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The Conventional (Fake) Derivation of Einstein's**

Mass-Velocity Equation.

Copyright © 2004 - 2013 David V Connell.

**General.**

The "Conventional" method (below, source unknown) of deriving the relation between the original mass and the increased mass in terms of speed does not assume a constant speed of light in its derivation, yet it produces the same equation as Einstein's invalid method :-

*M/M*_{o} = (1 - *V² / c²*)^{-½}

**and so must also be faulty**.

** The Derivation.**

With the usual notations, using the subscript zero to designate "at rest" values, the conventional method is as follows:-

energy
*E = Mc*²
(1)

momentum
*p = MV* (2)

from (1) and (2),
*E = pc² / V* (3)

For an energy change, (where **dE, dx, dp, dt ** are partial differentials, *F* is a Force, *x* and *t* are distance and time),

**dE** = *F***dx** = **dp.dx**/**dt** = *V*.**dp**
(4)

Multiply (3) by (4),
*E***dE** = *c²p*.**dp** (5)

Integrate (5),
*E*² = *c²p² + E*_{o}² (6)

where *E*_{o} is the integrating constant for initial conditions (*V* = 0)

From (3),
*cp = EV / c*
(7)

from (7) and (6),
*E² = E²V² / c² + E*_{o}²
(8)

giving, *E²* (1 - *V² / c²*) = *E*_{o}² (9)

whence
*E / E*_{o} = *M / M*_{o} = (1 - *V²/c²*)^{-½} (10)

** The Flaw**

When the method was examined the flaw was found to be that the equation for momentum was incorrect. It must be obvious that when energy is applied to cause **unhindered** motion the speed is a maximum as all the added energy appears as KE and none can be adsorbed by the moving object, so the mass stays constant at *M*_{o}. Therefore the equation for momentum should be
*p = M*_{o}V, not *MV* . To eliminate *M*_{o} requires the use of equation *E*_{o} = *M*_{o}c², instead of eq.(1), and these are the first two equations of this method. Following the same method used, with the corrected equations, leads to an equation identical to NR eq.(13) - the first order approximation of eq.10 above. If mass *M* **is** used (as it was), one of the partial differentials is not valid as *M* is assumed to be a constant there. It becomes valid when *M*_{o} is used.

If a reader of this article knows who is the author of the above derivation, would they please let the author know by e-mail to admin@betterelativity.com.

**References.**

1. A.Einstein, "Zur Elektrodynamik Bewegter Korper", Annalen der Physik 17, 891-921(1905).

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