= Solution
For the stipulated <Strong-shock Rankine-Hugoniot conditions>, $\rho_d=4\rho_u$ and $v_d=v_u/4$. The <ideal gas law> is $P_d=\rho_d k_BT_d/\mu$, with $\mu$ a dimensional <mean particle mass>. The <specific enthalpy> of a nonrelativistic <monatomic gas> is $5k_BT/(2\mu)$. Dividing the <energy flux> equation in <1/d> by the common <mass flux> gives
$$
\frac{v_u^2}{2}=\frac{v_u^2}{32}+\frac{5k_BT_d}{2\mu}.
$$
Therefore
$$
\frac{5k_BT_d}{2\mu}=\frac{15v_u^2}{32},\qquad
\boxed{k_BT_d=\frac{3}{16}\mu v_u^2.}
$$
As an independent check, <conservation of momentum> gives $P_d=\rho_uv_u^2-4\rho_u(v_u/4)^2=3\rho_uv_u^2/4$, and division by $\rho_d=4\rho_u$ gives exactly the same <temperature>. The numerical coefficient relies on $\gamma=5/3$ and the absence of dynamically important radiative or magnetic stresses.
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