Hands-On Examples of Exponential Phenomena

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Hands-On Examples
of Exponential Phenomena

Introduction. All throughout our daily lives we encounter exponential actions, and a number of these avail themselves readily to taking into the classroom for purposes of learning how to treat them mathematically, and to differentiate them from linear and steady state phenomena. While this activity may seem difficult at the start, this author has had 12 year-olds awed at their own handiwork as they moved data from linear graph paper to "funny looking" semi-log graph paper. "Neat!" and "Wow!" are not uncommonly heard in the classroom from suddenly inspired young minds. A whole new world of mathematical application becomes available to them in their future studies in all the sciences, AND in business, economics, historical studies of population and corporate growth, and on and on....

GRAPHING: Because learning the basic concepts of logarithms so often runs into mental blocks, this author prefers to bypass those hurdles and jump right over to the use of semi-logarithmic graph paper. ("Semi-log" means that the paper is laid out linearly in one direction and logarithmically in the other direction as shown in the 3-cycle example to the right.) Click on the graph paper, and a new webpage will appear that will go into the details of using semi-log graph paper (or click button).

One way to slice the PHYSICS "pie" is into the two sections called kinetics (or dynamics) and statics. In kinetic systems there is motion involved - such as how fast something accelerates; in statics there is no motion involved - such as how much force is being exerted.

  1. KINETIC PHENOMENA

    1. EXPONENTIAL PHENOMENA

      1. Troublesome examples usually to be avoided.
        Teachers will want clean examples that don't get their students lost and distracted by convoluted techniques. Such would happen if you wanted to follow the growth of some microbe, which takes a lot of expensive materials and equipment as well as knowledge of aseptic techniques (othewise you will likely get stringy, slimy growth of molds). Daily living is too full of exponential examples to mess around with such distractions.

      2. Approaching Equilibrium - in many instances these might better be called "decay" or deceleration reactions. You will see why in several of the following examples.

        1. The rate of cooling of a container of hot water
          Measure the temperature periodically of a cooling bucket of hot water. Again it will cool fastest at first and then more slowly later as it asymptotically approaches room temperature.

        2. The rate of emptying of a container of water with a too-short syphon.
          Another exponential graph can be obtained by following the progress of a syphon which is set up so that it cannot completely drain the reservoir. The syphon will run fastest at first, and then slower and slower as "h" falls in the picture to the left of the title. The flow will finally stop when the outlet is at the same level as the water in the reservoir, at h=0.

        3. The decay of amplitude of a pendulum
          Look at the figure to the right of the title above. Each time a pendulum swings, the weight does not reach the height of the previous swing due to frictional forces. Because short pendulums swing too fast for accurate data gathering, you will want a long one. A pendulum hung in a stairwell should be good besides getting up some excitement. Use strong monofilament fishing line, and as heavy a weight as possible to minimize the effects of friction with the air - you want the decay to take at least 10 minutes so lots of measurements can be made for graphing. It is suggested that you install some sort of grid in the background, and that you snap a photograph at the end of every 4th (or whatever number you choose) swing. From the photographic record you can very accurately ascertain the height of the weight by comparing it to the background grid. (Question: If you snap your pictures on every 4th swing, are you or are you not taking your photographs at constantly timed intervals?)

      3. Acceleration experiments

        1. The acceleration of a heavy ball rolling down a long smooth slope

          Galileo worked on the accelerating (exponential!) speed of a falling body. The problem was that acceleration due to gravity is so fast and he did not have any high point from which to drop things. Thus he found that he could not measure a truly falling body with any accuracy. He then turned to using massive metal spheres that were rolling down a very gently sloping plane. Friction and air-drag were insignificant, but he could rather accurately time how long it took the ball to roll past each meter-line he had drawn on the slope. What was his timer? He made up songs such that the beats matched the passage across each line. Then a musical friend translated the songs to measured notes.(Oddly, he noted, if he made the slope of sandpaper, the balls rolled FASTER. He later determined it was due to the ball's bouncing and minimizing the friction of continuous contact with the slope.)

          But finding a sufficiently long constant slope might be a problem. What about using hills? And what kind of ball could you roll? Basketballs would quickly reach terminal velocity due to their light density and wind resistance. Got any old cannon balls rusting out in the barn? Maybe an old bolling ball? If the slope is gentle enough, you ought to be able to use stopwatches to time the intervals between the start and the line they are assigned to. Better: place friends with cameras along the route. At timed intervals, each friend will click the shutter and thus photograph the ball. Later you can use a tape measure to ascertain the exact distances between the starting line and some object seen in the background of the photo (Thus you need to have a route that has varied backgrounds. You might want to placed numbered stakes along the route.)

