= Solution
For fixed finite velocities, the <nonrelativistic limit> $c\to\infty$ gives $\gamma^2=(1-v^2/c^2)^{-1}\to1$. Hence the <relativistic rocket equation> reduces to
$$
\boxed{mv'+um'=0},
$$
which is the nonrelativistic <rocket equation>.
The conserved quantities called energy in the two calculations include different contributions. The nonrelativistic calculation considered only <kinetic energy>, which increases as fuel releases <internal energy>; total energy including that internal reservoir is still conserved. Relativistic energy includes <rest energy>, and therefore includes the fuel's stored internal energy through the rocket's <rest mass>. That rest mass decreases as energy is transferred into the exhaust. Thus <conservation of relativistic energy> and the nonconservation of kinetic energy express the same physical conversion.
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