$2.(25\%)$ A sand$-$water slurry is pumped continuously through a $0\mathrm{.}5$ m diameter opening at the bottom of a barge into a lake. The jet velocity is $2$ m$/$s and the waste material has a relative density of $1\mathrm{.}05.$ The discharge produce a line thermal of negatively buoyant material as the vessel is moving with a velocity of $0\mathrm{.}5$ m$/$s relative to the lake water. $($a$)$ Find the buoyancy force per unit length of the line thermal. $($b$)$ Assuming that the radius of thermal, R$,$ spreading at a rate of $\backslash$beta $=0\mathrm{.}34$ in proportional to the vertical velocity, w $=$ dz$/$dt$,$

dR$/$dt $=$beta dz$/$dt$,$ R $=$ beta depth, mass $=$ pi$*$beta$*$z$=$pi$*$r$^2$

find the velocity and radius of the cloud after the sand$-$water mixture has reached a depth of $20$ m below the water surface.