The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the relationships between time, veloity, and the rate of aeleration for all material objets. When properly formulated as given in this work, these relationships establish what appear to be two new natural laws of physis. These laws, to be referred to as the Law of onstant aeleration, and the Law of relative aeleration are in omplete onformane with the priniples of both, the time and energy theory, and the millennium theory of relativity. 1
The Laws of Aeleration Copyright 2001 Joseph A. Rybzyk 1. Introdution The purpose of this paper is to identify the dual nature of onstant aeleration and to establish the two laws that define its harateristis. Whereas the first law deals with the onstant aspet of suh behavior, the seond law deals with the relative aspet. These two laws taken together establish all of the important relationships between the individual properties of time, veloity, and the rate of aeleration. It is also noted that these laws onform to the priniples of both, the time and energy theory 1, and the millennium theory of relativity 2, and represent the true relationships of the three stated properties for all objets undergoing onstant aeleration. We begin with the Law of onstant aeleration, first introdued in the time and energy theory. 2. The Law of Constant Aeleration As was previously established in the time and energy theory (* exept for the identifying subsript added to the variable, a) the Law of onstant aeleration is expressed as, v a (1) t` where, a, is the rate of onstant aeleration, and v, is the resulting veloity over time interval, t`. Time, t`, is then defined as, 2 2 v t` t (2) where, t` represents time relative to the moving objet, t, represents time in the stationary frame of referene, v, is the instantaneous veloity of the aelerating objet, and,, is the speed of light. (* The subsript is needed to distinguish between the previously mentioned, two different aspets of aeleration.) Speifially, the Law of onstant aeleration states that the time interval during whih aeleration takes plae is diretly proportional to the resulting veloity. At first, this statement may seem ontraditory beause we are austomed to thinking that, t`, beomes smaller as, v, 2
inreases. While it is true that, t`, beomes smaller as a unit of time in relation to an inrease in veloity, it is also true that the interval, t`, inreases to show the total amount of time that has passed during the aeleration. This onept is more easily understood if we revert for a moment to the lassial, albeit inorret, formula for onstant aeleration, v a. (3) t In equation 3, beause of the inherited (from the slowing of time) asymptotial nature of veloity, v, the rate of aeleration, a, annot be onstant. In other words, as the inrease in veloity, v, grows smaller, time, t, ontinues to inrease at the original rate. One, t, is replaed with, t`, the inrease in time is also asymptotial and the formula orretly defines a onstant aeleration. In view of the preeding analysis, it is apparent that equation 3 is invalid as a formula for showing onstant aeleration. From a pratial standpoint, however, it is used for most everyday appliations, beause the asymptoti nature of veloity is insignifiant at ommonly experiened speeds. From a purely tehnial standpoint, however, equation 3, in its intended use, has very limited appliation. Consider, for example, an observer with a wath on his wrist. Sine the wath moves with the observer, it essentially has zero veloity relative to the observer and therefore always gives orret time relative to his frame of referene, even when he moves. But, the moment the observer applies the time given by the wath to another objet that is moving relative to him, the time is no longer orret. The only truly orret time in suh ases, is time t`. In other words, the observer must onvert time, t, given by the wath, to time, t`, given by equation 2. With a little thought, it beomes evident that equation 1, is always orret, and equation 3 is nothing more than a speial ase of equation 1. Whereas, equation 3 works satisfatorily at low veloities beause the asymptotial nature of, v, is not yet apparent, equation 1, on the other hand, works at low veloities beause at suh veloities time, t`, essentially equals t. Atually, t` = t, only when, v = 0. At any other veloity it is smaller than t. In plain words, time, t`, is always orret at any veloity, while time, t, is only orret at 0 veloity, and therefore equation 3 annot be a orret formula for onstant aeleration. It turns out, however, that in spite of this, equation 3 has a very important role to play involving relative aeleration. This role will be overed shortly under the heading, Law of Relative Aeleration. 3. Defining Aeleration As innoent looking as equation 1 is, it raises serious questions about the proper definition of aeleration, and at the same time it provides us with yet another tool for testing the validity of relativisti priniples. Experimenting with the equation brings to light an apparent ontradition regarding the definition of aeleration. Beause of the asymptotial nature of veloity, the higher the rate of aeleration the more apparent the ontradition beomes. For example, it will be found that if the rate of aeleration, a, reahes the extreme rate of.999 m/s, the veloity, v, reahed in one seond will not exeed.7075 m/s. Moreover, when the rate of aeleration is inreased beyond the given value, the ahieved veloity will slowly redue and never exeed.707 m/s regardless 3
