= Solution
Work in the rest frame of the <shock>. The normal <momentum flux> is conserved, so $p_1+\rho_1u_1^2=p_2+\rho_2u_2^2$. In the strong-shock limit $p_1\ll\rho_1u_1^2$, while <mass conservation> gives $\rho_2u_2=\rho_1u_1$. The given <Strong-shock Rankine-Hugoniot conditions> therefore yield
$$
p_2\simeq\rho_1u_1^2\left(1-\frac{u_2}{u_1}\right)
=\frac{2\rho_1u_1^2}{\gamma+1},\qquad
\rho_2=\rho_1\frac{\gamma+1}{\gamma-1}.
$$
Using the downstream <ideal gas law> $p_2/\rho_2=k_BT_2/m$ gives
$$
\boxed{\frac{k_BT_2}{m}=\frac{2(\gamma-1)}{(\gamma+1)^2}u_1^2.}
$$
For a monatomic <perfect gas>, $\gamma=5/3$ and the coefficient is $3/16$. This is the immediate post-shock <temperature>, before substantial <radiative cooling>; ionization, dissociation or unequal <electron> and ion <temperatures> require additional physical modeling.
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