You are asked to review a space elevator. This elevator can be idealized as a chamber $($mass $m)$ supported from above by a linear spring $($stiffness $k)$ and a linear viscous dashpot $($damping constant $c)$ which are both attached to a set of rockets $($see itlustration below$).$ The rockets fly up vertically at a constant speed $v_o.$ Ignore the effect of gravity.

a$)$ Write the governing differential equation of motion for the elevator chamber.

b$)$ Solve the differential equation of motion to find the expression for the distance travelled by the elevator chamber, i$.$e$.z(t),$ as the rockets lift off vertically with a speed $v_0=50\frac{m}{sec}.$ Assume that the chamber has a mass $1000kg$ and is initially at rest. Use $k=4000\frac{N}{m}$ and $c=2000N*s\frac{ec}{m}.$