### 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 (v_{1}/v_{2}) 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)

If t_{1} = t_{2} = unit of time, we obtain:

(s_{1}<s_{2})

(2)

If s_{1} = s_{2 }= equal distances or units of length, we obtain:

(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**

**
For s = unit of length (UL) with x = 1 is true:
**(v = speed, UL = unit of length, t

_{UL}= time, used for the unit of length travelled with the speed

**v**)

(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:

(6)

**x . t _{UL} = 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:

(7)

**x . UL = s = distance and x . t _{UL} = t = time or unit of time**

The formulae (7) show very clearly, that the single unit of length **UL** at the time **t _{UL}** and the total distance

**(x.UL)**per unit of time

**(x.t**are travelled with the identical speed. This is not only very remarkable, it is also important to understand what speed really means.

_{UL})** **

**The comparison of speed**

The different speed v_{1} < v_{2} of uniform motions, with t_{1,2} = 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 t_{UL} **(v = UL/t _{UL})** and the total distance per unit of time

**(v = x.UL/t = x.UL/x.t**are travelled with the same speed (Fig.1).

_{UL})**Fig.1 ©2010Johann Rottensteiner
**

**v _{1} < v_{2}** means: x

_{1}. UL < x

_{2}. UL and t

_{1}= t

_{2}as well as t

_{UL1}> t

_{UL2 }(s

_{1}< s

_{2}) (x

_{1}.t

_{UL1}= x

_{2}.t

_{UL2 })

**For the speed v**_{1} is true: s_{1} = x_{1} . UL and t_{1} = x_{1} . t_{UL1} = unit of time

_{1}is true:

(8)

**For the speed v**_{2} is true: s_{2} = x_{2} . UL and t_{2} = x_{2} . t_{UL2} = unit of time

_{2}is true:

(9)

For different speed **(v _{1} < v_{2})** which are travelling different distances in equal times or units of times, the following conditions must be true:

x

_{1}. UL < x

_{2}. UL and x

_{1}. t

_{UL1}= x

_{2}.t

_{UL2}as well as t

_{UL1}> t

_{UL2}

_{ }

The comparison of speed gives the following result:

The comparison of speed gives the following result:

(10)

The comparison of speed **v _{1}/v_{2}** (with v

_{1}<v

_{2}and x

_{1}<x

_{2}), conspicuous in the formulae (10), gives a definite result and allows the following interpretation:

Owing to the cancelling of x

_{1}and x

_{2}, 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 **v _{1} < v_{2} ** (

**with**s

_{1}< s

_{2}and t

_{1 }= t

_{2}or

**even if**s

_{1}= s

_{2}und t

_{1}> t

_{2}) 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 x _{1} . UL < x_{2} . UL and x_{1}. t_{UL1} = x_{2}.t_{UL2} **

_{ }

**becomes true:**UL = UL as well as t

_{UL1}> t

_{UL2}

(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/t_{UL}) **multiplied by **the number of units of length travelled per unit of time (x.UL/t).

(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 v _{1} < v_{2} must be true:** x

_{1}.UL <

**x**

_{2}.UL t

_{1}= t

_{2}t

_{UL1}

**>**t

_{UL2}

(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).

**Fig.2 ©2010Johann Rottensteiner **

**v _{1} < v_{2}** means:

**x**

_{1}. UL < x

_{2}. UL and t

_{1}= t

_{2 }as well as t

_{UL1}> t

_{UL2 }

**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

I find it plausible and I couldn´t find anything that is wrong with your calculations. But I am no expert. Hopefully more people will take their time to study “your theory”. I wish you best of luck to attract the right people. Which would be people who dare to raise to question A. Einstein.

Best regards

First of all the word “speed” is a singularia tantum!

Even if your calculation would be right. There is no connection between your calculation and the lorenz-transformation or the special-relativity. If you are a mathematician, you should know, how an argumentation works.

How can you say: “All speeds of uniform motions must be compared by the square of the respective speeds.” Why? There is no reason at all.

And if you say: “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.”,

please read http://www.physik.uzh.ch/~strauman/physik-a/srt.pdf

(this is the Lorentz-Transformation and Einsteins special theory of relativistic) and show us what you would change and please tell us why!

please read more scientific books!

cheers Pacman

[…] Thank you for your critical comment. You are quite right in many points. But I have written this weblog for a wide range of readers and I wanted to do that as clear and as short as possible. In fact, it should only be pointed out the main problems with the comparison of speed and made clear the need of a new speed-conception. […]