Relativistic Kinetic Energy Simplified Copyright 207 Joseph A. Rybczyk Abstract The relativistic form of the kinetic energy formula is derived directly from the relativized principles of the classical form of the kinetic energy formula. This, in turn, involves the formulas for inertial force, constant acceleration, and the distance traveled during such acceleration. The end result is the simplest derivation of the relativistic kinetic energy formula, that when added to the rest mass energy gives E = mc 2.
Relativistic Kinetic Energy Simplified Copyright 207 Joseph A. Rybczyk. Introduction It was apparent to the author from the very beginning of his research involving relativistic kinetic energy that it would be difficult to present the findings in an easy to comprehend manner. This was due to the relational complexity of all of the variables involved. This was especially true with regard to equating such relativistic principles to the classical principles to which they are related. Only now, upon making a final discovery involving the distance traveled during relativistic acceleration, was he able to show a direct relationship to principles of the classical form without exception. 2. Relativistic Kinetic Energy Starting with the classical relationships for kinetic energy, we have FFFF () CCCCCCCCCCCCCCCCCC RRRRRRRRRRRRRRRRRRhiiiiii where K is kinetic energy, F is inertial force, and D is the distance over which the force is applied. For relativistic inertial force we have FF = mm 0 vv2 aa (2) RRRRRRRRRRRRRRRRRRRRRRRR IIIIIIIIIIIIIIII FFFFFFFFFF where the classical mass is replaced by relativistic rest mass m0 and the fractional term known as the relativistic gamma factor. This is followed by the acceleration rate term a that is then defined by aa = vv TT (3) CCCCCCCCCCCCCCCC AAAAAAAAAAAAAAAAAAAAAAAA where a is the rate of acceleration, and v is the speed reached during time T. For the distance term of equation () we have DD = vvvv + vv2 (4) AAAAAAAAAAAAAAAAAAAAAAAA DDDDDDDDDDDDDDDD where the fraction that includes the Lorentz factor vv2 (5) LLLLLLLLLLLLLL FFFFFFFFFFFF replaces the ½ fraction used in the classical formula. The complete formula is then given as 2
mm 0 vv2 vv TT + vv2 vvvv (6) where the acceleration term a in Relativistic Inertial Force equation (2) is replaced by the v/t term of Constant Acceleration equation (3). This is then followed by the Distance equation (4) just given. And, since the relativistic inertial mass is given by mm = mm 0 vv2 (7) IIIIIIIIIIIIIIII MMMMMMMM where m is the total mass, we obtain, by way of substitution into equation (6) that further simplifies into giving mm vv TT + vv2 + vv2 vvvv (8) mm vv vvvv (9) TT mmvv 2 + vv2 (0) RRRRRRRRRRRRRRRRRRRRRRRR KKKKKKKKKKKKKK EEEEEEEEEEEE as the final formula for relativistic kinetic energy. 3. Other Versions of the Kinetic Energy Formula Depending on the nature of the investigator s research, three other versions of the just derived kinetic energy formula might prove useful. That is, if the rate of acceleration is of interest, or the time during which the acceleration takes place, then these other versions of the formula may be used. They all give identical results for the same speed v reached. The three other versions are as follows: mm(aaaa) 2 + vv2 () RRRRRRRRRRRRRRRRRRRRRRRR KKKKKKKKKKKKKK EEEEEEEEEEEE 3
