A bomb explodes at time $t=0$ in a uniform, isotropic medium of density $\rho$ and releases energy $E$, generating a spherical blast wave. The radius $R$ of this blast wave varies with time $t$ as
KVPY 2017, Advanced
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$(b)$ Velocity of propagation of blast wave is $\quad v=\sqrt{\frac{\gamma p}{\rho}} \quad \dots(i)$
$\gamma=\frac{C_{p}}{C_{V}}$, where $p=$ pressure
and $\rho=$ density.
As, $\quad p=\frac{n R T}{V}=\frac{n R T}{\frac{4}{3} \pi r^{3}}$
and $\quad v=\frac{d r}{d t}$
So, by Eq. $(i)$, we have
$\frac{d r}{d t}=\sqrt{\frac{\gamma n R T}{\frac{4}{3} \pi r^{3} \cdot \rho}}$
$\Rightarrow \quad \frac{d r}{d t} =k \cdot r^{-3 / 2}$
$\Rightarrow \quad k d t=r^{3 / 2} d r$
Integrating above equation, we get
$\Rightarrow k \int d t =\int r^{3 / 2} d r$
$\Rightarrow k t =\frac{2}{5} r^{5 / 2}$
So, $\quad r \propto t^{\frac{2}{5}}$
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