of how muh higher the rate of aeleration is. The question then beomes, whih of the two values is the rate of aeleration? If we go by the amount of energy used, we ould say it is the value we assigned to, a. On the other hand, we ould also say that the rate of aeleration is the veloity reahed at the end of one seond. Either an be onstrued as the orret answer, but for onsisteny, and to avoid onfusion only one should be used. Another solution would be to learly define whih value is being used in eah partiular ase. For the purpose of this paper, the value assigned to, a, is the rate of aeleration. Although the ase given above is extreme, the priniple disussed is always present at any rate of aeleration. To atually ahieve in veloity, the rate of aeleration indiated by, a, in terms of a given distane traveled per unit of time, will always take more than a unit of that time. This of ourse presents itself as a valid test of relativisti priniples, beause equation 1 predits how muh time is required for the resulting veloity to equal the rate of aeleration. 4. The Law of Relative Aeleration Whereas the lassial formula, equation 3, is inorret for onstant aeleration, it is orret for defining relative aeleration and is repeated here in modified form as the Law of relative aeleration. Thus, where, a r, is the rate of relative aeleration, and v, is the resulting veloity over time interval, t, equation 4 shows the orret relationship between these three values for all objets undergoing relative aeleration. v a r (4) t The use of equation 4 in this unintended appliation is supported by the following logi: Sine veloity, v, is asymptoti, as previously noted, and time interval, t, isn t, it follows that the rate of aeleration, a r, given by equation 4 will drop over time as the veloity inreases. In fat, as the veloity approahes, the rate of aeleration approahes zero. Whereas, it is obvious that, a r, annot be the rate of onstant aeleration, it is equally obvious that it is the rate of relative aeleration between the aelerating objet and an observer in the stationary frame of referene. Moreover, the fat that equation 4 uses stationary time, t, and not moving frame time, t`, gives it a vital role to play involving both the rate of aeleration and the resultant veloity of an objet undergoing aeleration. Without equation 4, we have no way of using equation 1 in determining either the rate of onstant aeleration, or the instantaneous veloity of an objet under observation. The reason is simple. To use either equation, we need to be able to first determine the veloity or a hange in veloity over an interval of time. Beause, t`, and, v, are interdependent variables, equation 1 annot be used diretly for this purpose. However, sine, v, has the same value in both, equation 1, and equation 4, and sine, t, is not interdependent on, v, we an determine, v, using equation 4, and then substitute it into equation 1, along with t`, determined from, t, to find onstant aeleration, a. In fat, unlike the ase with equation 1, in the ase with equation 4, the relative aeleration, a r, over a unit interval of time, t, always equals the instantaneous veloity, v. 4
5. Alternate Forms of the Two Laws There are many relationships that may be developed through the integration of the two laws of aeleration. Of these, perhaps the two most important are the alternate forms of the laws themselves. The alternate form for onstant aeleration is arrived at as follows: From equation 4, we get, a r t v (5) the left side of whih an be substituted in plae of, v, into equation 1. We an then also substitute the right side of equation 2 in plae of, t`, in equation 1, to arrive at, a art, (6) 2 2 v t whih simplifies to, a ar. (7) 2 2 v This is not only an alternate form of the Law of Constant Aeleration, but also shows learly the relationship between onstant aeleration and relative aeleration. The following orollary of equation 7, of ourse, is the alternate form of the Law of Relative Aeleration: 2 v ar a (8) 2 6. Conlusion When we begin to understand the true nature of time, the impliations are profound. Not only does it beome apparent that the asymptotial nature of veloity is nothing more than a funtion of the asymptotial nature of time, but it also beomes apparent that many of our equations that depend on lassial time are atually inorret. However, as we learned with the new kineti energy formula, other hanges may also be neessary. Analysis of how the kineti energy formula was updated, should lead to a better understanding of what other hanges might be neessary in the updating of other formulas. 5
REFERENCES 1 Joseph Rybzyk, Time and Energy, Unpublished Work, (2001) 2 Joseph Rybzyk, Millennium Theory of Relativity, Unpublished Work, (2001) The Laws of Aeleration Copyright 2001 Joseph A. Rybzyk All rights reserved inluding the right of reprodution in whole or in part in any form without permission. Note: If this doument was aessed diretly during a searh, you an visit the Millennium Relativity web site by liking on the Home link below: Home 6