+ (aaaa)2 + (aaaa)2 mm(aaaa) 2 mmvv 2 (2) RRRRRRRRRRRRRRRRRRRRRRRR KKKKKKKKKKKKKK EEEEEEEEEEEE (3) RRRRRRRRRRRRRRRRRRRRRRRR KKKKKKKKKKKKKK EEEEEEEEEEEE This completes the simplified derivation process for the relativistic form of kinetic energy. Now let us take a moment to look at its relationship to Einstein s E = mc 2 equation 2. 4. The Relationship of K to E = mc 2 As the reader likely knows, Einstein s total energy equation EE = mm (4) EEEEEEEEEEEEEEnn ss TTTTTTTTTT EEEEEEEEEEEE is actually a simplified version of the more complete version given as EE = mm 0 vv cc (5) EEEEEEEEEEEEEEnn ss TTTTTTTTTT EEEEEEEEEEEE where the total mass m is replaced by the rest mass m0. The kinetic energy version of Einstein s equation is then given by mm mm 0 (6) EEEEEEEEEEEEEEnn ss KKKKKKKKKKKKKK EEEEEEEEEEEE which then becomes mm 0 vv cc mm 0 (7) EEEEEEEEEEEEEEnn ss KKKKKKKKKKKKKK EEEEEEEEEEEE where the kinetic energy K equals the total energy E minus the rest energy. In the case of the author s kinetic energy equation, which was derived directly from the principles involved, we have to add the rest energy to obtain the total energy E. This gives EE = KK + mm 0 (8) RRRRRRRRRRRRkk ss TTTTTTTTTT EEEEEEEEEEEE for the simplified version of the author s total energy equation, and EE = mmvv 2 + mm 0 + vv2 (9) RRRRRRRRRRRRkk ss TTTTTTTTTT EEEEEEEEEEEE 4
for the complete version of the author s total energy equation, using equation (0) above for K. 5. Conclusion Contrary to what was claimed in an earlier paper 3 by this author, this was the most direct derivation of the author s kinetic energy equation to date. For those who want further clarification of the fraction in acceleration distance equation (4) that replaces the fraction ½ in the classical version of the acceleration distance formula, please refer to Section 4 of the author s recent paper, The Ultimate Clarification of Relativistic Kinetic Energy 4. Also, as pointed out in the author s very first paper involving kinetic energy, and all of the succeeding papers on kinetic energy to date 5, the author s version of the kinetic energy formula, and subsequently his version of the total energy formula gives identical results to Einstein s formulas with one important exception. The author s kinetic energy equation gives accurate results even at extremely low speeds where Einstein s formula becomes erratic. Re: Appendix at end of paper. This does not mean that Einstein s formula is incorrect, only that it is not in the most idealistic form for computing results at that low Newtonian level of speed. Appendix Comparison of Rybczyk s Kinetic Energy Formula to Einstein s and Classical REFFERENCES Joseph A. Rybczyk, Time and Energy - The Relationship between Time, Acceleration, and Velocity and its Affect on Energy (200), Available from the top of the home page of the Millennium Relativity website at http://mrelativity.net 2 Albert Einstein, The energy equation is first mentioned in the paper, Does the inertia of a body depend upon its energy-content? (905), It is unclear when the equation was first given in the form E = mc 2 3 Joseph A. Rybczyk, Most Direct Derivation of Relativistic Kinetic Energy Formula (200), Available from the Millennium Briefs Section at the bottom of the home page of the Millennium Relativity website at http://mrelativity.net 4 Joseph A. Rybczyk, The Ultimate Clarification of Relativistic Kinetic Energy (206), Available from the bottom of the home page of the Millennium Relativity website at http://mrelativity.net 5 Joseph A. Rybczyk, Time and Energy, Inertia and Gravity (2002), Relativistic Motion Perspective (2003), The Relationship between E = Mc 2 and F = ma (2007), Newtonian Mechanics Solution to E = mc 2 (205), The Ultimate Clarification of Relativistic Kinetic Energy (206), All available from the home page of the Millennium Relativity website at http://mrelativity.net From the Briefs Section of the website, we have: Most Direct Derivation of Relativistic Kinetic Energy Formula (200), A Mathematical Discrepancy in Special Relativity (2009), Using Special Relativity s Gamma to Derive the Millennium Relativity Kinetic Energy Formula (2003) 5
Appendix Comparison of Rybczyk s Kinetic Energy Formula to Einstein s and Classical 6
Relativistic Kinetic Energy Simplified Copyright 207 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Millennium Relativity home page 7