Consider the case of free fall with a damping force proportional to the velocity, \(f_{D}= \pm k v\) with \(k=0.1 \mathrm{~kg} / \mathrm{s}\).

a. Using the correct sign, consider a \(50-\mathrm{kg}\) mass falling from rest at a height of \(100 \mathrm{~m}\). Find the velocity as a function of time. Does the mass reach terminal velocity?

b. Let the mass be thrown upward from the ground with an initial speed of \(50 \mathrm{~m} / \mathrm{s}\). Find the velocity as a function of time as it travels upward and then falls to the ground. How high does the mass get? What is its speed when it returns to the ground?