      4. Other examples which can also be used to show differences in growth rates, which can be designated by "doubling times".
        1. Get out the history books and following the decennial population numbers of the USA, or
        2. the GNP or of the Dow Jones Industrial Average, or
        3. even a stock's price over a long period of time.

      5. Amplification experiments

        1. Landslides
          Imagine the collapse of a pyramid of neatly stacked oranges. Long ago the Scientific American had a mathematical article on that subject of pyramids of metastable objects.

          Suppose, for example, you had a wide sticky sheet of plywood. On it were loosely stuck marbles about an inch apart. If the sheet is tilted to some subcritical angle, the marbles don't move, but their sticking in place is tenuous. Now if you dropped a marble onto the sheet up near the top, it would roll down (its momentum overcomes the stickiness) and hit and displace another marble. Now there are two moving marbles that could hit four more and those hit eight and so on.

    2. EXAMPLES OF LINEAR PHENOMENA

      1. A light beach ball rolling down a smooth slope.
        A light beachball will reach terminal velocity very quickly and thereafter it should roll at a constant speed.

    3. EXAMPLES OF COMPLEX PHENOMENA

      1. The rate of deceleration of a coasting, out-of-gear car.
        Get a car going fast down a LEVEL, STRAIGHT highway - say at 60 mph. At time = zero, shift to neutral and coast. Every 5 seconds, record the speed. There is a problem in this that the kids should ponder because it won't yield a nice exponential decay graph. Reason: the car has two frictional forces acting to slow it - wind drag and rolling friction (bearings and axel/transmission rotation - but not engine drag since the engine was disconnected); at higher speeds, wind drag predominates fades to zero as the car slows, but there is always an underlying rolling friction component that finally predominates as the car slows down. This would thus be a lesson in the complexities inherent in reality. So without mentioning wind drag and rolling friction, see if the kids can discover these things for themselves.

      2. The rate of growth of children
        Perhaps you or some of your friends have a place in your homes where Mom has been recording your heights through the years on some wall or door jam. If a few of can can bring that data to class, the class can graph it. This should be the last thing the done on this page because it will be complex - growth spirts, slowing down and asymptoting - things the class can discuss to try to make sense of it.

  2. STATIC PHENOMENA

    1. TARGET THEORY
      Within this realm are a number of inherent exponential and arithmetic phenomena. Now that you know how to use semi-log graph paper (as well as ordinary or bi-linear graph paper, you should be able to determine which types are represented by the following examples.

      1. The Descartes/Poisson Machine - the roulette wheel
        The roulette wheel was constructed after a discussion between Rene Descartes another Frenchman, Poisson, who developed the "Poisson Distribution" (yes, I know "poisson" = fish in French, but this doesn't have any ecological implications for the fishing industry!). Poisson's work pertained to calculating the coefficients in an infinitely expanded binomial. But Descartes saw another use - target theory. Using Decartes' original roulette wheel with 36 holes and not 0 or 00, you imagine that you have made 36 spins of the ball, and kept note of where the ball fell each time. While the ball always dropped into a hole, it did not always drop into a different hole. Sometimes a particular hole was hit twice, or three times or more, and rather often a hole was not hit at all. What Decartes discovered was that Poisson's equation rather accurately predicted how many holes got hit once and only once, how many twice and only twice, and so on.

        But what was really interesting was that the equation somewhat accurately predicted the proportion of holes missed! (Why 'somewhat'? Because Poisson's work dealt with an infinity of holes and an infinity of balls, while the roulette wheel example had only the finite number of 36 of each. Thus some error is involved when applying Poisson to real world situations. Actually, the error is only 10% when there is a roulette wheel with 10 holes and 10 rollings of the ball.) Now let's look at Poisson's equation to the right. In this equation, "m" is the ratio of balls to holes (1 in the above example); "x" is the number of hits in a given hole (zero for missed holes), and Px is the proportion of holes getting x and x only hits (so Decartes was interested in Po. So when looking for proportion missed, Po = e-m). This became very important in ecology when we look at how prey species (primarily plants) are able to survive when their populations are being "hit" so many times by predators (mainly non-flying bugs). And you might suggest that "hits" by hurricanes or tornadoes fits into this.

