= Solution
In an adiabatic outflow without additional heating, the reservoir's <specific enthalpy> supplies both escaping <kinetic energy> and the work against <gravity>. For a monatomic <ideal gas>, $\gamma=5/3$ and $h=c_s^2/(\gamma-1)=3c_s^2/2$, where $c_s$ is the reservoir's <adiabatic sound speed>. The <Bernoulli equation>, measured from a reservoir initially at rest at radius $r$, gives
$$
\frac{v_\infty^2}{2}+h_\infty
=\frac{c_s^2}{\gamma-1}-\frac{GM}{r}.
$$
The maximum terminal speed uses all available <specific enthalpy>, so $h_\infty\to0$ and
$$
\boxed{v_{\infty,\max}=\sqrt{3c_s^2-\frac{2GM}{r}}.}
$$
A freely escaping adiabatic flow needs a nonnegative radicand; equality is marginal escape. If the radicand is negative, this reservoir cannot supply an unheated escaping outflow. The initial rest condition is a stagnation-reservoir approximation: an exactly stationary finite-density surface cannot simultaneously carry a nonzero steady <mass flow rate>. Further heating, radiation <forces> or an imposed confining <pressure> would change the terminal-energy budget.
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