1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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

Yes, it comes from the forst equation, you cannot just erase it.

What is the change in momentum of an air molecule that hits the sail in terms of ρ?

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.

How could I translate this into an equation of motion like I wrote above?

Yes you can, but I do not remember how, I gonna search it and update the asnwer

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.