Velocity Sum 
Systems in Releative Motion
An issue that confounds new students of AD is the velocity SUM equation, because of their entrenched SR mind set. They firmly believe the SR thesis regarding "systems in relative motion."
New AD students have, in general, some concerns regarding the AD derivation that finds the "simplified Lorentz equations", or Carezani equations
and
when three or two "systems in relative motion" are reduced to only one. When the discussion moves to the difference between Kinematics and Dynamics, that is, without expending energy and expending energy, respectively, the idea is not clear enough in relation to "systems in relative motion." (Galilean relativity). Confronted with the SR velocity SUM equation (Kinematics) and the AD equation (Dynamics), the attitude is "ecstasy," especially at high velocities.
When the velocity is very low, the confusion is TOTAL. Because AD gives different values than Classic Mechanics (CM) and SR, the first reaction, of course, is to think that AD is wrong.
A real example follows. The issue will be explained below, but it is interesting to follow in detail all the difficulties and alternatives that a "real" physicist undergoes.
Question 1:
According to the Web page, the velocity sum equation in AD is:
Bsum = sqrt(1  (1B1^2) * (1B2^2) * ... * (1B3^2))
Well, I plugged in the following scenario:
B1 = 0.926 * 10^9 = 1 km/hour B2 = 0.926 * 10^9 = 1 km/hour
Going through the equation, I obtain:
Bsum = 1.31 * 10^9 = 1.41 km/hour
If the velocity sum equation means anything like the velocity equation in SR, then this is telling me that if I am walking a 1 km/hour, and I see Bob pass me at 1 km/hour, then Bob is moving at 1.41 km/hour.
What' s wrong here? Is the equation misstated? Am I misinterpreting the equation (and if so, what does the equation mean)? Is AD invalid at speeds <<c? Is Bob actually moving at 1.41 km/hour and we don't realize it for some reason?
Answer:
***You are right. There is no mistake in your calculation.
Classic Mechanics give 2 km/h
SR gives 1.9999 km/h
AD gives 1.41 km/h
At small velocities SR is closer to Classic Mechanics, but at large and at small velocities AD conserves energy and momentum, and SR doesn't.
I remind you that Classical Mechanics doesn't apply to relativistic velocities because the velocity sum is larger than c. In SR and AD the velocity is always less than c. The difference between SR and AD is in energy and momentum conservation.***
This answer is not clear enough, as we realized later. The "mystery" of the difference between CM, SR and AD continues to be a mystery that, of course, drives the reader to the conclusion that AD is wrong, as you can see in his "angry" answer:
Actually, I don't care whether AD is close to Newtonian (Classical) Mechanics or not. What I do care about is whether it's close to reality or not. (BTW: it's not true that AD conserves momentum and SR doesn't  see below).
That is the whole point of theories  to try to explain and predict what happens in the real world. A theory is interesting only to the extent that it agrees with the universe we happen to find ourselves in.
Now, in the universe we happen to find ourselves in, if I am traveling at 1 km/hour, and I see someone pass me at 1 km/hour, then a third person on the ground will measure his speed at approximately 2 km/hour. This is consistent with both Newtonian Mechanics and SR (the difference between the observed value and the predicted value is within measurement error in both cases), but it is not consistent with what AD predicts
Hence, at least in this case, AD is wrong.
What is more illustrative is the following:
Besides, energy is not conserved (for your rather odd definition of energy conservation) for velocity sums at nonrelativistic speeds  this is easily verified (just compute the energy for the sample above) Why would we expect it to start to be conserved once we start moving at relativistic velocities?
As we said before, our answer was not clear enough for a physicist saturated with "SR's systems in relative motion." He cannot realize that the third person, or body, in the "ground," doesn't conserve energy regarding CM or SR, because the phenomena are unrelated. Of course, the difference is not in the high (relativistic) or low velocity, the difference is only conceptual, or more precisely, physical. As will be explained, the difference is that there is no connection among the three phenomena or the three different "Bobs." If a connection is established between them through SR's systems in relative motion, the consequence is that there is no energy conservation. Also, this confirms overwhelmingly AD's thesis that a "system" is formed by the phenomenon and observer.
A short program in qBasic was constructed, to get values to support our arguments. Our answer is the following:
*** Any theory is not intrinsically right or wrong. Regarding your statement "What I do care about is whether it's close to reality or not," we can say that "reality," as an absolute concept, doesn't exist. A theory is closer to "reality" when more experimental or observational results can be explained. This depends on the equations, and of course, there will never be a "perfect" equation to explain the phenomenon completely.
Newtonian gravitation is almost perfect, but the power is not 2. I don't remember exactly the proposed correction, but it is 1.99999998 or 2.00000002. 2 is used to calculate the planetary motion and the "errors" are corrected by the theory of perturbations, including approximate calculations.
Your idea of "reality" or "theory" is too "narrow" to be true regarding our historical knowledge.
But you confirm what I said before with your following phrase:
That is the whole point of theories  to try to explain and predict what happens in the real world. A theory is interesting only to the extend that it agrees with the universe we happen to find ourselves in.
