I'm a little stuck with a problem and I was hoping that you guys could help.
Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, the model is given by, $$ \frac{d^2 x}{dt^2}= -\frac{gR^2}{(x+R)^2} -\frac{k}{(x+R)}\frac{d x}{dt} \\ x(0)=0\\ x'(0)=v_0 $$ Where g is acceleration due to gravity, R is the radius of the earth, k is the coefficient of friction, and x is the height above the earth.
So, I first want introduce a small $\epsilon$ by scaling this problem (assuming $v_0$ is small), and second I want to find the $\epsilon$ series expansion as $\epsilon \rightarrow0$.
What I've done so far:
To scale the problem, let $t=t_c\tau$ and $x=x_cy$, where $t_c$ and $x_c$ are characteristic time and length. So we have, after a little rearranging $$ \frac{dx}{dt}= ...= \frac{x_c}{t_c}\frac{d y}{d\tau} \\ \frac{d^2 x}{dt^2}= \frac{x_c}{t_c^2}\frac{d^2 y}{d\tau^2} $$ and so $$ \frac{x_c}{t_c^2}\frac{d^2 y}{d\tau^2}=-\frac{gR^2}{(x_cy+R)^2} -\frac{k}{(x_cy+R)}\frac{d x}{dt}\\ \frac{x_c}{t_c^2}\frac{d^2 y}{d\tau^2}=-\frac{gR^2}{(x_cy+R)^2} -\frac{k}{(x_cy+R)}\frac{x_c}{t_c}\frac{d y}{d\tau} (*)\\ $$ Now, if this was a projectile question that ignored air resistance I would pick $x_c$ to be the max height it reached if it had a constant acceleration of -$g$ (and $t_c$ to be the time to reach $x_c$, givinf $x_c=\frac{v_0^2}{g}, t_c=\frac{v_0}{g}$). ie I am using the simplified projectile problem in this area and will eventually test it with the perturbation method (right?). So here is where I'm confused. If I get to assume $R^2\approx(x+R)^2$ then what happens with the $k$ term? If I did not have air resistance I would just use $x_c=\frac{v_0^2}{g}, t_c=\frac{v_0}{g}$. I'm not sure what the equivalent would be here.
Once I had that I would plug in $t_c$ and $x_c$ so that terms would disappear from (*) and get a simplified $\frac{dy^2}{d^2\tau}$, then chose an appropriate $\epsilon <<1$, use a taylor series expansion around $\epsilon$? So from here, if you show me how to do the correct rescaling (and why) and how to to the perturbation of it, it would be much appreciated. Thank you for any and all answers!