A small spherical bubble of radius a containing carbon dioxide rises in water due to a buoyancy force $\rho g V$, where $\rho$ is the density of water, $g$ is gravitational attraction and $V$ is the volume of the bubble. The drag on a bubble moving at speed $u$ is $6 \pi \mu a u$, where $\mu$ is the dynamic viscosity of water, and an accelerating bubble acts like a particle of mass $\alpha \rho V$, for some constant $\alpha$. Find the location at time $t$ of a bubble released from rest at $t=0$ and show the bubble approaches a steady rise speed

$$U=\frac{2}{9} \frac{\rho g}{\mu} a^{2}$$

Under some circumstances the carbon dioxide gradually dissolves in the water, which leads to the bubble radius varying as $a^{2}=a_{0}^{2}-\beta t$, where $a_{0}$ is the bubble radius at $t=0$ and $\beta$ is a constant. Under the assumption that the bubble rises at speed given by $(*)$, determine the height to which it rises before it disappears.