I have been working on a ODE homework which involves modeling the velocity of a drop of water falling from the sky. The ODE that models its velocity is given by:
$$ mv'=kv^2-mg, \qquad k=\frac{1}{2}C_{d}\rho _{a}, $$ $C_{d}$: friction, $\rho_{a}$: air density, $A$: transversal section of the water drop.
I have had to find the theoretical velocity limit of the water drop as a function of $A$ by solving directly the ODE and compare these results with Euler and Runge-Kutta IV methods on MATLAB. I have done all of that.
The last question of my homework is an open question: It asks to modify the ODE presented by imagining now that the water drop losses a fraction of mass (by evaporation) while it is falling. I will have to apply Euler and Runge-Kutta IV on this new ODE. So I am looking for suggestions to improve the equation. Mass $m$ now is going to be a function of time, it has to decrease. I have been thinking to assume that the water drops are spheres and by using $density * volume=mass$.
Thank you very much in advanced!
I think you should just be able to write that the mass $m$ is a function of time and perhaps assume a linear one, namely
$$ m(t)=m_0-st,$$ where $m_0$ is an initial mass and $s$ is the rate of loss (positive) in units of $\frac{mass}{time}$. Then your equation is no longer separable (but who cares if you're going to solve in numerically) and would look like
$$ v'=\frac{k}{m(t)} v^{2} -g, $$ or
$$ v'=\frac{k}{m_0-st} v^{2} -g, $$
Hope this helps some. Is this what you're looking for?
Cheers,
Paul Safier