        Descartes/Poisson work leads directly into discussions of Venn Diagrams. If you want to go on safari and hunt elephants with a machine gun that shoots paint-balls or kill bacteria with viruses, click button. What is shown in the Venn diagram to the right is a case where you have - say - 400 elephants, and your paint-ball machine gun is loaded with 200 blue paint-balls and 50 pink paint-balls. It is assumed that EVERY paint-ball hits an elephant, and that some elephants get hit more than once, and some not at all.


      2. Epidemiology

        Extending beyond this simple use of semi-log graph paper, above, we can move on to a more sophisticated level of inquiry into the central nature of "target theory" - one dealing with "survival curves", and its components of (a) the minimum number of lethal hits, and the average dose that kills 50% of the subjects (also known as LD50). These inquiries explain many things that hold colorful images for young minds: the teacher's throwing tests down stairs and grading according to the steps upon which the papers land, the risks of smoking, radiation, getting killed in a shoot-out on a battle field, or getting run over by a Mack truck. Click button for directions as to how to use semi-log graph paper to analyze such things with what are called dose plots.

        Let's devise several quick "card" experiments dealing with "infections" that require one, two, or more "pathogens" to get one sick.

        You should start out with one deck of playing cards per five students. Shuffle all the decks together. Tell the students that picture cards (J Q K) are "bad" or "lethal pathogen" cards.

        1. Expt #1:

          1. This is a model of an extremely virulent pathogen such as ebola, where encountering just one is enough to kill you. The students are told that if they get merely one single picture card, they are "dead" Each student is to draw one card (this is equivalent to the "dose" or "exposure"). How many and what proportion of the students die? How many and what proportion survive? What is the "minimum number of lethal hits"? (Super-simple and obvious: ONE: any student who draws a picture card dies.)
          2. What if you had upped the dosage to drawing two cards; with any single picture card's being lethal?
          3. Pretend that you had done this experiment many times - each time upping the dosage by drawing additional cards. Make a dose plot - actually a dose versus log of the fraction of survivors. (See how handy the semi-log paper is: you don't need to determine the logs, only the fraction of survivors.) Again: the "fraction of survivors" equals the number of survivors divided by all participants (living and "dead").

        2. Expt #2:

          • The students are now told that it takes two or more picture cards to be lethal. Do a dose plot of what you would expect of this. (Hint: obvious with doses of zero and one, no one dies. At dose = 2, only a low percentage of the students will draw two picture cards, so the rate of death is low. At higher doses, there is more and more likelihood of drawing two picture cards, and so the curve descends ever more steeply.

        3. Expt #3:

          • It is estimated that for a person to catch typhoid fever, one must ingest at least 10,000 typhoid bacteria. Pretend you were the scientist who came up with this 10,000 figure. What would your dose plot look like. (Hint: don't forget about shoulders, and extrapolations to the vertical axis!)
          • we are constantly being irradiated with cosmic rays, which can lead to cancer (lethal). However, suppose that there are a thousand sensitive sites in our bodies. If one is knocked out by a cosmic ray, the others can repair it within a day or so. The only way to be killed is to have ALL thousand sites knocked out knocked out within a short time. Thus the minimum number of lethal hits is 1,000. However, most cosmic rays that hit our bodies miss the sensitive sites. Thus "almost all" people who get hit with 2,000 cosmic rays survive and are on the shoulder. What does this curve look like?

        4. Expt #4:

          • In WW2 the statistics were that half of the soldiers who were hit by four bullets in a short span of time died. Thus the lethal dose for 50% of the subjects was 4 (bringing in the concept of LD50). Still there were some soldiers who got hit seven times and survived because the bullets hit them in non-lethal places. BUT what is the minimum lethal dose? ONE! (to the brain or heart, etc.) Thus LD50 and minimum lethal dose are often different.

            (Hint: Graphs of this and extrapolation of the downward slope back to the vertical axis gives the minimum lethal dose, while the dose needed to kill 50% can be read off the horizontal axis.

        5. Expt #5:

          • In summary, if you see a dose plot where the line is straight and descends out of the origin, what does that mean? What is different about that "pathogen", bullet, or whatever, than the type that causes dose plots with shoulders.

            Homework assignment: What sort of dose plot do cigarettes have?

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