.72 3.75 3.82 3.78, 3.78
3.76 3.78 3.97 3.79, 3.79
3.80 3.83 4.13 3.81
3.81 3.90 4.13 3.87
3.85 4.10 4.31 3.87
3.88 *miss* *miss* 3.91
4.05 *miss* *miss* 3.99
Median: 3.80 Median: 3.78 Median: 3.97 Median: 3.79

The last set of data, which show less variation than the others, were taken in a fashion advocated by Earl Zwicker (IIT).  Namely, we moved the hand that held the stopwatch up and down in synchronization with the motion of the ball, and punched the buttons on the watch when our hands were at the right place.  He suggested that this technique leads to a reduction of effects of our reaction time.  It is not reasonable to conclude from the data that these numbers are really different in the four cases of interest, although more precise measurements might indicate that the ball bounced differently on the table versus the floor.

11 December 2001: John Bozovsky (Bowen HS, Physics): Rough Rider
John
showed us the Rough Rider Car (available at Walgreens, WALMART, etc), as well as another toy car.  He used these vehicles so that we could make a qualitative and visual comparison of uniform motion (motion with constant speed v) and uniformly accelerated motion from rest.

Type of Motion distance d versus time t
Uniform d = v t
Uniformly Accelerated from rest d = 1/2 a t2

He set up these two motions by having somebody push one car across the table, and he released the other car down slightly inclined plane just when the two cars were at the same horizontal position, using his carefully calibrated manual reflexes acquired by years of practice as a physics teacher.  They started at the same place, and car moving with constant speed went ahead at first, but the uniformly accelerated car caught up with it at some distance D and time T, both of which were measured.  At this time the cars were at the same position, so that

D = v T = 1/2 a T2

Thus, we may calculate the velocity v = D / T  and the acceleration a = 2 D / T2 = 2 v2 / D.  He drew the graph of d versus t, as a visual presentation of the motion of the two cars, showing where the accelerated car caught up car going with constant speed.  Nice! Keep shopping for more good toy cars, John!

22 October 2002: Ann Brandon [Joliet West, Physics]     Projectile Motion
Ann
brought in Pasco Projectile Launcher Projectile Mini-Launcher ME-6825, which shoots small ball bearings. [The more powerful model , ME-6800, which shoots plastic balls, is also available; for details see the Pasco Website http://store.pasco.com/.] The Mini-Launcher has three settings, and we shot the balls with the most energetic setting.  First she launched the projectile horizontally at a height of 94.5 cm from the floor, and we measured the distance it traveled before striking the floor to be 2.36 meters.  Since the ball took a time Ö (2 h/g) = Ö (2)(.945)/(.98) = 0.43 seconds to hit the ground, it left the muzzle at a speed of 5.4 m/sec.  She then launched the ball at an angle of 60º to the horizontal, and measured its horizontal distance of travel back to the launch table, obtaining the range R = 2.48 metersAnn compared this with the range formula R = v02 sin 2 q  /g = 2.50 m.

Ann also mentioned the following items:

Good stuff, Ann!

10 December 2002: Karlene Joseph [Lane Tech HS, Physics]      Synthesizing Planar Motion with x- and y- coordinates
Karlene
used three white board sheets (about 40 cm ´ 60 cm) to generate motion in the plane as a superposition of independent motions in the x- and y-directions. The bottom (first) white board was held fixed on the table, the middle (second) white board (lying flat on the first board) was moved along the long direction of the table to represent x, and the top (third) white board (lying flat on the second board) was moved perpendicular to the long direction to represent y-motion. A board marker, which was held at the intersection line of the second and third boards as they moved, traced out the "trajectory" on the bottom board.  If the (middle) second and (top) third boards were moved at constant speed, the marker traced out the trajectory, and produced a straight line in a "slanted" direction on the bottom board.  If the second board (x-motion) was moved with constant speed, while the third board (y-motion) went from fast to slow (deceleration), a downward arc of roughly parabolic shape was obtained.  In addition, if the second board (x-motion) again was moved at constant speed, whereas the third board (y-motion) went from slow to fast, and upward arc of rough parabolic shape was obtained.  Finally, if the second board (x-motion) went with constant speed, but the third board (y-motion) went from fast to slow to "stop", and then from slow to fast in the opposite direction, a full arc of roughly parabolic shape was obtained.  Great!  Having shown the independent motions in the x- and y-directions in an direct, visual, interactive fashion, Karlene then wrote the equations describing the general motions being considered:

It was suggested that a parabola be drawn first on the first white board, and the second board be moved at a constant rate, as always.  Then, the person moving the third board (y-direction) would see what has to be done in order for the marker to trace the parabola; i.e. y-accelerationDriving is believing, or something like that!  An exciting presentation of old concepts in new packages, Karlene!

