# ODE water drop modeling question

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