Uniform motions and the comparison of their speeds

(by Johann Rottensteiner)

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

Speed is distance per time. That means, different speeds are corresponding with different distances travelled in equal units of time. But the comparison of different speeds 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 speeds 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)

(1)

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

(2)

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

               (t₁>t₂)

(3)

The analysis of the speed formula

To clear up the problematic nature of the comparison of speeds, 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

(4)

For s = unit of length (UL) 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, which in 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, which in a certain number x of units of length is travelled in the time or in the unit of time t, is true:

                   or    

(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_UL) 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 speeds

The different speeds v₁ < v₂ 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_UL) are travelled with the same speed (Fig.1).

Fig.1

v₁ < v₂  means:   x₁ . UL  <  x₂ . UL   and   t₁ = t₂   as well as   t_UL1  >  t_UL2
                                           (s₁  <  s₂)          (x₁.t_UL1 = x₂.t_UL2 )

For the speed v₁ is true:   s₁ = x₁ . UL     and     t₁ = x₁ . t_UL1 = unit of time

(8)

For the speed v₂ is true:        s₂ = x₂ . UL     and     t₂ = x₂ . t_UL2 = unit of time

(9)

For different speeds  (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. t_UL1 = x₂.t_UL2       as well as       t_UL1 > t_UL2

The comparison of speeds gives the following result:

            ->            

(10)

The comparison of speeds 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 speeds) are never included in the calculations and are therefore never taken into account.

The comparison of the speeds 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 speeds taking into account also their different distances travelled in equal times or units of time.

Instead of  x₁ . UL < x₂ . UL  and  x₁. t_UL1 = x₂.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).

      multiplied by      

(12)

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

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

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

Fig.2   v₁ < v₂ means:    x₁ . UL  <  x₂ . UL       and      t₁= t₂           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 speeds, in question the comparison of speeds, 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

Formula Drift Co-Founder Jim Liaw recently spent some time talking to Bill Wood of Tech House & Speed. They cover the hot topic of Irwindale Speedway’s bankruptcy and cancellation of the racing season. Jim sheds a lot of light on how Formula Drift may or may not be affected by this with the information that is available thus far.

We’ll be sure to keep everyone informed on this topic as more information is released from the venue.

CLICK HERE to listen to the interview.