Question

A bomb explodes at 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 $t$ as:
A) $t$
B) ${t^{2/5}}$
C) ${t^{1/4}}$
D) ${t^{3/2}}$

Answer 1

The waves produced by the explosion are longitudinal in nature. Therefore consider a small spherical element of wave and then use the concept of velocity of a longitudinal wave in medium and compare it with the basic derivative definition of velocity i.e. $v = \dfrac{{dr}}{{dt}}$, where $r$ is the radius of the wave from centre of the explosion. On integration, it will give the relation of the radius with time.

Complete step by step solution:
Step1: Since due to the explosion the waves so formed is longitudinal in nature hence the speed of longitudinal waves in a medium is given by-
$v = \sqrt {\dfrac{{\gamma P}}{\rho }}$ ………………(1)
Where $\gamma$ is the ratio of two specific heats, $P$ is the pressure and $\rho$ is the density of the medium
Also from the Ideal gas equation, we have,
$PV = \gamma RT$
Where P = pressure, V= volume, $\gamma$= ratio of two specific heats, R= universal gas constant, T= temperature.
Therefore, $P = \dfrac{{\gamma RT}}{V}$
Put this in equation (1) we get,

$$v = \sqrt {\dfrac{\gamma }{\rho } \times \dfrac{{\gamma RT}}{V}} \\\ v = \gamma \sqrt {\dfrac{{RT}}{{V\rho }}} \\\$$

For sphere $V = \dfrac{4}{3}\pi {r^3}$
As $\rho$,$\gamma$ ,R and T all are constant let them together as another constant k , Therefore
$v = \gamma \sqrt {\dfrac{{RT}}{{\left( {\dfrac{4}{3}\pi {r^3}} \right)\rho }}}$
$\Rightarrow v = \dfrac{k}{{{r^{3/2}}}}$
Where,
$k = \sqrt {\dfrac{{3RT}}{{4\pi \rho }}}$
Step2: Also The velocity of propagation of this wave is given by –
$v = \dfrac{{dr}}{{dt}}$
Combining both equations we get, $\dfrac{{dr}}{{dt}} = \dfrac{k}{{{r^{3/2}}}}$
${r^{3/2}}dr = kdt$
Now integrating both sides we get,
$\int {{r^{3/2}}} dr = \int k dt$
$\dfrac{r^{5/2}}{5/2}=kt$
Neglecting the constants
$r \propto {t^{2/5}}$

$\therefore$ The correct option is (B).

Note:
As here we are finding a relation between two quantities so the constant of integration can be neglected. Also we need to remember the equation of velocity of a longitudinal waves in a medium i.e. given by $v = \sqrt {\dfrac{{\gamma P}}{\rho }}$