Sunday, November 3, 2013

The Roller Coaster Lab and Chaos Theory

Last week, I ran a lab with Honors and AP where they were to design a roller coaster track for a marble to travel down.  The coaster had to have a bunch of little things, such as a loop, a jump, a couple of hills, and it had to land in a cup.  I gave each group 3 tries to sink the marble.  Most groups made it at least once, but few made it more than once.  We may ask "How did we not obtain the same result each time?"

Scientists assume that each trial in an experiment was done in a manner that changed no more than 1 variable.  Otherwise, changing more than 1 variable could result in a change as a result of 2 changes, neither of which can be separated by itself.  The goal is to separate exactly 1 variable, so everything else must be done exactly the same.

Try doing this.  Try dropping a bouncy ball from the height of your head.  Record the location of the third bounce on the floor.  Now try doing the exact same thing to achieve the exact same result.  It's harder than you think.  The ball and floor have all kinds of hidden variables that we can't measure, such as irregularities in the surface of the ball, slight .001 degree tilt of the floor, or inconsistencies in the amount of air resistance encountered as it falls.

In our lab, we dropped the marble, and for the most part, nobody made any real changes in the design from trial to trial.  Some groups moved the cup; others tweaked the track a bit, but most, if not all, groups, experienced some variation from trial to trial, despite making no changes.  We may ask how this could have happened.

Suppose you dropped the marble a few millimeters away from the starting point.  This would slightly change the amount of initial potential energy, changing the kinetic energy at future points in the coaster.  At some points, that change can be enough to cause the marble to hit an uneven bump in the track, causing a variation in the amount of friction encountered.

Another possibility is that you put some sort of spin on the marble.  You will learn that a spinning object has "rotational kinetic energy," which adds to the total energy of the marble.  This slight change can create a major change later on in the coaster.

Chaos theory states that changing one tiny, seemingly insignificant detail can lead to a series of increasingly noticeable changes that eventually change a final result.  An example of this is in Ray Bradbury's short story, A Sound of Thunder.  A hunter who travels back in time accidentally steps on a butterfly, changing things, such as how words are spelled or how people support politicians.  Although we might find this example to be a bit of a stretch, the idea is quite sound.

Killing one butterfly means slightly less prey for a natural predator.  Maybe an animal needed that butterfly to make the difference between survival and death of itself.  Perhaps such animal would have given birth to more species, one of which could have had a necessary mutation to evolve into a modern species that we humans encounter.

Perhaps this species influences our culture or behavior by affecting our ecosystem, killing unwanted disease-ridden animals, or by controlling the population of a disliked animal.  Maybe the politician that a person would have supported had a position regarding how to handle the population of a species that no longer exists all because a guy stepped on a butterfly.  Hence, chaos theory (sometimes called the butterfly effect; Bradbury did not invent the term, he used a butterfly to make reference to the term).

I'm interested in hearing about other possible changes that could have caused the marble to behave differently.


  1. How do you turn this (interesting) speculation of how events would be altered into more concrete scientific/mathematical models?

    Is chaos theory solely based on explaining the past and present, or is it used for predictions of the future?

    1. Very good question. I think the idea of chaos theory inspires one to design more accurate experiments that do not have sources of error that can accumulate to much larger measured errors. Satellites that are designed with even a slight error in the curvature of the focus result in very large errors in concentrating waves from very far away sources. Although we round all the time in physics class, engineers generally do not like rounding because of the large error in calculating large answers.

    2. Ah so chaos theory is less a realm of physics and more a pragmatic consideration when conducting research or making observations? Are there people who specialize specifically in chaos theory? If so, do they spend their time further exploring the theory or rather applying it for academic/corporate/governmental research and design?

    3. Physics is more of a means through which we understand chaos theory. Chaos theory applies in mathematics. If you wanted to take an irrational number, say pi, and raise it to the 20th power. You should get a noticeably different answer than if you took 3.14 to the 20th power. The percent difference between pi and 3.14 is not even a hundredth of a percent, whereas after raising both to the 20th power, it is now about 1 percent. Apply this to any situation requiring successive multiplication, and rounding error creates a reality far different than predicted, despite the minimal initial error.

    4. And to answer your other question, I am not sure... Try googling "chaos theory." Wiki will give you background information regarding its origin. In the modern world, you might find interesting answers with google scholar.