What are the extra terms in the k.e expression

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Discussion Overview

The discussion revolves around the expression for kinetic energy, specifically focusing on the first six terms in the context of classical and relativistic physics. Participants explore the distinctions between classical and relativistic kinetic energy, as well as the significance of higher-order terms in the expansion of kinetic energy.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Ian requests clarification on the first six terms in the classical expression for kinetic energy.
  • Some participants question the classification of the terms as "classical," suggesting they are more accurately described as "relativistic."
  • Njorl provides a derivation starting from the relativistic total energy and discusses the binomial expansion, identifying the first term as rest energy and the second as the classical kinetic energy term (1/2)mv².
  • There is a suggestion that the higher-order terms could be referred to as corrections to the classical formula, though no standard terminology exists for them.
  • One participant notes that the higher-order terms do not correspond to any named quantity in classical physics and suggests that their significance is limited.
  • Another participant humorously remarks that these terms are generally ignored in practice.

Areas of Agreement / Disagreement

Participants express differing views on the classification of the kinetic energy terms, with some arguing for a relativistic interpretation while others maintain a classical perspective. The discussion does not reach a consensus on the terminology or significance of the higher-order terms.

Contextual Notes

There is an acknowledgment of the ambiguity in terminology regarding kinetic energy expressions, and the discussion highlights the lack of standard naming for the higher-order terms in the expansion. The relevance of these terms appears to be context-dependent, particularly in relation to the limits of classical mechanics.

Ian
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Can someone please tell me what the first six terms are in the classical expression for kinetic energy.
Thanks,
Ian.
 
Mathematics news on Phys.org
1) The kinetic energy of what?
2) Why did you post this in the general math forum?
 
Start with the relativistic total energy (rest and kinetic):

E2=(mc2)2+p2c2

p=mv

substitute and simplify to E=mc2(1+v2/c2)1/2

then use the binomial theorem. The first term of the binomial expansion will be the rest energy, the next term will be the commonly used (1/2)mv2. The second through seventh terms are the ones you want.

Njorl
 
Maybe it is a matter of semantics, but I wouldn't call that the "classical expression for kinetic energy", rather the "relativistic expression for kinetic energy". I guess that it's just a matter of taste...
 
Originally posted by suyver
Maybe it is a matter of semantics, but I wouldn't call that the "classical expression for kinetic energy", rather the "relativistic expression for kinetic energy". I guess that it's just a matter of taste...

I agree. I don't think there is a standard terminology for the next few terms of the expansion. I suppose they could be called the first through fifth order corrections to the classical formula for kinetic energy.

Njorl
 
Originally posted by Ian
Can someone please tell me what the first six terms are in the classical expression for kinetic energy.
Thanks,
Ian.

They do not correspond to any named quantity in classical physics.

This needs not be surprising. You can write any function as a linear combination of other functions in many ways (much in the way you can use different coordinate systems for describing vectors).

It turns out that, when you expand the K.E. in powers of v/c, you gain some understanding of why things look like classical mechanics when v/c<<1, but the (infinitely many) higher-order terms do not have a wide enough use to get them a name of their own; actually, had relativity not prompted this peculiar expansion of KE in terms of powers of (v/c), those terms would probably never had shown up elsewhere.
 
I guess for the most part I would simply call them, ignored!
 

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