A balloon is made of a material of surface tension S and its inflation outlet (from where gas is filled in it) has small area A. It is filled with a…

A balloon is made of a material of surface tension $\displaystyle S$ and its inflation outlet (from where gas is filled in it) has small area $\displaystyle A$. It is filled with a gas of density $\displaystyle \rho$ and takes a spherical shape of radius $\displaystyle R$. When the gas is allowed to flow freely out of it, its radius $\displaystyle r$ changes from $\displaystyle R$ to $\displaystyle 0$ (zero) in time $\displaystyle T$. If the speed $\displaystyle v(r)$ of gas coming out of the balloon depends on $\displaystyle r$ as $\displaystyle r^a$ and $\displaystyle T \propto S^\alpha A^\beta \rho^\gamma R^\delta$ then