= Solution
Take $T>0$ and a positive prescribed distance $L$; if $L=0$, the zero-speed journey has zero cost. Minimize $E[V]=\int_0^T(aV^2+b\dot V^2)\,dt$ subject to $V(0)=0$ and the <isoperimetric constraint> $\int_0^TV\,dt=L$. Introduce a constant <Lagrange multiplier> $\lambda$ and the integrand $aV^2+b\dot V^2-\lambda V$. For a variation $\eta$ with $\eta(0)=0$, <integration by parts> gives
$$
\delta E_\lambda=\int_0^T(2aV-\lambda-2b\ddot V)\eta\,dt+2b\dot V(T)\eta(T).
$$
The interior <Euler-Lagrange equation> is
$$
\boxed{b\ddot V-aV=-\frac\lambda2.}
$$
When terminal speed is free, the same calculation also imposes the <natural boundary conditions for a free endpoint>:
$$
\boxed{b\dot V(T)=0.}
$$
One cannot drop this boundary term when interpreting an extremal as a genuine minimum. If $b=0<a$, the interior condition makes $V$ constant; with a continuous zero initial speed and $L>0$ it cannot be attained. In that case <Cauchy-Schwarz inequality> gives the infimum $aL^2/T$, approached by speeds nearly constant except in a short initial layer. For $b>0$, the derivative cost controls such layers and the natural terminal condition matters in parts (b) and (c).
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