Marbles or Hot Wheels?

One of the conceptual themes that threads through SC 130 Physical Science at the College of Micronesia-FSM is the theme of the mathematical models. Many theories in physical science are framed as mathematical models, equations that make predictions about the behavior of a system.

Many of the laboratories center on linear mathematical models. There is no mathematical pre-requisite for the course. Students in the course may be in developmental mathematics courses. At best, they are either in college algebra or have completed college algebra.

Although the students enter the course with weak mathematical backgrounds, the intent of the course for mathematically weaker students is to introduce them to mathematical models through observable systems. The hope is that the physically observable system will provide cognitive hooks for the abstract mathematical models.

Yet even while introducing linear models, the course seeks to expose the students to more complex mathematical vistas. To take the student higher on the mountain of magically mathematical entities, to let them glimpse other models.

While laboratory three includes a simple quadratic relationship in the fall time for a ball, the following week potential and kinetic energy are used to produce a square root relationship. A marble is released on a banana leaf ramp and the speed of the marble after leaving the bottom of the ramp is measured using a meter stick and a stop watch. The marble is started low on the ramp and with each increase in elevation the class is asked to predict the next velocity. The students tend to make a linear assumption about the speed. The resulting plot, however, is not linear.

At this point I introduce the theory that will lead to the model that I will propose for the velocity. I believe that sometimes one should see things that they do not understand, that are just beyond one's comprehension. This particular system trades potential energy for kinetic energy. The complication with using a marble, however, is that potential energy goes into both linear kinetic energy and rotational kinetic energy. Conservation of energy - the underlying concept - asserts:

potential energy = kinetic energy
potential energy = linear kinetic energy + rotational kinetic energy
mgh = ½mv² + ½Iω²
In cgs units the result is velocity = 37.42√height

The details including the rotation inertia I for a sphere are worked out in detail in section 041 of the course companion.

The rotational energy of the marble does not change the model, but the additional energy term introduces another layer of complexity. The upside is that the system can be reproduced almost anywhere in the Federated States of Micronesia. That is another intent of the course. Many students go on to be elementary school teachers for the nation. Science equipment budgets for the rural elementary schools is frequently zero. Demonstrating physical science concepts with expensive, high precision equipment would make science appear to be out of the reach of an elementary school teacher. In addition, the technology could get in the way of understanding the universality of the underlying concepts. The students could come to believe that physical laws only apply to exotic systems measured with mysterious black boxes.

This past fall the division acquired a Hot Wheels® set that included track and a car. During the winter break I explored the height versus velocity relationshp for the car on the track.

The cars have very small, light wheels that permit neglecting the rotational component of the kinetic energy. This also means a slightly different constant in the mathematical model. The car is predicted to have a velocity:
velocity = 44.27√height

A comparison of the marble velocity , the marble theoretic velocity, the Hot Wheels® velocity, and the Hot Wheels® theoretic velocity is depicted in the following chart.

Note that the marble theoretic velocity (green line), is a better match to the observed marble velocity (blue line) than the theoretic velocity without taking into consideration the rotational kinetic energy (orange line). The adjustment for the rotational kinetic energy provides a better match to the observed data for the marble.

An initial test of the Hot Wheels® system (red line) suggests that the system does not fit as well as the model. The exponent is larger, closer to one, the system appears to be more linear than the marble on the banana leaf ramp.The system is still a non-linear system, but less obviously non-linear than the marble data.

A difference noted during the actual measurements was that the Hot Wheels® car tended to slow down more quickly than the marble. Although the cars are well designed to roll with minimal friction on their bright orange tracks, there is still a frictional component between the wheels, axles, and axle mounts. The marbles, being axle-free, roll will almost no significant frictional component.

Given the above, I am likely to go ahead and demonstrate both systems in class, but plot only the marble data.

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