GATE 2023 ME – Question 65
An explosion at time $t = 0$ releases energy $E$ at the origin in a space filled with a gas of density $\rho$. Subsequently, a hemispherical blast wave propagates radially outwards as shown in the figure.
Let $R$ denote the radius of the front of the hemispherical blast wave. The radius $R$ follows the relationship $R = k\,t^aE^b\rho^c$, where $k$ is a dimensionless constant. The value of exponent $a$ is ___________.
(Rounded off to one decimal place)
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Correct answer: 0.39 to 0.41
Explanation
Use dimensions: $[R] = L$, $[t] = T$, $[E] = ML^2T^{-2}$ and $[\rho] = ML^{-3}$. Equating the powers: mass, $b + c = 0$; length, $2b - 3c = 1$; time, $a - 2b = 0$. From the first, $c = -b$, so $2b + 3b = 1$ and $b = \frac{1}{5}$, $c = -\frac{1}{5}$. Then $a = 2b = \frac{2}{5} = 0.4$. So $R = k\left(\frac{Et^2}{\rho}\right)^{1/5}$.