# Ordinary differential equations - boat example

1. Sep 17, 2010

### Anabelle37

Question:
A boat of mass m is travelling with the velocity v0. At t=0 the power is shut off. Assuming water resistance proportional to v^n, where n is a constant and v is the instantaneous velocity, find v as a function of the distance travelled. (Note that you need to consider the two cases).

I'm having trouble setting the problem up. Do I need to have a force balance equation? What would it be as the one I originally thought of doesn't make much sense: mdv/dt=v0 - kv^n.

Thanks

2. Sep 18, 2010

### Staff: Mentor

After the power is shut off, the only force on the boat is the resistance, so your equation is m*dv/dt = -kvn. v0 is the velocity at time t = 0: it's the initial condition.

3. Sep 18, 2010

### Anabelle37

Ok, thanks heaps.
How do I solve for v(x) not v(t)??

I've started:
dv/dt= -(k/m)v^n
dv/dt= -av^n where a =k/m
integral (1/v^n) dv = intgeral (-a) dt
(v^(-n+1))/-n+1= -at + c
???

4. Sep 18, 2010

### mugaliens

Or you could just attach a lightweight floating plumb line, such as polypropylene, paying it out for about three minutes until it comes to rest, and backwards solve...

Sorry for the intrustion, folks. I've been out of academia and in the real world way too long. I'll let you be.

I would ask you, though, if you'd like some ideas on how to model situations so as to obtain data required for parameterized equations required for backsolving solutions, I'd be happy to work with you. I'm sort of interested in getting back into anything having to do with Dffy-Q's - it was the only higher math class in college I in which I earned a solid A.

As for the physical situations upon which they're based, I've experienced them in droves, and on many different scales, from micro-robotics to aerial transport.

5. Sep 18, 2010

### Staff: Mentor

Why do you think you want v(x)? Velocity is usually in terms of time, not distance.
So v^(-n + 1) = (n - 1)(at - c'), where c' = -c

To solve for v, raise each side to the power 1/(1 - n).

6. Sep 19, 2010

### Anabelle37

Ok thanks heaps.
"Why do you think you want v(x)? Velocity is usually in terms of time, not distance."[/I-mark
Because the question says find v as a function of the distance travelled?

7. Sep 19, 2010

### Staff: Mentor

Sorry, I overlooked that piece of information.

I think this would work - You have v as a function of t. If you integrate v (with respect to t) you get distance as a function of t. From v = v(t) and s = s(t), you should be able to solve algebraically for v as a function of distance.