The Forces Affecting a Sailboat

Question

I've recently read an article about a practical example for " The Forces Affecting a Sailboat " .. I know the standard formula for the wind force which is : Wind force = Wind pressure * Sail area Wind pressure = 1/2 * Air density * Wind speed^2

The strange thing is that the wind force was calculated based on the following formula Wind force = 𝛥𝑃 *𝑀/ 𝜌𝑎𝑖𝑟 *𝑑

where 𝛥𝑃 is the change in pressure, 𝑀 is the combined mass of the sailboat and the sailors, 𝜌𝑎𝑖𝑟 is the density of air, and 𝑑 is the distance between the two pressures

I would like to know more about this formula , Where did it come from? this is the link for the article https://scholarcommons.usf.edu/ujmm/vol8/iss1/2

Answer 1

The formula comes from matching accelerations. Call $F$ the wind force, $M$ the mass of ship and sailors, and let $a$ be the acceleration of the boat. We get $$F=M a$$ and also the same formula for the air, but now per unit volume. $$\nabla p = \rho a$$ The accelerations are the same, and we can estimate $\nabla p$ by $\Delta p$ divided by the distance for the pressure drop $d$ (assuming linearity of the decrease of the pressure). Then equating the accelerations gives $$\frac{F}{M}=\frac{\Delta p}{d\cdot \rho}$$ and moving our mass over gives the formula that you quote: $$F=\frac{\Delta p\cdot M}{d\cdot\rho}$$.