Most Direct Derivation of Relativistic Kinetic Energy Formula Copyright 2010 Joseph A. Rybczyk Abstract The millennium relativity form of the kinetic energy formula is derived through direct modification of the Newtonian formula in as brief a manner as practical. Explanation of the underlying relationships involving momentum and acceleration is then presented in the simplest terms practical. 1. Introduction Direct modification of Newton s kinetic energy formula is conducted in the fewest steps practical in the derivation of the millennium relativity 1 (MR) form of the relativistic kinetic energy formula. The resultant formula is then analyzed for improved understanding of the relativistic principles involved. 2. Direct Modification of Newton s Kinetic Energy Formula Given the Newtonian kinetic energy formula in the form 1 2 1 and the millennium relativity gamma factor 2 in the form 2 we derive 1 1 3 for the millennium relativity form of the relativistic kinetic energy formula 3 where k is the kinetic energy of mass m moving at velocity v, and c is the speed of light. 3. Expanding the Formula into its Related Components The just derive kinetic energy formula (3) can be expanded into the form 1 1 4 then rearranged into its components 1 1 5 1
that can be separated into 6 where p is its momentum component, and 1 1 7 where D is its instantaneous relativistic constant acceleration distance component. In the context given, instantaneous distance is defined as the distance an object under any kind of motion would travel at that instant if an instant didn t consist of zero time. Thus, an instant as defined here is an infinitely small unit of time but not zero time. It some ways it is similar in meaning to a snapshot of an object in motion. Although the object is in motion, its image captured in the snapshot isn t. This concept will be discussed in detail as we progress with the analysis. 4. Evaluation of Results The first important step in the evaluation process is to recognize that gamma operates on velocity v in both components of the resultant equation, showing that mass does not increase as a result of acceleration to speed v. This can be interpreted to mean that the effect that velocity v has on momentum is amplified by the gamma factor as speed v approaches light speed c. To analyze the effect of velocity on the instantaneous distance component we begin by comparing it to the Newtonian distance for constant acceleration. Thus, given the Newtonian formula 1 2 8 where D is the distance traveled during time interval T at the constant rate of acceleration a as given by 9 where v is the speed at any instant, we derive by way of substitution of the right side of equation (9) for acceleration rate a in equation (8) 1 2 10 for the alternate form of the formula. Then, given 11 for the distance D u traveled at uniform motion speed v, it is apparent that the distance D given by formula (10) is 1/2 the distance D u traveled at a uniform rate of speed v during the same time 2
interval T. If we assign time interval T an infinitely small unit value we can eliminate it from both equations, (10) and (11) giving 1 2 12 for the instantaneous distance traveled at a constant rate of acceleration. This simplified formula can be interpreted as showing that the distance D traveled at constant acceleration at the instant speed v is reached equals 1/2 v x 1. It can also be interpreted as showing that the distance D traveled at any instant at a constant rate of acceleration equals 1/2 the distance that would be traveled at a uniform rate of speed v for that same instant. Now let us continue with a modification to equation (12) in the form of 1 1 1 13 where the main fraction now varies from 1/2 to 1 as v varies from 0 to c. In this interpretation, the distance traveled at any instant during constant acceleration varies from 1/2 the distance traveled at a uniform rate of speed to 1 x the distance traveled at a uniform rate of speed during that same instant depending on the instantaneous value of v as v varies from 0 to c and γ varies from 1 to. By multiplying the top and bottom of the main fraction in equation (13) by γ and rearranging we obtain the form 1 1 14 as used earlier in equations (5) and (7). 5. The Complete Relativistic Kinetic and Total Energy Formulas By substituting the right side of equation (2) for gamma in equation (5) we derive 1 1 15 that simplifies to 16 and 17 3
for the complete form of the millennium relativity kinetic energy formula. If we then add the internal energy term mc 2 to the right of formula (17) we obtain 18 for the millennium relativity formula that gives the total energy E of an object in motion. 6. Advanced Understanding of Acceleration To fully understand the physical behavior underlying relativistic kinetic energy it is necessary to understand the behavior underlying relativistic acceleration. Whereas the Newtonian form of constant acceleration as modified by gamma was adequate for use in deriving the relativistic kinetic energy formula, it does not truly represent the underlying nature of relativistic constant acceleration embodied in the resultant kinetic energy formula. For a final understanding we need to review the two forms of millennium relativity constant acceleration 4. We begin with the formula for constant acceleration as experienced in the acceleration frame. Thus, where t is the dilated time in the uniform motion frame represented by the velocity v reached at any instant during the acceleration, T is the corresponding time in the stationary frame that velocity v is referenced to. The relationship between the two time intervals is then given by 19 that in turn gives 20 and 21 where the millennium relativity form of gamma was previously given by equation (2). Now that we have defined the relationship of t to T, we can express the constant acceleration form of relativistic acceleration as 22 where a c is the rate of acceleration experienced in the frame of the accelerating object. From this it follows that 23 4
were a r is the corresponding relative rate of acceleration measured in the stationary frame to which T and v are referenced for the period in which the acceleration takes place. The relationship of a c to a r is then given by that in turn gives 24 25 and 26 or by way of substitution of the left side of equation (20) for the fraction T/t in equation (26) 27 for the complete relationship of constant acceleration rate a c to relative acceleration rate a r. The distance for relativistic constant acceleration is then given by 1 1 1 28 whereupon substitution of the right side of equation (22) in place of a c and simplifying gives 1 1 29 which is the exact equivalent of equation (28) used in the Time and Energy, Inertia and Gravity paper. Similarly, the distance for relative acceleration is given by 1 1 1 30 whereupon substitution of the right side of equation (23) in place of a r and simplifying gives 1 1 31 which is the exact equivalent of equation (19) used in the Time and Energy paper. Dropping the t and T intervals from equations (29) and (31) respectively then gives 5
1 1 32 for the instantaneous relativistic constant acceleration distance. This is the same equation previously given as equation (14) in the present paper. Since the instantaneous version of the equation essentially relates the acceleration distance to the corresponding uniform motion distance that uses the same time interval, dropping the interval has no effect on the resultant instantaneous relativistic constant acceleration distance. 7. Conclusion Derivation of the relativistic kinetic energy formula through direct modification of the Newtonian formula using gamma is straightforward and accomplishable in only three steps as demonstrated in this paper. Explaining the underlying nature of the formula is another matter entirely. Not only are many more steps required, but some are very abstract and difficult to comprehend. Yet, the resulting kinetic energy formula has been shown to be another form of the special relativity formula and is subsequently supported by the same evidence. Assuming the formula is correct then, we should focus our attention on the underlying relationships embodied in it. The investigator has done his best in the present paper to present those relationships in the simplest terms practical. REFERENCES 1 Joseph A. Rybczyk, Millennium Theory of Relativity, (2001), http://www.mrelativity.net 2 The Millennium Relativity gamma factor is simply another form of the Lorentz gamma factor used in Einstein s Special Theory of Relativity 3 The Millennium Relativity form of the kinetic energy formula has been shown to be another form of the relativistic kinetic energy formula given in Einstein s Special Theory of Relativity 4 Joseph A. Rybczyk, The Laws of Acceleration, (2001), http://www.mrelativity.net Most Direct Derivation of Relativistic Kinetic Energy Formula Copyright 2010 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 6