= Solution
Write $R=CE^a\rho_0^bt^d$ for a uniform ambient <mass density>. Equating <mass>, length and time dimensions gives $a+b=0$, $2a-3b=1$ and $-2a+d=0$. Thus the <Sedov-Taylor blast wave> has
$$
\boxed{R=\xi\left(\frac{Et^2}{\rho_0}\right)^{1/5},\qquad V_s=\dot R=\frac{2R}{5t}}.
$$
The dimensionless coefficient depends on the gas's <adiabatic index> and on whether $R$ denotes the shock radius. Solving the similarity profile for a monatomic ideal gas gives $\xi\simeq1.15$, so taking it as unity is an order-of-magnitude approximation rather than an extra dimensional result.
Timed photographs of the 1945 atomic explosion provide a direct scale test: the measured fireball radius follows $R\propto t^{2/5}$ over the relevant blast-wave interval, and inserting the independently established explosion <energy> and atmospheric density gives a coefficient of order unity. The published photographic analysis tested the radius-time law and used its normalization to infer an explosion <energy> of the correct nuclear-test scale. This illustrates both the success and the limits of dimensional reasoning: the power law is fixed dimensionally, whereas the numerical coefficient requires a profile calculation or measured normalization.https://doi.org/10.1098/rspa.1950.0050[Part II, photographic radius-time measurements and explosion-energy estimate; the 20–185 m interval tests the predicted similarity scaling]{ref}
Neglecting the exterior <temperature> requires a strong shock, $\mathcal M_s=V_s/c_{s,u}\gg1$, or $\rho_0V_s^2\gg p_u$. The swept-up <mass> must dominate the initial ejecta for this energy-conserving similarity stage, while radiative losses, gravity, ambient gradients and continuing <energy> input must be unimportant. A very hot exterior or a late weak disturbance introduces a sound-speed scale and invalidates the cold-upstream similarity assumptions.
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