Lesson Plan

Lecture/Lab Version

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.
- Computer(s) equipped with Adobe’s Atmosphere
Browser for access to the FasTrack simulator at http://www.umsl.edu/~fraundor/fastrak.html
.

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)

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}
[sec^{2}] . Hence the equation
is (Dt)^{2}
= (Dt)^{2}
– 10^{-18} [sec^{2}].
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.