Boat slowing down with velocity-dependent friction force

[rammer]

Homework Statement

A boat has mass m=1000 kg and is moving at velocity v0=324 ms^-1. Friction force btw the boat and water is proportional to velocity v, Fd=70*v. How long it takes to slow down to 162ms^-1 ?

Homework Equations

I'm not sure which function should I integrate, because acceleration is not constant.

The Attempt at a Solution

I understand Friction force and acceleration as functions of v, but I have no idea how to express these as functions of time, since acc is not constant. Then I would integrate a(t) with respect to time and substitute final velocity for v(t) and from that I'd get the answer.

 

[omoplata]

Draw a force diagram, and apply Newton's 2nd law. You can get the acceleration that way. Then use $a = \frac{dv}{dt}$ to get a differential equation. Solve it and find $v$ in terms of $t$. Solve for $t$ and substitute correct value of $v$ to find the time.

 

[rammer]

a=-70v/m
∫dv=∫(-70v/m)dt
v=-70/m ∫vdt - the problem is I do not know v in terms of t :(

 

[omoplata]

Instead of your second line, do this,
$$\int{\frac{dv}{v}}=\int{\frac{-70 dt}{m}}$$

 

[rammer]

I have to correct given informtion v0=25 ms^-1 and v=12.5 ms^-1

Ok, I tried your suggestion and from that I get:

t = (-m*(ln(v) + v0))/70 -what doesn't seem right

After substituting v = 12.5 I get t = 393 s and that is wrong (correct answer should be 9.9s)

 

[omoplata]

t = (-m*(ln(v) + v0))/70

This does not agree with my final answer.

Check whether you applied the initial boundary condition (v = v0 when t=0 ) correctly.

 

[rammer]

At t = 0, I'm pretty sure, the integration constant is equal vo (=25).

 

[omoplata]

Not ln(vo) ?

 

[rammer]

Yes, you're right, thanks. I finally got it correct. My mistake was I put the constant directly from initial conditions, not solving from integrated function.