# Ordinary differential equations - boat example

1. Sep 21, 2010

### Anabelle37

URGENT! ordinary differential equations - boat example

1. The problem statement, all variables and given/known data

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).

Thanks

2. Relevant equations

force balance equations

3. The attempt at a solution

I've gotten my ODE to be: m*dv/dt = -kv^n with the initial condition v0 at t = 0.
It says find v as a function of the distance travelled.
So i went on to say dv/dt= (dv/dx)*(dx/dt)= v*(dv/dx) so now x is the independent variable not t.

so, m*v*(dv/dx) = -kv^n
v*(dv/dx)= -(k/m)*v^n
v*(dv/dx)= -av^n where a =k/m
v dv = -av^n dx
integral (v/(v^n)) dv = integral (-a) dx
integral (v^(1-n)) dv= -ax + c
(v^(2-n))/(2-n) = -ax + c (don't know if the integral on the LHS is correct?)

but x=o at t=0 so v(t=0)=v(x=0)=v0

so c= (v0^(2-n))/(2-n)
therefore,
(v^(2-n))/(2-n) = -ax + (v0^(2-n))/(2-n)
v^(2-n) = -(2-n)ax + v0^(2-n)
v^(2-n) = (n-2)ax + v0^(2-n)
therefore, v= [(n-2)ax + v0^(2-n)]^(1/(2-n))

but a =k/m
so v(x) = [(n-2)kx/m + v0^(2-n)]^(1/(2-n))

does this seem right at all because it seems a little messy?? Also it says to note that you need to consider the two cases. What are the two cases as I've only considered one???

2. Sep 21, 2010

### lanedance

Re: URGENT! ordinary differential equations - boat example

haven't been through the working, but few standrad checks are
- see if it behaves as you expect, ie monotonically decreasing
- substitute back into the equation and check it satsifies the orginal DE

for the 2 cases have a look at the integration step, and consider n = 2