11 February 2003: Chris Etapa [Gunsaulus Academy]      Forces and Motion / Cars and Hovercrafts
Chris
presented an activity described on the Look · Learn and Do Publications website: http://www.looklearnanddo.comChris successfully used the lesson contained there in her eighth grade class.  First, Chris reviewed the meanings of the terms distance, velocity, and acceleration.  Using boxes equipped with primitive, home-made wheels, the class was divided into groups of 4, and each group designed  and built a car.  Next, the students tested their cars by giving them a push and measuring the distance D traveled over a given time interval T. They then calculated the velocity V = D / T. This completed the activities described on that webpage. 

Next, her students made hovercrafts from one liter water bottles and balloons --- an activity based upon previous SMILE miniteach presentations [ph8901.html; pl95m7.html].  We did a variation of this activity during today's class.

After dividing  into groups of 3-4 participants, we stood the bottle vertically on its base, and cut around the top portion of the bottle at its shoulder, forming an inverted cup-like structure.  We stretched the lip of the balloon over opening of the bottle, with the cap removed, and blew into the opening at the shoulder, inflating the balloon.  When the balloon was inflated we held the air in it by pinching it just above the cup.  We set the apparatus upright on the table, with the shoulder rim resting on the flat surface, and released the pinch.  The air rushed out of the balloon, into the inverted bottle-cup, and out at the shoulder opening or rim, and the hovercraft rose slightly off the table.  In fact, the craft began to move slightly across the table --- presumably because of residual asymmetries. Ken Schug modified the apparatus by taking the  plastic bottle cap, punched a small hole in its center, and then put it on the bottle-cup.  When the balloon was re-attached, re-inflated, and released the outflow of air into the room was reduced, and the motion of the bottle was more stable.  This modification was suggested by Larry Alofs [Kenwood HS, physics], a visitor from the Math-Physics SMILE program.

Great job, as always, Chris!

25 February 2003: Michelle Gattuso [Sandburg HS, Orland Park, Physics]      Kinetic and Potential Energy / handout
Michelle
showed us a laboratory experiment that involved attaching special tape to a ball of mass m. The tape passed through a spark timer, and when the ball was released from rest, a record of its motion was made.  She used the Nakamura Electronic Spark Timer, which is listed at item P1-180 for $112 in a recent Arbor Scientific Catalog; see their website, http://www.arborsci.com. The timer operates at two settings, 60 Hz and 10 Hz.  According to Arlyn van Ek, there seemed to be considerably less friction than with the older timers containing carbon paper. When the ball is released from rest at an initial height H, its velocity v at height h should satisfy the condition of conservation of mechanical energy:

Etotal = P E + K E
m g H + 0 = m g h + 1/2 m v2
Since both sides of the second equation are independently measurable, energy conservation can be tested. [A variation of the experiment is to determine g, the acceleration due to gravity, with this apparatus.]  Here is a summary of her Procedure

  1. Attach the timer tape to the object.  Place two lab stools on top of each other and then place them on top of the lab table.  Measure the height and record this value in your data table.
  2. We know the time between sparks to be 1/60 second (it sparks 60 times every second).  By measuring the distance on the tape between spots produced by the sparks, we can calculate the average speed vavg = Dx/ Dt, since the object travels a distance Dx in time Dt.  The average velocity between two points in  vavg= (v1 + v2)/2, where v1 and  v2 are the initial and final velocities, respectively, over the interval.  Since the initial velocity on the first time interval was zero, we can calculate the velocity at every spark location.
  3. Record the distance fallen (by reading the position on the tape) in your data table, including the units.  Now record the speed of the object after the object has fallen this distance.
  4. Using your object and the speeds you have calculated, find the Kinetic Energy K E = 1/2 mv2 at each height, and record it in the data table.
  5. Record the distance between the object in the ground, h, by subtracting the initial height H from the distance the object has fallen.
  6. Recall that the stored potential energy is P E = m g h.  Calculate this stored gravitational potential energy for each height h, and record that potential energy in the data table.
  7. Finally, draw an energy versus time graph, noting the potential energy, kinetic energy, and their sum--  the total energy -- at each data point.  Is the total energy conserved?

You dropped the ball, but didn't drop the ball, Michelle!  Great job!

