Show that the kinetic energy coefficient, \(\alpha\), for the "power law" turbulent velocity profile of Eq. 8.22 is given by Eq. 8.27. Plot \(\alpha\) as a function of \(R e_{\bar{V}}\), for \(R e_{\bar{V}}=1 \times 10^{4}\) to \(1 \times 10^{7}\). When analyzing pipe flow problems it is common practice to assume \(\alpha \approx 1\). Plot the error associated with this assumption as a function of \(R e_{\bar{V}}\), for \(R e_{\bar{V}}=1 \times 10^{4}\) to \(1 \times 10^{7}\).