A boat moving because of the wind

[sapz]

Homework Statement

A boat has a sail that has a surface S towards the wind.
The wind blows at constant velocity u.

Also given the density of the air, p.

Find the velocity of the boat as a function of time.
What will be the velocity of the boat after a long enough time? What exactly is this "long enough" time?

Homework Equations

The Attempt at a Solution

I figured I could write the forces equation, since I know that the force that the wind asserts on an object is proportional to its surface and to the velocity^2.

k is some constant,
So F = k*S*(u-v)^2 = ma

But then I get a diff. equation with v, v dot, and v^2, which I don't know how to solve.
Is there an easier way to solve this?

 

[haruspex]

Please post the equation you get. If you get what I get, you should find that it is easily turned into a quite straightforward integral.

 

[SteamKing]

Is the force from the sail the only force acting on the boat? Think about what happens after the boat begins to move.

 

[sapz]

This is the equation I get:
k*S*(u-v)^2 = ma
k*S*(u^2-2uv+v^2)= ma

How does that become a nice integral?

 

[haruspex]

k*S*(u-v)^2 = ma
How does that become a nice integral?

u is constant, a = dv/dt:
k*S*dt = m(u-v)^-2dv

As Steamking points out, in the real world there is also a drag related to v. But typically that is taken to be a quadratic in v also, so the equation doesn't change much.