How Does Friction Affect the Deceleration and Distance Traveled by a Motorboat?

[prehisto]

Homework Statement

When engine was turned off,boat with mass m was moving with speed v_0.
The force of friction F=-\alpha\nu-\betav^2.
How long it would take to drop speed of boat 3 times?
Find the distance which the boat will travel in this time?

Homework Equations

The Attempt at a Solution

I think i should try to solve differental equation in form of
m\dot{v_0}=-\alpha\nu-\betav^2

But I really don't know what to do next,could someone,please help me with some steps or tips?

 

[prehisto]

Ok,now I am pretty sure that i have to solve DE :
\int(\frac{dv}{ \alpha v+\beta v^2})=\int dt

And integration bondaries from v=v_0 to v_0/3 ?
And for other intergral offcourse its fromt=0 to t_0

Either way, how can I find the distance?

Please help .

 

[vela]

Hint: Use
$$a = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx}.$$ This is just the chain rule.

 

[haruspex]

Ok,now I am pretty sure that i have to solve DE :
\int(\frac{dv}{ \alpha v+\beta v^2})=\int dt

And integration bondaries from v=v_0 to v_0/3 ?
And for other integral of course its from t=0 to t_0

You dropped a minus sign (as well as a factor m). You could fix the sign by changing the v limits to being from v₀/3 to v₀.
I note that in the OP you gave the force as being $-\alpha \nu -\beta v^2$. I assume you meant $-\alpha v -\beta v^2$