Resisted motion involving densities?

Question

A space craft in the shape of a cylinder has mass $N$ and the area of its cross section is $B$. It is moving at constant velocity but meets a dust cloud, with the dust sticking to the spaceship. If the dust has density $d$, how far does the vehicle travel before it has half the velocity of the original?

Answer 1

Drag force $F=\frac12dv^2CB$, where $C$ is drag constant and $v$ is the velocity.

Let the initial velocity $v_0$.

\begin{align} &F=-N\frac{dv}{dt}=\frac12dv^2CB\\ &\frac{v'}{v^2}=-\frac{dCB}{2N}\\ &v(t)=\frac{1}{C+\frac{dCB}{2N}t}\\ &v(0)=\frac1C=v_0\\ &\therefore v(t)=\frac{1}{\frac1{v_0}+\frac{dCB}{2N}t}\\ \end{align} Let the time of half velocity $t_1$. \begin{align} &v(t_1)=\frac{v_0}2=\frac{1}{\frac1{v_0}+\frac{dCB}{2N}t_1}\\ &t_1=\frac{2N}{dCBv_0}\\ \end{align} Total distance until $t_1$ is \begin{align} &\int_0^{t_1}v(t)dt=\int_0^{t_1}\frac{1}{\frac1{v_0}+\frac{dCB}{2N}t}dt=\left.\frac{2N}{dCB}\ln\left(\frac1{v_0}+\frac{dCB}{2N}t\right)\right]_0^{\frac{2N}{dCBv_0}}\\ &=\frac{2N}{dCB}\left(\ln \frac2{v_0}-\ln \frac1{v_0}\right)=\frac{2N}{dCB}\ln2 \end{align}