11 March 2003: Betty Roombos [Gordon Tech HS, Physics] and Karlene Joseph [Lane Tech HS, Physics]     Skating Around the Issue
Betty
and Karlene showed us how a student could gain insight as to how an object moves when dropped out of an airplane. We watched .Karlene skate in line across the room holding a soccer ball, which she threw into the air and then caught.  Karlene threw the ball straight up, and caught it as it came straight down, as viewed from her reference frame.  However, in our frame we saw the ball travel up and down along an inverted parabolic arc.  Karlene then dropped the ball from above her head, while rolling across the room at roughly constant speed.  We saw the ball fall in a parabolic arc. Karlene next tried a bombing run, in which she held the ball high and then dropped it while moving in order to hit a fixed target on the floor.  On the third try she gauged the proper release point and hit the target, a styrofoam™ cup -- which shattered -- which won our applause!  Stupendous shooting, Karlene!

Next Betty pulled Karlene across the room with a piece of bungee cord that she kept stretched by a calibrated amount, thereby applying a constant force to KarleneBetty had to move faster and faster to maintain this state of constant force, which produced a constant acceleration. Betty suggested other experiments  with (a) two bungee cords to double the force, and/or (b) pulling two kids to double the mass. Also, Betty mentioned that the amount of frictional drag actually could be measured. 

What great ways for students to gain insights into Newton's Laws.  Pretty stuff, Betty and Karlene!

22 April 2003: Christine Scott [Beethoven School] and Lilla Green [Hartigan School, retired]       Soup
Chris and Lilla
began by leading us through a free expression session about our ideas and concepts relating to soup.  Then, they broke the 'orrible news:  we weren't going to get to eat some soup, but to roll soup cans down inclinesChris and Lilla produced several different cans of soup [broth, cream, chunky] and set up plywood ramps [about 4 feet long --- 1.2 meters, and 12 inches wide --- 30 cm].  These ramps were supported on a horizontal surface with a short stack of books at one end.  Each regular group rolled each can down the ramp three times, and measured how long it took-- starting from rest -- to travel the length of the board. Results (averages of the three trials) were then compared, the broth can moving fastest for all three groups  [Note: If more books had been put under the elevated end of the ramp, the soup cans would have rolled more quickly.]

A control group was first given a long plastic jar that was about 1/3 filled with powdered cleaner --- they found that it would roll for only a short distance -- after which it stopped, requiring a strong push to roll further.  That control group next took a jar filled with a liquid, which rolled easily without additional pushing. They took a little scouring powder from a nearly empty can, and rubbed it on the jar to make suds. The jar rolled faster as a consequence. Finally, the control group rolled the empty open scouring powder can down the plane, and it wasn't particularly fast.

Various contributing factors were discussed --- diameters of the cans, masses of the cans, viscosity [flowing ability] of the contents of the can, and friction of the cans on the board.   The discussion  revealed that the nature of the contents in a sealed can were very important, although friction did determine whether the can would or would not roll without slipping down the incline.

Next they took two long Inertia Rods (a red one and a blue one), which were shown to be of equal weight by taping them to each end of  a meter stick, and then finding that the stick balanced when supported at the middle --- the 50 cm mark.    Two hapless volunteers were given the task of holding one of the rods in the middle, and then of rotating the rod back-and-forth in a plane perpendicular to their outstretched arms; i.e. wiggling the rod in a torsional mode.  The blue rod was decidedly more difficult to wiggle, according to the volunteer who received it.  We felt that the weight inside the blue rod was more unevenly distributed --- with more mass closer to the ends of the rod, than for the red rodWow -- wee!

We concluded that the broth can rolled fastest because, like the red bar, its mass was uniformly distributed.  In the chunky soup can there presumably were chunks near the perimeter of the inside of the can, like the blue bar, which slowed its rolling.  Is that correct, or are there other reasons as well?

True or False:  What's soup for the goose is soup for the gander!
You had quite a roll today.  Thanks, Chris and Lilla!

28 September 2004: Don Kanner [Lane Tech HS, Physics]           A Quick Graph
Don described a quick way to get rather accurate data of a falling object, by dropping that object alongside a vertical meter stick, and recording the fall with a video camera.  Using the "freeze frame" display feature, the position of the top of the the falling object is recorded at a rate of 30 frames per second.  You just read the data directly off the image, and then draw the graph.  Neato!  Porter Johnson mentioned that a bucket dropped into the hand-dug well over 100 meters in depth at the Hohenzollern Medieval Castle in Nuremberg, Germany took 5-6 seconds to hit the water level -- kerplunk!  The Tiefer Brun (deep well) was essential for defending the castle during times of siege!  For details see the website http://www.oldandsold.com/articles13/travel-125.shtml. Thanks for sharing this, Don!

