Hello I would like to see how this differential equation solves to give the result on the picture.
$c$ is a constant and I believe $\frac{dm}{dt}=-k$ Obviously they are dividing by $m$ and then integrating w.r.t. $t$ yet I can't get the result out myself. Any help.
Here is the pdf by the way it is an MIT document on the rocket equation for reference: http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec14.pdf
Thanks!
$$\ m\frac{dv}{dt}=-mg-c\frac{dm}{dt}$$ Multiply both members by $\ dt $ and divide by $\ m$: $$\ dv=-gdt-c\frac{dm}{m}$$ Now integrate to get the result $$\ \int_{v_0}^v dv=-g\int_0^t dt-c\int_{m_0}^m \frac{dm}{m}$$ $$\ v-v_0=-g(t-0)-c\ln(\frac{m}{m_0})$$ $$\ v=v_0-gt-c\ln(\frac{m}{m_0})$$ Where I supposed that $\ t_0=0$