# A sailboat and wind, to find $v(t)$

Consider a sailboat of mass $m$ that lies on still water. A wind blows the sailboat with velocity $v_w$ perpendicular to the sailboat. If the boat has a sail with area $A$, and the air density is $\rho$, find the velocity of the boat as a function of time, assuming that the sail is tight. Neglect friction with the water.

Well, I understand that the wind exerts dynamic pressure on the sail: $$p=0.5 \rho v^2$$ so that the force exerted on the sail is $$F=pS=0.5p\rho v^2$$ With $F=ma$ we get the equation of motion: $$S\rho /2(v_w-v(t))^2=m \frac{dv}{dt}$$ separating variables and integrating from $0$ to $t$ and from $v_0=0$ to $v(t)$ gives the desired $v(t)$.

Now I am asking 2 things:

• Is there a really factor $1/2$ here that I should take in calculation?

• What is the change in momentum of an air molecule that hits the sail in terms of $\rho$? How could I translate this into an equation of motion like I wrote above?

the change in momentum does not depend on $\rho$, because $\rho$ only retaltes the averge number of molecules in a given volume, and you are interested in only a single collision. Rather inmportant is the temperature, that measures the average kinetic energy per molecule of air.