Ann Brandon and Debbie Lojkutz [Joliet West HS, physics]           Accelerometers 
Ann and Debbie first showed us the Peanut Butter Jar Accelerometer, and demonstrated how it works.  They then gave each of us a plastic straw (with a small slit at one end), a piece of string, a big paper clip, and an arrow drawn on a small piece of paper.  We tied one end of the string to the big paper clip, and tied a knot at the other end.  Then we pushed the knotted end of the string through the unslit end of the straw and out through the slit end.  Then we pushed the string through the slit, so that the knot would be caught.  Next we attached the arrow to the paper clip, so that it pointed at the straw.  We had thereby constructed our very own accelerometers, which we tested.  They worked!

09 November 2004: Ann and Debbie handed out the following sheet:

    Accelerator Homework
  1. Take your accelerometer with you for a ride in the car. (Please -- have someone else drive!)
    When going forward, what happens to the accelerometer when the driver (a) steps on the gas pedal? (b) steps on the brake?
    When going in reverse, what happens to the accelerometer when the driver (a) steps on the gas pedal  [What is the indicated direction of acceleration?] (b) steps on the brake [What is the indicated direction of acceleration?]
    What happens to the accelerometer when the driver makes a right-hand turn? [What is the indicated direction of acceleration?]
    What about a left-hand turn? According to your accelerometer, what is the direction of acceleration?
  2. Teach your mom or dad about using the accelerometer to determine (1) the direction of acceleration and (2) whether acceleration is taking place. Have your parent fill out their report.
  3. Dear Parent,
    It's Physics Demo time for your child. They have learned a concept in Physics that is easily demonstrated and their assignment is to demonstrate and explain this concept to you.  To verify that they have done this, you should write a couple of sentences describing what they (the student) did and said.  The very bottom of the page is for any comments about this assignment or for a message to me.  Would more of these "student teaching" assignments be OK with you?  Thank you for your cooperation.
  4. My son/daughter taught me about using his/her accelerometer to determine the direction of acceleration. This is what I learned:

Ann and Debbie also called our attention to two items in a recent catalog of Frey Scientifichttps://www.schoolspecialtyonline.net/browseProducts.jsp?categoryID=0&URLID=1004.  The first item, an Impact Car (#05578611, $7.75) [http://store.schoolspecialtyonline.net/OA_HTML/xxssi_ibeSearchResults.jsp?type=search&minisite=10029&query=impact+car] permits the measurement of the maximum force at impact.  The second, a Large Lens Kit (#05527379, $64.55) [http://store.schoolspecialtyonline.net/OA_HTML/xxssi_ibeSearchResults.jsp?type=search&minisite=10029&query=large+lens+kit] contains several lenses with magnets on their backs, which are suitable and convenient for blackboard optics.  

Thanks, Ann and Debbie!

26 April 2005: Bill Shanks [Joliet Central, physics -- retired]                           Meeting of the Board + A One Watt Flashlight
Bill
recently was driving North on Interstate 75 in Georgia, on a quiet section between Macon and the metropolitan Atlanta traffic snarl.  Suddenly, he saw a large wooden board moving through the air in front of him, coming right at him.  Fortunately, he was able to swerve to avoid major impact, receiving only a minor dent on the right front bumper.  After this moment of terror had passed, Bill began to wonder whether he had just been lucky, or whether there had been time to react to the situation.  Aha!  Another good physics question!  The board might have been lying on the road initially, where it would be struck by a big truck and knocked high into the air.  If it rose to an approximate height h = 3 meters above the roof of the car before Bill saw it and it began to descend, a time t ~ 0.8 seconds would elapse before it came down to the level of the roof:

h = 1/2 g t2 ~ 1/2 ´ 10 ´ (0.8)2 = 3.2 meters
Bill might might have had time to react!

Excellent real-life physics experience! Thanks, Bill.

Comment by Porter Johnson: By contrast, an object dropped from a viaduct above the road is even more dangerous, since you probably won't see it until somewhat after it is dropped, and at that point it is moving more quickly through the field of view.

13 December 2005: Carl Martikean [Proviso Math and Science Academy]    Poetic Kinematics
Carl
had been discussing motion with his freshman class and asked them to plot the motion described by the first two stanzas of Paul Revere's Ride by J W Longfellowhttp://eserver.org/poetry/paul-revere.html. Here is the first stanza:

"Listen my children and you shall hear
Of the midnight ride of Paul Revere,
On the eighteenth of April, in Seventy-five;
Hardly a man is now alive
Who remembers that famous day and year."
The meter changes from a gallop in the first stanza to more of a march or walk in the first part of the second stanza. Carl showed both Distance versus Time and Velocity versus Time curves that described this motion.