Lesson Plan

Lecture/Lab Version

Modeling Motion at Any Speed (Kinematics/Dynamics)

Objective:

Kinematics students will:

• Experience how, at high speeds, the glider time Dt elapsed between gates (separated by 1 foot) differs from the lab time Dt elapsed between gates.
• Try to find a rule or rules which explain these behaviors quantitatively.
• Optional, only if the concept of proper velocity is a content objective:  See how proper velocities (Dx/Dt) and coordinate velocities (Dx/Dt) compare.

Dynamics students will:

• Determine how the time Dt spent by a spring-driven glider between timing gates changes with the work W put into spring compression before launch.
• Find how kinetic energy K is related to speed, if work W is converted to K.
• Determine the time elapsed between gates, before and after inelastic collision with a stationary object of equal mass.
• Find how momentum p is related to speed, if p is conserved in collision.
• Optional followup:  After student conclusions have been turned in, consider and discuss what historical figures like Aristotle, Galileo, Newton, Einstein, and Minkowski, might have said about the results.

Prerequisite Knowledge:

Students are able to:

• Record data (like energies and elapsed times) in a spreadsheet.
• Graph various functions of the data, like W versus (Dx/Dt)2.
• Attempt to explain data with equations & underlying physical mechanisms.

Resources:

• Lecture version assumes that your classroom has only one computer, from which you can lecture.
• Lab version assumes that you have enough computers for all students, either working individually or in groups.

Lesson Preparation:

Before conducting this lesson, be sure to read through it thoroughly, and familiarize yourself with the Adobe Atmosphere Browser and the FasTrack simulator as well.  Verify that the activities play correctly on all machines in your class area.  You may want to bookmark the FasTrack page for your students, and make copies of the worksheet for each as well.

Lesson Outline (e-mail pfraundorf@umsl.edu for a current draft)

Motivation

We experience space and time very differently.  Therefore it came as quite a surprise, in the early 1900’s, when humans discovered that the rate at which we travel through space affects the rate at which our pocket watch ticks (e.g. in comparison to pocket watches in some other state of motion or non-motion).  Even more surprising was the fact that observers in different states of motion experience simultaneity differently as well, i.e. which events at separate places happen at the same time depends on your state of motion!

The main reason these effects were not discovered earlier is because they are often small and easy to ignore at low speeds.  (One exception to this is magnetism, a result of combining Coulomb’s law and relativistic length contraction which is quite noticeable at low speeds.)  By providing access to equipment that can access high speeds (or simulations thereof), students are given a chance to experience these effects first hand, and to try and figure out on their own what they might mean.  We hope this will provide:  (i) insight into how space and time are tied together,  (ii) a curiosity how others have explained something that students hereby are given a chance to struggle with on their own, and  (iii) an opportunity to struggle with the paradigm change process itself:  “How do you deal with effects which your perceptions have not encountered before, and which your language has not been fully modified to take into account?”

Creating a dataset:  How much and what to put in?

To carefully examine the transition from normal to relativistic speeds, we recommend taking data like that in Table 2 of the paper, i.e. data in which time elapsed on the glider clock (between gates separated by a foot) ranges between 0.1 and 10 nanoseconds.  That’s because lightspeed (the constant that relates our units for distance with our units for time) is about 1 foot per nanosecond.  Perhaps six data points in this range, with only a few points lying outside, will suffice (unless experimental precision is something the student wants to work on explicitly).

Analyzing the data:  What to try plotting?

The basic data consist of the energy used to compress the spring (this you control), and the times elapsed between gates separated by 1 foot on both laboratory (map) clocks, and on (traveler) clocks affixed to the moving glider.  For the students studying kinematics the energies used are only important in that they allow one to vary the experiment (and the resulting times).  The first thing to plot, therefore, might simply be times elapsed (say glider clock time t versus laboratory clock time t) during passage through the first set of gates.

As you can see below (for the data given in Table 2), the data points are almost on a straight line (namely the line described by Dt = Dt) when the elapsed times are much larger than a nanosecond.  However, as the time elapsed on the glider clock dips into and below the nanosecond range, the time elapsed on the map clocks seems reluctant to decrease below a nanosecond.  What’s going on here, and more importantly, can we plot these variables so that a simpler curve (e.g. a straight line) results? At this point, of course, there are many possibilities.  One might for example plot powers or roots or products or ratios of the measured variables, versus other powers or roots or products or ratios.  Two examples follow.  The first, plotting the square of time on the glider clock, versus the square of time on the lab clocks, during a gate traverse indeed gives a straight line… Given a straight line, of course, one can begin to write down equations.  Hmm.  If a student is very observant, they might note that this is a line of slope 1 with an x-intercept at 10-18 [sec2] .  Hence the equation is (Dt)2 = (Dt)2 – 10-18 [sec2].  What can this mean?

It’s obvious that an “Aha!” experience at this point might help, although no particular result (even the discovery of this equation, let alone it’s interpretation) is required or expected.  Hermann Minkowski’s idiosyncratic answer, of course, might have been “This is nothing more or less than the space-time version of Pythagoras’ theorem, written in the form (Dt)2 = (Dt)2 – (Dx/c)2”, but even Einstein didn’t see that as a particularly insightful answer at first glance.  Only later did the bridge it provides, to the geometric understanding of gravitation and other forces, become clear.

Totally different analysis paths are possible as well.  For example, were the student to take the tact of plotting various ratios of the experimental parameters (in effect, speeds of various sorts), they might have stumbled upon the circular plot below.  This illustrates an alternate interpretation of the metric equation, namely that we travel at the speed of light “through time” when we are not moving through space, and as our speed through space (dx/dt) increases the “rate at which we travel through time” (liberally defined here in velocity units as c times dt/dt) is decreased so that the sum of squares of the two remains at the speed of light.  Thus traveling through space at lightspeed brings the traveler’s clock (relative to times measured in the lab frame) to a standstill. Similar options are open to students who wish to explore the way that the conserved quantities energy and momentum behave at high speeds.

Connecting data to underlying mechanisms:  Concepts to explore.

In the kinematics part of this exercise, the movement to finding meaning in the data should be quite interesting.  Above, already, we’ve mentioned differing results which might emerge depending on whether relationships between elapsed times, or between speeds (interval ratios), are the focus.  A different set of plots and relationships might emerge if students decided to draw inspiration from classical undergraduate relativity texts (like French).  In the history of motion studies, the concepts used underwent quite a few changes as one moves from the ancient Greeks, through Galileo and Newton, to the present.

Take a look at the concepts used by the students in “explaining” the results.  Are they loosely defined, or precisely defined?  Do they follow the strategy of others, or are they a bit off the wall?  Do they allow one to predict what would happen in future experiments with the same gate separations?  How about with different gate separations?  Do the “explanations” yield predictions which might be tested by future experiments, or which are relevant to phenomena in everyday life?  Do the explanations show too little, or too much, caution?  All of these are questions about scientific assertions that the students might want to learn to ask themselves.

Conclusion:

The question here is perhaps not “Did the students identify the correct relationships, or did they use the correct concepts?”.   Consider instead “What processes did they use, and how far did they take those processes?”.   Try applying this question to relative student performance in:  (i) data acquisition, (ii) data analysis, and (iii) data interpretation.  Of course, then be ready to provide them with your own answers, and perhaps answers others might have provided, to these challenges as well.