You also confirm what we said before: Classic Mechanics fails at high velocities and SR works when the accelerated particle receives external energy, but fails when applied to decay phenomena. AD doesn't fail when it is applied to decay phenomena and its KE equation includes the SR KE equation when the external energy is taken into account.
You don't need to trust us or AD. You can calculate with its equations and you will confirm what AD says. See page 16 "VELOCITY SUM" in the "A Special Report for Professional Supporters of SR and GR" in the WWW. Thanks.
Regarding your statement (BTW: it's not true that AD conserves momentum and SR doesn't  see below), you didn't show anything about this. We will show you, with numbers, that what we say is true. We do the calculation for energy but it is the same for momentum.
We did some calculations to show you something interesting. This was done 50 years ago by Carezani, a young student, using tables of logarithms with 20 or more decimals.
I am so sorry, and I regret to tell you this, but you are wrong and we can demonstrate it with numbers.
We take B = 9.26 10^7 (1000 Km/h) not 9.26 10^10 (1 km/h) because the qBasic program that we are using will not show significative numbers when a quantity very close to 1 is subtracted from 1. Nothing will change.
You need to realize something very simple: if the velocity is 1, and the body mass is 1, the KE1 of body 1 is, in classic mechanics:
KE1 = 1/2 * 1 * 1 = 1/2
If the velocity is 1, and the body mass is 1, the KE2 of body 2 is
KE2 = 1/2 * 1 * 1 = 1/2
The total KE involved is
KET = KE1 + KE2 = 1/2 + 1/2 = 1
Repeating your words "......., then a third person on the ground will measure his speed at approximately 2 km/h."
Okay, He will measure KE3:
KE3 = 1/2 * 1 * 2^2 =Â 4/2 = 2 (Applying CM or SR because his velocity is equal to 2 km/hour)
Now the KE3 for the "third person" is double regarding the KE SUM of person 1 plus person 2.
What is the "reality"? Do we expend 1 unit of energy or 2 units of energy?
In AD everything is compatible. This is Carezani's great discovery. In the Universe, there is no Kinematics. In the Universe, Dynamics is valid. At the end of this paper, we will show you the revolutionary conclusion that we obtain from these results and observations.
What follow are the results of our little program: mo = 10, ad = AD, sr = SR, cm = CM, KE = Kinetic energy, mv = motion mass.
Autodynamics
B = 9.26E07 (Velocity in fraction of c) mv1ad = 9.999999999995712 (Motion mass of body 1) KE1AD = 4.28738E12 (KE of body 1) mv2ad = 9.999999999991424 (Motion mass of body 2) KE2AD = 4.28738E12 (KE of body 2) KEADT = 8.57476E12 ========>>>>> Total kinetic energy Bsad = 1.309562E06 ( Velocity from velocity SUM equation) KEADS = 8.57476E12 ========>>>>> Total KE with velocity SUM equation.
Classical Mechanics
KE1CM = KE2CM = 4.28738E12 (KE of body 1 and KE of body 2) KECMT = 8.57476E12 ========>>>>> Total kinetic energy adding KE of body 1 plus KE of body 2. Bscm = 1.852E06 (Velocity from velocity SUM equation) KECMS = 1.714952E11 =========>>>>> Total KE with velocity SUM equation.
Special Relativity
mv1sr = 10.00000000000429 (Motion mass of body 1) KE1SR= 4.26738E12 (KE of body 1) mv2sr = 10.00000000000858 (Motion mass of Body 2) KE2SR = 4.28738E12 (KE of body 2) KESRT = 8.57476E12 ========>>>>> Total kinetic energy. Bssr = 1.852E06 (Velocity from velocity SUM equation) KESRS = 1.714952E11 ========>>>>> Total KE with velocity SUM equation.
(By the way: You can see the fictitious creation of energy in SR through mass increment).
You can see that in AD, the Total kinetic energy(KEADT) = (KEADS)Total KE with velocity SUM. This is not true in SR and CM. And you can see something very illustrative: In AD, CM and SR the KE sum are equals. The AD velocity sum equation represents dynamics phenomena; meanwhile CM and SR only represent a Kinematic phenomena.
Those phenomena don't exist in the Universe.
Carezani's revolutionary discovery is that the THIRD system, or yet the SECOND system in relative motion doesn't exist! Only the phenomenon and observer EXIST.***
Even though he didn't reply, we sent another email for the following reason:
*** We are sending you a more detailed explanation, because reading the last one we sent to you, we realized that it is not clear enough.
In Classic Mechanics, the body with mass equal 10, traveling at B = 9.26 10E7 has a KE = 4.28738E12. Body 2 has the same value, but it is not related "relativistically" to body 1. Body 3(really the observed by the third person on the ground) traveling at B = 2 * 9.26E7 has an energy 4 times larger than the KE of body 1 or the KE of body 2, equal to 1.714952E11, that is, 2 times larger than the KE of body 1 and body 2 added, but it is not related "relativistically" to body 1 or body 2.
