I want to correct myself at one point. Contrary to what I had expected in my previous message, the approximation K=pv suggested by JD for the high-speed kinetic energy is more accurate than K=(p/v)c^2 suggested by me. But if we return instead to the general K=E-mc^2 and express it in terms of p, v as((p/v)-m)c^2, it will still be sufficiently simple and, in addition, it is mathematically exact for all |v|<c. For a free monochromatic photon, it will just reduce to K=pc=E.
Moses Fayngold,NJIT
On Sunday, September 27, 2015 4:16 PM, Moses Fayngold <moshfarlan@yahoo.com> wrote:
http://www.av8n.com/physics/thermo/energy.html . This is a good site. But I want to suggest two corrections.
1) In Sec. 1.3 there are 2 equations for kinetic energy K in terms of p and v: K=(1/2)pv at low speeds (1.2f) and K= pv at high speeds (1.2g) The exact expression for kinetic energy is K=E-mc^2 = ((p/v)-m)c^2 = (gamma(v)-1)mc^2. It reduces to (1.2f) at low speeds. But at high speeds it is approximated by K--->mc^2 gamma(v)=(p/v)c^2. It coincides with (1.2g) only at v=c (when p is infinite for any massive particle). But analytically, it is quite different from (1.2g) at all v other than c, and I expect it to be more accurate numerically.
2) In the beginning of Sec. 1.5 we read: "...suppose an electron meets a positron. The two of themannihilate each other, and a couple of gamma rays go flying off, with 511 keVof energy apiece. In this situation the number of electrons is not conserved (true!),the number of positrons is conserved (false!), and mass is not conserved (false!). However,energy is conserved (true!)." (Notations in italics are mine - MF) The first falsehood here is an obvious typo. But the second one may represent a widely-spread (and fashionable!) misconception confusing non-additivity of the rest mass with its non-conservation. The latter would contradict the very tenet of the discussed site: non-conservation of the rest mass would mean non-conservation of the rest energy of the system. The corollary of this argument: even though the rest mass of a single photon with definite momentum is zero, the rest mass of the two photons with non-parallel momenta is not. In the considered case, the pair of the produced photons has the center of mass, and its rest mass is equal to 2 X 511 keV /c^2. I will not go into farther details, since this topic has been extensively discussed on this Forum without any agreement. Moses Fayngold NJIT
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