# Classical Mechanics,motorboat

## Homework Statement

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

## The Attempt at a Solution

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

But I realy dont know what to do next,could someone,please help me with some steps or tips?

## The Attempt at a Solution

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Ok,now Im 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?

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

1 person
haruspex
Homework Helper
Gold Member
Ok,now Im 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 $-\alpha v -\beta v^2$

1 person