You are the *observer* and each body, 1, 2 or 3 is an unrelated phenomenon. The "Conventional Wisdom" will tell you about "Galilean Relativity," but this is not true, because body 3 "is not" traveling in a system in relative motion with respect to body 2 and this with respect to body 1. You are the *observer* and each phenomenon is totally independent of the others.
Of course you can ask yourself: What is the traveling velocity of body 3, to have an equal KE as the KE SUM of body 1 and 2?
v = SQR ((2 * (2 * KE)) / m) = 1.309562E6
That is, of course, the AD velocity, Bsad.
This is the Classic Mechanics solution. But mixing with this solution the wrong solution given by SR (systems in relative motion), is totally irrelevant. Classical Mechanics is meant for small velocities, and it is true under these conditions. Any Engineer, to find the energy of each system, will not mix Classical Mechanics with SR "systems in relative motion"
What is the AD interpretation?
AD applies to DECAY cases. Body 1, with mass equal 10, and traveling at B = 9.26 10E7 decays into body 2, with mass 9.999 999 999 995 712, that, traveling at the same velocity, decays into body 3, with mass = 9.999 999 999 991 424.
What is the velocity of a body, with the last mass, having a KE that is the KE SUM of the KE of body 1 and the KE of body 2? You know the answer: The velocity is 1.309 562 10E6.
In AD, also, the successive phenomena are observed by the "same observer." There only is one observer. There are not "systems in relative motion."
Where is the "magic" of AD?
Body 1 starts with mass = 10, and after two successive decays, at the same velocity of 9.26E7 c, the mass in motion is:
9.999 999 999 991 424
If you subtract this mass from 10 we, have:
m lost = 0.000 000 000 008 576 = 8.576E12
that is, the KE of the particle (KEADT) = (KEADS) through massenergy equivalence E = mo c^2. There is mass transformation into kinetic energy through a decay process. There is energy conservation, and of course, momentum conservation.
The other AD "magic" is that the equation works perfectly at low velocities and at high velocities.
You can analyze now, by yourself, what happens with SR, where the "systems in relative motion" meet "reality" and lead to mistaken solutions.
The power of AD is that it can demonstrate overwhelming, and definitely, that the SR idea about "systems in relative motion" doesn't work. If body 1 is in a system with velocity B1 with respect to body 2 on other system with velocity B2 with respect to an observer in system 3, SR transformation make no sense.
KE, that is, a Dynamic process, has no relation with Kinematics, that doesn't involve energy, and is not related to systems in relative motion.
Of course, you will now have some "mental constraint" regarding high velocities. At high velocities the two solutions are much closer because the two solutions have the same limitation: The increasing velocities increase asymptotically to the limit, that is, the value of c.
As the Kinematics SUM velocity cannot be larger than c, that is, the light velocity, at high velocities it is very close to the Dynamics SUM velocity, that also cannot be larger than c.***
What follows is the same as what we said above, but with numbers.
CLASSICAL MECHANICS
1 km/hour + 1 km/hour = 2 km/hour (2 km/hour) / (1 km/hour) = 2 times For the example, two velocities' values will be taken: 0.8 c and 0.96 c.
SPECIAL RELATIVITY (0.8)
0.8 * 300 000 = 240 000 Km/sec (0.8 + 0.8) / (1 + (0.8 * 0.8)) = 1.6 / 1.64 = 0.975 609 756 0.975 609 756 * 300 000 = 292 682.926 8 292 682.926 8 / 240 000 = 1.219 512 195 times (1)
AUTODYNAMICS (0.8)
{ 1  [ (1  0.8^2) * (1  0.8^2)]}^1/2 = 0.932 952 303 0.932 952 303 * 300 000 = 279 885.691 279 885.691 / 240 000 = 1.166 190 379 times (2)
Dividing result (1) by result (2) the values is going close to 1.
1.219 512 195 / 1.166 190 379 = 1.045 723 08
SPECIAL RELATIVITY (0.96)
0.96 * 300 000 = 288 000 (0.96 + 0.96) / (1 + (0.96 * 0.96)) = 1.92 / 1.9216 = 0.999 167 361 0.999 167 361 * 300 000 = 299 750. 208 2 299 750.208 2 / 288 000 = 1.046 799 334 times (3)
AUTODYNAMICS (0.96)
{ 1  [ (1  0.96^2) * (1  0.96^2) ] }^1/2 = 0.996 921 983 0.996 921 983 * 300 000 = 299 076.594 9 299 076.594 9 / 288 000 = 1.038 460 399 times (4)
Dividing result (3) by result (4) the value is closer to 1
1.046 799 334 / 1.038 460 399 = 1.008 030
If the phenomena are connected through SR "systems in relative motion", the consequences are wrong, that is to say, the results are wrong.
If the phenomena are connected through AD "only one system" or successive phenomena observed by the same observer, the consequences are right, that is, the results are right.
The "relativity" of "space" and "time" is given by the light velocity constancy between the phenomenon and the observer, that together, form the "relativistic system," the only one system that exist.


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Storm In Physics (2005) 
2nd book published by Dr. Carezani.

