Pace Car 1 : Sound & Light

Theatre of Her Mind

A scene is acted out in the theatre of her mind as kilometer post after kilometer post silently count the length of her journey.  It is the principle of relativity that struts and storms across the stage.  According to which, all laws of motion take their simplest form within a single inertial reference frame when that reference frame is attached to a body.  In any other reference frame considered attached to a moving observer then there is defined a Galilean transformation which defines a mathematical relationship between the object and the observer with one or the other moving, or at rest; while the other is considered moving, or at rest. These two relative reference frames are the foundation of the Newtonian description of motion of physical objects.

She pictures herself and a compatriot traveling down a long level straight section of turnpike, on a windless day. She imagines that if she paces her friend’s auto, adjudging that she is neither gaining ground, nor losing ground, she can safely assume that both autos are traveling at the same velocity; remaining at a constant distance, L, apart as the pair rolls down the highway.  With this conclusion in mind, she can regard that the pair of autos have formed a tandem, which is open to the mathematics that was presented before.  Amidst the soybean fields along the turnpike, a football field is parallel and aligned with the roadway.  So she notices, that when he is at one goal line, she is at the other goal line, thus establishing L, their distance apart (there are many methods for establishing L)..

In the reference frame attached to the autos, the vehicles are at rest relative to one another.  Another observer is seated on a wooden bench, motionless, at rest, in a roadside rest area. In his reference frame, he sees the tandem traveling down the turnpike at the velocity, v. From his viewing position he factors in the addition of velocities that occurs in the auto reference frame.

If a projectile were launched (as from a front or rear mounted machine gun a la James Bond, addition of velocities by transfer of momentum, applies to each bullet) from either auto to the other, then the addition of velocities would be hidden and unobservable by either driver.  So if she sends her light signal to the forward auto, then he fires a single bullet back towards her auto.  The bullets velocity, as she sees it is, v = d / t with addition of velocities masking transfer of momentum.

Tokio Drift in Japan

From within the reference frame attached to the rest area, however, the observer would include this in his calculations, by the Galilean transformation.  From his vantage point, he observes that L, is the distance between the tandem autos; and during the same universal Newtonian time, t, that the pair is traveling at the velocity, v.

Then her imaginings shifts scenes.  For sound, along with no change to their vehicle speeds, the sound pulse acts as a projectile through the still air, but different.  It is in the nature of sound waves (and light waves) to violate Galilean invariance; in that the velocity of the source or receiver does not under normal measuring conditions, have an impact on the wave speed.

Thus in still air, the equation of motion would include, by algebra, the velocity of her vehicle L = ct + vt. (her vehicle has the velocity, v and the speed of sound is, c and each begins its movement at the ends of L).   This formula factors in the aspect that the auto moves forward through the still air, while at the same time, the sound pulse continues to travel at the same speed.  The auto meets the sound pulse somewhere within the distance, L.  The auto travels forward the distance vt, with v as the velocity of her auto; while at the same time the sound pulse travels rearward, the distance ct, with c as the speed of sound through the windless air. This all takes place during the same Newtonian universal time, t.

This is identical in form to the Galilean transformation as used by the rest area observer.  She has made a quantum leap across reference frames; working completely from within the reference frame attached to her vehicle.  The air (medium is motionless) resides in the rest area reference frame; it is disconnected from the autos reference frame.  However, it is somehow interacting and mixing within the two reference frames; it is interwoven in a mathematical way that is different from the rules of engagement, but forward by the principle of relativity.

Danica Patrick

So, she imagines, if she sends the light signal from her headlights forward to her compatriot’s auto and begins her clock at the same moment, then, disregarding his reaction time, he could fire a single imaginary bullet from his imaginary gun back towards her auto.  When the bullet pierces her front radiator she stops her clock.  The time she measures for the bullets airtime to arrive at her auto would be the simple Newtonian law of motion, t = L / c.  The bullet slows down or speeds up depending on the guns direction of motion, so as to appear to arrive in the same time as if it were at rest.

However, for a sound pulse this would not be the case.  The time would be t = L / (c + v) after algebraic rearrangement.  This is different from the simplest form of the Newtonian equation of motion; it can be solved for the velocity, v, the velocity of her automobile.  This is not absolute motion, nor strictly a limited relative motion, but is some kind of intermediary motion.  Peeling back and revealing a previously unobservable layer of reality.

She has found a means to determine her own velocity not just relative to her friend’s auto, which is in motion relative to the roadway, but at rest relative to her; but also she has found her velocity relative to the observer at rest.  If suddenly, by some alien magicks they find themselves on a featureless landscape, plunged into an eerie darkness, but at the same distance apart; in the dark the Newtonian or Galilean principle of relativity method, does not visually work. But, by a sonic phenomena combined with the light method I have described here, she could still find the velocity of her auto.