EXTREME PHYSICS UNIT II - "ONE MAP, TWO CLOCKS"

Instructional Goals

1.   Offer observation challenges involving properties of space and time that are difficult to share viscerally in a lab experiment.

Begin with a web-based airtrack that allows students to generally detect differences in the behavior of clocks in different states of motion, and in particular to discover Minkowski's space-time version of Pythagoras' theorem, the metric equation (c dτ)2 = (c dt)2 - (dx)2 and some of its consequences.

Direct "single frame" consequences of the metric equation include:

• the familiar expression for speed of map-time (γ ≡ dt/dτ) in terms of coordinate velocity (v ≡ dx/dt e.g. in lightyears per map year):

γ = 1/√[1-(v/c)2]

(ii) that lightspeed, the units conversion for the number of meters in one second, calibrates the tradeoff between travel through space and traveler time because it's the root sum of squares of these two speeds, i.e.

c2 = (dx/dt)2 + (c dτ/dt)2, and

(iii) the concept of a proper velocity (w ≡ dx/dτ e.g. in lightyears per traveler year) that has no upper limit, and turns out to be proportional to momentum as well.

If energy is used as the independent variable, students can also discover the relativistic expression for kinetic energy from the experiment.

Simulations further allow one to slow lightspeed to laboratory rates (e.g. 55mph) to provide an even more visceral experience, and to add inelastic collision to the experiment, letting students discover by experiment the proportionality between momentum and proper velocity.  A bit of space-time curvature might further modify the metric equation investigated by the air track, allowing students to explore the effects of such curvature on earth and in the neighborhood of more compact objects.

Spacetime Connection Lab

Apparatus

SpaceTime Explorer Applet on computers in the classroom.

Pre-lab discussion

Let the vehicle move across track and ask for observations. List observations and then ask which items are quantifiable. Lead them to observe that the glider moves at constant speed; i.e., that it travels equal distances in equal lab time (mapTime) intervals.

The independent variable is energy (E). Emphasize that we are dealing with kinetic energy transferred to the glider by a spring, or some other energy source.  Gate separation (x) is another important physical parameter, but it is fixed (hence a constant) for this experiment.

The dependent variables are mapTime (t) and travTime (τ). Emphasize these as a clock reading and not as interval of time.

Spacetime Momentum Lab

Apparatus

SpaceTime Explorer Applet on computers in the classroom, collision-enabled.

Spacetime Curvature Lab

Apparatus

SpaceTime Explorer Applet on computers in the classroom, collision-enabled.