The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in the figure below. 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 vb below the hydrofoil.
(a) Ignoring the buoyant force, show that the upward force exerted by the water on the hydrofoil has a magnitude given by the following equation.
F ≈ ½ (n2 − 1) ρ vb2 A

(b) The boat has mass M. Show that the liftoff speed is given by the following equation. (Do this on paper
v ≈

2Mg
(n2 − 1)Aρ

Question

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in the figure below. 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 vb below the hydrofoil.
(a) Ignoring the buoyant force, show that the upward force exerted by the water on the hydrofoil has a magnitude given by the following equation.
F ≈ ½ (n2 − 1) ρ vb2 A

(b) The boat has mass M. Show that the liftoff speed is given by the following equation. (Do this on paper
v ≈

2Mg
(n2 − 1)Aρ