A useful approximation for the \(x\) component of velocity in an incompressible laminar boundary layer is a parabolic variation from \(u=0\) at the surface \((y=0)\) to the freestream velocity, \(U\), at the edge of the boundary layer \((y=\delta)\). The equation for the profile is \(u / U=\) \(2(y / \delta)-(y / \delta)^{2}\), where \(\delta=c x^{1 / 2}\) and \(c\) is a constant. Show that the simplest expression for the \(y\) component of velocity is

\[\frac{v}{U}=\frac{\delta}{x}\left[\frac{1}{2}\left(\frac{y}{\delta}\right)^{2}-\frac{1}{3}\left(\frac{y}{\delta}\right)^{3}\right]\]

Plot \(v / U\) versus \(y / \delta\) to find the location of the maximum value of the ratio \(v / U\). Evaluate the ratio where \(\delta=5 \mathrm{~mm}\) and \(x=0.5 \mathrm{~m}\).