Final answer:
The student is tasked with deriving the expression for Z(s), assuming zero initial conditions, as part of an engineering problem related to the static equilibrium condition of a car's suspension system. Static equilibrium and mechanical energy conservation are significant factors in the derivation process.
Step-by-step explanation:
The question at hand pertains to the equilibrium conditions of a suspension system, which is modeled using differential equations in an engineering context. The student is asked to derive an expression for Z(s), which is a common technique in control systems engineering involving the use of Laplace transforms to analyze dynamic systems.
When considering the equilibrium of a system such as a car's suspension, the net external force acting on the system should be zero to satisfy static equilibrium. The relevant equations would typically involve forces and moments acting on the system, but in this case, the student is asked for a solution where the initial conditions, such as initial velocity and spring compression, are zero. This significantly simplifies the equations, allowing for an easier derivation of the expression for Z(s).
In this instance, the student may need to apply concepts of mechanical energy conservation to solve for variables such as the height h and spring compression x, which are part of the system's energy equation. The question hints at energy transitions from potential to kinetic energy, which may occur in a dynamic suspension system during operation.