Question #42041

A rigid balloon of constant volume $V_0$ and mass $m = \rho_0 V_0$ is released from rest at a height $y = H$ in a static gas column. The gas column rests on the ground ($y=0$) where its pressure is $P_0$ and density is $\rho_0$. The gas obeys a polytropic relation $P = K \rho^2$ everywhere, where $K$ is a positive constant. Assuming the balloon's volume is small enough that the gas density is locally uniform around it, and ignoring viscous drag, what is the speed $v$ of the balloon just before it hits the ground? (Assume $H < \frac{2 P_0}{\rho_0 g}$)