a $20-$kg mass mi hangs from a spring whose spring constant is k $=15$ kN$/$m$.$ A second mass m$\_2=10$ kg drops from a height h $=0\mathrm{.}1$ m and sticks to mass m$\_1.$ Subsequently the two vibrate up and down together. $(1)$ Determine an expression for the motion u$($t$)$ of the two masses after the moment of impact. Measure the displacement u$($t$)$ positive downward from the rest position of mass m$\_1.(2)$ Determine the natural frequency and the maximum displacement of the two$-$mass system.