Newton's laws in (1+1)D

Traveler-point descriptions of unidirectional motion using proper velocity at high speed give us an expression for differential-aging (gamma) factors which don't involve division by the root of a difference between nearly equal numbers. They also result in reasonably simple equations for constant proper acceleration thanks to the use of a single metric, or reference frame used to define both distances and simultaneity. We also get equations for net force which generalize nicely to the (3+1)D case.

Another difference, as shown also in the table below, is that momentum is simply proportional to proper velocity w. Also, in general we distinguish the 3-vector proper or "felt" force ξ (with frame-invariant magnitude detected by a traveler's cell-phone accelerometer) from the frame-variant force f related to one's rate of momentum change, even though for proper forces in (1+1)D the two are equal.

For students in our non-calculus basic physics course, we've used a single class day (on a chapter about relativity) to show how the laws we teach are a low-speed approximation to laws that work at any speed. As suggested by the table below, length contraction is off the table because it requires two extended frames with synchronized clocks. So is relativistic mass. Our proper speedometry animations nonetheless allow students to see length contraction and other of Bill Shurcliff's “effective for the indicated frame only” (eifo) phenomena, as well as how the traveler-point perspective fits in to simplify the description of motion from a single frame’s viewpoint via a “hybrid kinematic”.

Possible empirical observation (e.g. takehome or modeling-workshop) problems might include acceleration measurements, particularly those using phone accelerometers, and relativistic airtrack simulator experiments and/or data analysis [1].

For students interested in doing more, here are links to wolfram alpha widgets for adding proper velocities and for constant proper acceleration trips in (1+1)D. More functional calculators in this context can be found on our calculators page here. Below, also, find a table of times and peak speeds for a trip (one way) to a destination x lightyears away, e.g. for using 1 ly/y^2 ~ 1-gee constant proper acceleration with a deceleration turn-around halfway there. This might e.g. be of interest to students interested in travel between stars.

Footnotes:

1. P. Fraundorf et al. (2001) "An experience model for anyspeed motion", arXiv:physics/0109030.