# Motion with a changing acceleration

1. Dec 30, 2008

### fizics

1. The problem statement, all variables and given/known data
A boat, whose mass is m, is traveling at a speed of v_i when its engine is shut off. The magnitude of the friction force f between boat and water is proportional to the speed v of the boat: f=kv. Find the time t for the boat to slow down to speed v_f

2. Relevant equations

3. The attempt at a solution
I think there should be some transformations with the help of calculus, but I really don't have any ideas.

2. Dec 30, 2008

### Staff: Mentor

Start by writing the general equation of motion relating F, m and a. Now, you have an additional equation above due to the varying resistance force as a function of v. These two equations are related.

Now look at the general kinematic equations of motion in a constant acceleration with the usual forms of v= and x= .... Those are simplifications where integrations had simple answers because the acceleration was constant. Re-write the general forms of those equations, and look at what changes when the acceleration is not constant.

That should get you started. Show us your calculations as you work through this....

3. Dec 30, 2008

### fizics

f= ma= kv, so a= (kv)/m. Thus,
v_f - v_i = ∫a dt (from 0 to t) = (k/m)* ∫v dt (from 0 to t) = ?

4. Dec 30, 2008

### glueball8

Hi,

$$F=kv=ma$$

$$\frac{dv}{dt}=\frac{Kv}{m}$$

$$dt=\frac{m}{kv} dv$$

$$t=\int_{vi}^{vf} \frac{m}{kv} \ \ dv$$

since K and m are constants therefore

$$t=\frac{m}{k} \int_{vi}^{vf} \frac{1}{v} \ \ dv$$

then you have to integrate it...

right?

5. Dec 30, 2008

### fizics

I don't have the answer but I think it's right. How do you come up with this? Is it because in $$F=kv=ma$$, there's v, and you want to have a dv in the expression?

6. Dec 30, 2008

### glueball8

hmm you said that... "f= ma= kv"

basicly you use the fact
$$a=\frac{dv}{dt}$$

can you integrate it and get the equation?

7. Dec 30, 2008

### fizics

t= (m/k)*ln(v_f/v_i)