Chapter 14, Problem 14.83AP

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.48. The hydrofoil has a shape like that of an airplane wing. Its area projected onto a horizontal surface is A. When the boat is towed at sufficiently high speed, water of density ρ moves in streamline flow so that its average speed at the top of the hydrofoil is n times larger than its speed v_b below the hydrofoil. (a) Ignoring the buoyant force, show that the upward lift force exerted by the water on the hydrofoil has a magnitude

$F=\frac{1}{2}(n^2-1)\rho v_b{}^{2}A$

(b) The boat has mass M. Show that the liftoff speed is given by

$v=\sqrt{\frac{2Mg}{(n^2-1)A\rho }}$

Figure P14.4 8