= Solution
Interpret $v$ as the normal upstream speed in the shock rest frame and $\rho$ as its upstream <mass density>. If the quoted jet speed is in another frame, the shock speed must first be supplied to obtain this relative <velocity>. Use a calorically perfect gas of mean particle <mass> $\bar m=\mu m_u$.
In the strong-shock limit, $\mathcal M_1\gg1$. The <Strong-shock Rankine-Hugoniot conditions> give
$$
\rho_2=\frac{\gamma+1}{\gamma-1}\rho,\qquad
u_2=\frac{\gamma-1}{\gamma+1}v,\qquad
p_2\simeq\frac{2}{\gamma+1}\rho v^2.
$$
The <pressure> expression also follows from the supplied ratio by inserting $\mathcal M_1^2=\rho v^2/(\gamma p_1)$. The <ideal gas law> then gives the immediate postshock <temperature>
$$
\boxed{T_2\simeq\frac{2(\gamma-1)}{(\gamma+1)^2}\frac{\mu m_u}{k_B}v^2.}
$$
The upstream thermal contribution is negligible at this order. This is a <temperature> in kelvin, rather than a <temperature> change per unit time.
To turn the postshock thermal energy into a rate, the processed mass per unit time is $\dot m=\rho vS$. Its downstream <specific internal energy> is
$$
\epsilon_2=\frac{k_BT_2}{(\gamma-1)\mu m_u}
=\frac{2v^2}{(\gamma+1)^2}.
$$
Thus the \b[thermal internal energy carried into the shocked gas per unit time] is
$$
\boxed{\dot E_{\rm int}=\dot m\epsilon_2
=\frac{2}{(\gamma+1)^2}\rho Sv^3.}
$$
Per unit area, divide this expression by $S$. Its dimensions are watts; the <mean molecular weight> cancels because the heat capacity per unit mass is inversely proportional to it, while $T_2$ is proportional to it.
There is a second common convention for shock heating power: the loss of directed bulk <kinetic energy> flux. The <energy fluxes of a strong adiabatic shock> give
$$
\boxed{\dot E_{\rm kin,loss}=\frac{\dot m}{2}(v^2-u_2^2)
=\frac{2\gamma}{(\gamma+1)^2}\rho Sv^3
=\dot m h_2.}
$$
This equals the downstream <enthalpy> flux, and includes <pressure> work as well as the advected internal energy. For $\gamma=5/3$, the compression ratio is $4$, $T_2=3\mu m_uv^2/(16k_B)$, $\dot E_{\rm int}=9\rho Sv^3/32$, and $\dot E_{\rm kin,loss}=15\rho Sv^3/32$. Specifying the energy convention avoids conflating these rates. A rate $dT/dt$ for a fixed slab cannot be inferred without its participating mass and thermal evolution; the supplied data fix the postshock <temperature> and the processed energy fluxes.
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