 ...An Exact Classical Mechanics leads toward Quantum Gravitation... Contents
2.2 Method 1: By Linear AccelerationAn undefined object of rest mass m0 is to be considered accelerated from rest with respect to a fluid i-ther to a speed v so that its inertial mass increases to m. Equation[4] for radiation pressure has already established
that kinetic mass mK is equivalent to kinetic energy EK as given by [3]. A
massive object, when accelerated from rest, also acquires kinetic energy and
so its inertial mass must increase as speed increases. It is reasonable to
assume that this inertial mass m is directly proportional to a total energy
E though with a constant of proportionality needing to be determined. This
assumption will be seen fully justifiable after method 2, based on the conservation
of angular momentum, has derived the identical end equation. No other assumption
could have provided this consistency. With B as the constant of proportionality
to be determined, we can write:
 | [5] |
Also Newton's second law, for accelerating force F with mass as an extra variable has to be expressed as the "rate of change of momentum", which can now be written in two forms:  | [6] |
Expanding [6] using differentiation by parts:  | [7] |
Now the work done by force F in an element of distance dx is Fdx and is equal to the gain in kinetic energy dE of the object. Hence multiplying both sides by dx we have:  | [8] |
Putting v = dx/dt this becomes:  | [9] |
Rearranging & noting that E = E0 when v = 0:  | [10] |
Now this is a standard form of integral readily solved by putting: z = 1 - Bv2, so that dz = -2Bvdv. Integration yields:  | [11] |
Which can be re-written:  | [12] |
Clearly as v is increased so E increases according to equation[12] until Bv2 = 1. At this point E becomes infinite and so no further increase in v is possible. Let this limit be defined as the speed of light c. Hence B = 1/c2. Substituting for B in equation[5] & [12] results in:  | [13] |
Note: The author has been informed that this method was at least partly anticipated about 100 years ago by Whittaker. Where c = 2.997925 X 108 m/s. The kinetic energy of the moving object is clearly given by:  | [14] |
With E given by equation[13]. So light must have E0 = 0 and m0 = 0. It is necessary to see how well these predictions are supported by observation and compare them with the achievements of an alternative classical approach. |