# Sailing Against The Wind

**Calculus**Level 3

Because the keel keeps the ship from moving sideways, only forces along the boat's axis will actually push the boat. By orienting the sail correctly, the projection of the wind's force onto the sail—which is then projected onto the axis of the ship—can create a movement which is partially *in the direction opposite the wind*.

In the diagram above, \( \mathbf{W}\) is the direction of the wind, \( l_s \) is perpendicular to the sail and \( l_a \) is along the ship's axis ( with \( 0 < \alpha < \pi/2, \) and \( 0 < \beta < \pi/2 \) ).

Assuming \( \alpha = \beta = \pi/3 \), what percent of the wind's force could the ship apply in the direction opposite to \( \mathbf{W} \).