In the following mechanical system a velocity input, \(v(t)=6\) for \(t \geq 0\), is applied as illustrated.

(a) Write the system's differential equation in terms of the velocity \(v_{1}(t)\).

(b) What is the time constant \(\tau\) and the steady-state velocity \(v_{1_{s s}}\) for the system?

(c) Given the information: \(v_{1}(0)=10, b_{1}=b_{2}=b_{3}=1\) and \(m=1\), obtain the system response.

(d) Sketch the system response.

(e) Solve the same differential equation in (a) with the data in (c) by using Laplace transforms.

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k m -f(t) b XXXX A Translational Mechanical System