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?


Paul Safier


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.