= Frictional heating
{title2=$\dot E_{\mathrm{thermal}}=\mu_kN|w|$}
In the ideal <kinetic friction> model, the loss of combined <mechanical energy> of the two contacting bodies becomes thermal <internal energy> at rate
$$
\dot E_{\mathrm{thermal}}=-f_k(v-U)=\mu_kN|v-U|\ge0.
$$
This follows by adding the contact <powers> $f_kv$ and $-f_kU$ from <Newton's third law>. It depends on <contact slip>, whereas the contact <power> received by either body separately depends on its <velocity> in the chosen <inertial frame>. Thus positive <frictional heating> is compatible with increasing <kinetic energy> of one body, as the moving-belt example in <kinetic friction> shows. Heating need not cause a <temperature> rise if thermal <energy> escapes by <heat transfer>.
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