Friction problem from Halliday and Resnick

[Farnak]

Hi, new member here ^_^

"A 1000kg boat is traveling at 90km/h when its engine is shut off. The magnitude of the frictional force f_k between boat and water is proportional to the speed v of the boat: f_k = 70v, where v is in meters per second and f_k is in Newtons. Find the time required for the boat to slow to 45 km/h."

From what I'm getting, the only force on the boat is the force of kinetic friction and as time passes the velocity decreases, so the force of kinetic friction's magnitude will decrease as well. So this will decrease the magnitude of the acceleration and cause velocity to decrease more slowly as time passes by ...

I'm having trouble putting all that together not being used to dealing with non-constant acceleration problems so could someone please help me organize my thoughts for this problem? Thanks!

 

[nicksauce]

This can be summarized neatly in the differential equation,

$$F = -kv = ma = m\frac{dv}{dt}$$

where k is the constant of proportionality in the problem. Along with the initial condition v(0) = 90km/h, this equation can be solved to give you v(t). Would you know how to do that? Does this make sense to you?

 

[Farnak]

Are you supposed to solve the equation through integration? If so, I think I'm messing up:

-kv = m (dv/dt)
Integrating both sides
(-k/2)v^2 + C = mv

But an equation like this gives me a constant value for velocity =( ... where did I mess up?

 

[aniketp]

-kv = m (dv/dt)
Integrating both sides
(-k/2)v^2 + C = mv

-kv=mdv/dt
therefore b*dv/v=dt where b= -m/k
therefore b*log_e(v_f/v_i)=t
where v_f and v_i are the final n initial velocities...

 

[Farnak]

Ahhh ok, so I guess I need to learn differential equations if I want to read more, sorry for the calculus stupidity, thanks!