$(32$ points$)$ Following a spacewalk gone horribly wrong, Dr$.$ Potter and Dr$.$ Stephens are floating through space. They are initially $9$ m apart but are connected by a $12$ m tethering cable as shown. A short burst from Dr$.$ Potter's jet pack gives him an initial velocity $v_0=4\mathrm{.}5\frac{m}{s}$ as shown. If the tethering cable is perfectly elastic, what are their velocities after they rotate through $180$ around each other? Their masses are $90$ kg and $72$ kg respectively. $($Hint: treat the astronauts as point masses.$)$