a. Show that for any pump the maximum liquid discharge velocity is given by

\[u=\sqrt[3]{-\frac{2 g_{\mathrm{c}} W_{\mathrm{S}}}{ho A}}\]

where \(u\) is the maximum liquid discharge velocity, \(W_{\mathrm{s}}\) is the pump shaft work, \(ho\) is the density of the liquid, and \(A\) is the pump outlet discharge area. Make sure you list your assumptions in your solution.

b. A 1-kW pump discharges water through a \(50-\mathrm{mm}\) (internal diameter) pump outlet. What is the maximum velocity of the liquid from this pump? What is the maximum discharge rate (in \(\mathrm{kg} / \mathrm{s}\) )?