From: henry@spsystems.net   
      
   In article ,   
   Makhno  wrote:   
   >Is this a valid step here? The increase in kinetic energy when an   
   >infinitesimal mass dm is accelerated out of the engine is clearly   
   >dE=0.5*dm*c^2. This takes place over a time period T, which is an unknown   
   >constant.   
      
   No, it takes place over an infinitesimal time dt.  So just divide both   
   sides by dt, and you get dE/dt = 0.5*(dm/dt)*c^2 as suggested.   
      
   Any mathematicians in the audience are probably having apoplexy at the   
   "just divide by dt" part, and will tell you at great length that something   
   like dm/dt is not really a quotient and you can't just casually treat dt   
   as if it were an ordinary variable.   
      
   To which I say:  if it was good enough for Leibniz, it's good enough for   
   me. :-) The dirty little secret of calculus is that just casually treating   
   dt like an ordinary variable gets you the right answer, every time... and   
   since the development of "nonstandard analysis", we know why.  You can   
   build a mathematically-rigorous non-standard number system in which dm/dt   
   *is* a quotient and dt *is* an ordinary variable, and it's provably   
   equivalent to the vastly more complicated epsilon-delta circumlocutions   
   that later mathematicians invented to explain why Leibniz's calculus   
   perversely insisted on working so well.  The engineers' deplorable habit   
   of just dividing dm by dt and getting dm/dt turns out to be rigorously   
   justifiable after all.   
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