Comparison of speed

by Johann Rottensteiner

The Michelson-Morley-Experiment and the Lorentz-Transformations are the experimental and mathematical foundation of the Theory of Relativity by Albert Einstein. The preoccupation of many years with uniform motions and their speed made obvious a logical-mathematical problem which occurs with the comparison of speed and with their additon and subtraction. The solution of this problem requires inevitably a new speed-conception and the need of revising and correction of the corresponding formulae in physics. Owing to a logical and mathematical fault, we have a wrong idea of the speed. In classical mechanics, the linear addition and substraction of speed is not possible and gives a wrong result. If Einstein had known this fault, he had never created the Special Theory of Relativity.

Uniform motions and the comparison of their speed

Copyright©2010Johann Rottensteiner

(Uniform motions and the comparison of their speed as PDF-Download)

Speed is distance per time. That means, different speed are corresponding with different distances travelled in equal units of time. But the comparison of different speed reveals specifically on this conditions – different distances in equal times – a logical and mathematical faulty result if we want to meet the speed-definition with all consistency. To see the problems, the simple speed-ratio (v1/v2) or the comparison of their different distance-time-ratios needs a thorough mathematical analysing.

The speed formula (speed = distance/unit of time) allows two possibilities comparing different speed of uniform motions:
1.  by means of the different distances which are travelled in equal units of time
2. by means of the different times which are needed to travel equal distances or units of length

(v = speed,  s = distance  and  t = time)

f2              (v1 < v2)                                                                 (1)

If t1 = t2 = unit of time, we obtain:
f3               (s1<s2)

(2)

If s1 = s2 = equal distances or units of length, we obtain:

f4               (t1>t2)

(3)

The analysis of the speed formula

To clear up the problematic nature of the comparison of speed, it should be taken an alternative representation of the speed formula.

(v = speed, x = number of the units of length, UL = unit of length, t = time, s = distance)
s  =  x . UL  =  distance

f5                           f6                                                                                                                                                                         (4)


For s = unit of length (UL) with x = 1 is true:
(v = speed, UL = unit of length,  tUL = time, used for the unit of length travelled with the speed v)

f7                                          f8

(5)

A)  For the time or unit of time, in which a certain number x of units of length is travelled with the speed v, is true:

f9                         f10

(6)

x . tUL  =  t  =  time or unit of time      and      x . UL  =  s  =  distance


B)
  For the speed, in which a certain number x of units of length is travelled in the time or in the unit of time t, is true:

f11             f12                 or              f13

(7)

x . UL  =  s  =  distance      and      x . tUL  =  t  =  time or unit of time

The formulae (7) show very clearly, that the single unit of length UL at the time tUL and the total distance (x.UL) per unit of time (x.tUL) are travelled with the identical speed. This is not only very remarkable, it is also important to understand what speed really means.

 

The comparison of speed

The different speed v1 < v2 of uniform motions, with t1,2 = 0 as the common starting-line, now should be compared with each other. Once again the statement is important, that the unit of length at the time tUL (v = UL/tUL) and the total distance per unit of time (v = x.UL/t  =  x.UL/x.tUL) are travelled with the same speed (Fig.1).

fig1

Fig.1  ©2010Johann Rottensteiner

v1 < v2  means:   x1 . UL  <  x2 . UL   and   t1 = t2   as well as   tUL1  >  tUL2
                                           (s1  <  s2)          (x1.tUL1 = x2.tUL2 )

For the speed v1 is true:   s1 = x1 . UL     and     t1 = x1 . tUL1 = unit of time

f14

(8)

For the speed v2 is true:        s2 = x2 . UL     and     t2 = x2 . tUL2 = unit of time

f15

(9)

For different speed  (v1 < v2) which are travelling different distances in equal times or units of times, the following conditions must be  true:
x1 . UL < x2 . UL        and       x1. tUL1 = x2.tUL2       as well as       tUL1 > tUL2


The comparison of speed gives the following result:

f16            ->             f17

(10)

The comparison of speed v1/v2 (with v1<v2  and  x1<x2), conspicuous in the formulae (10), gives a definite result and allows the following interpretation:
Owing to the cancelling of x1 and x2, the different distances travelled in equal times or units of time (with different speed) are never included in the calculations and are therefore never taken into account.

The comparison of the speed v1 < v2  (with  s1 < s2  and  t1 = t2  or  even if  s1 = s2 und t1 > t2)  will always be reduced to a comparison of the distance-time-ratios of the respective units of length. Therefore,we were never able to compare different speed taking into account also their different distances travelled in equal times or units of time.

Instead of  x1 . UL < x2 . UL  and  x1. tUL1 = x2.tUL2  becomes true:     UL = UL    as well as    tUL1 > tUL2

f18

(11)

The solution is:
A comparative value of the speed of uniform motions must therefore include the distance-time-ratio (= speed) of the individual unit of length and the number of units of length travelled per unit of time. We consequently obtain the right comparative value from the distance-time-ratio of the unit of length (UL/tUL) multiplied by the number of units of length travelled per unit of time (x.UL/t).

f19      multiplied by       f20

(12)

As you can see, both formulae (12) fulfill the definition of speed and therefore the comparison of different speed of uniform motions has to be done by the square of the respective speed.

For  v1 < v2  must be true:         x1.UL  <  x2.UL          t1 = t2             tUL1  >  tUL2

f21

(13)

Compulsory it must results a new understanding of the speed if we want meet the definition of speed with all consistency. A new definiton of speed should be: Speed is the distance-time-ratio over the distance per unit of time.


As a new finding must be true:
All speed of uniform motions must be compared by the square of the respective speed (fig.2).

fig2

Fig.2   ©2010Johann Rottensteiner

v1 < v2 means:    x1 . UL  <  x2 . UL       and      t1= t2           as well as       tUL1  >  tUL2

Final observation
A wrong meaning of the speed has influenced our logical suppose in a way, which cannot be confirmed with mathematical methods.

All calculations with speed, in question the comparison of speed, their additions and subtractions, must be controlled exactly. That means also the calculations to the Michelson-Morley-Experiment, the Lorentz-Transformations and the conception of the special theory of relativity by A. Einstein, which must be revised as well as partly corrected.

Johann Rottensteiner

Copyright©2010Johann Rottensteiner

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