Excellent, TALewis!
I was inspired to take the analysis a step further..
A typical choice of frictional modeling is:
\vec{F}_{fr}=-k||\vec{v}||\vec{v}
Since this yields a coupled system of non-linear diff eqs., it must probably be solved by numerical means.
However,somewhat surprisingly, by assuming a slightly different frictional law, we can solve for the range exactly!
Let:
\vec{F}_{fr}=-k_{x}|v_{x}|v_{x}\vec{i}-k_{y}|v_{y}|v_{y}\vec{j}
That is, we use an uncoupled system in the component velocities.
SOLUTION OF VERTICAL EQUATION OF MOTION:
1. First phase: Going up
We have v_{y}\geq{0} so:
-g(1+\frac{k_{y}v_{y}^{2}}{mg})=\dot{v}_{y}
We make the substitution:
u=\sqrt{\frac{k_{y}}{mg}}v_{y}
Rearranging, integrating and utilizing the initial condition v_{y}(0)=v_{y,0}
we gain:
v_{y}(t)=\sqrt{\frac{mg}{k_{y}}}{tan(tan^{-1}(\sqrt{\frac{k_{y}}{mg}}v_{y,0})-\sqrt{\frac{k_{y}g}{m}}t)
The maximum height is therefore gained at t=t_{1}
t_{1}=\sqrt{\frac{m}{k_{y}g}}tan^{-1}(\sqrt{\frac{k_{y}}{mg}}v_{y,0})
By integration, and using the relation y(0)=0 we gain:
y(t)=\frac{m}{k_{y}}ln(\frac{\cos(tan^{-1}(\sqrt{\frac{k_{y}}{mg}}v_{y,0})-\sqrt{\frac{k_{y}g}{m}}t)}{\cos(tan^{-1}(\sqrt{\frac{k_{y}}{mg}}v_{y,0}))})
We may then find the maximum height:
y_{max}=\frac{m}{2k_{y}}ln(1+\frac{k_{y}}{mg}v_{y,0}^{2})
2. Phase 2: Going down
Clearly the vertical equation of motion may be written as:
-g(1-\frac{k_{y}}{mg}v_{y}^{2})=\dot{v}_{y}
In this case, we arrive at the following expression for the velocity (we have chosen t=0 to represent the time at y_{max}) :
v_{y}(t)=-\sqrt{\frac{mg}{k_{y}}}Tanh(\sqrt{\frac{k_{y}g}{m}}t)
This yields the expression for the height in the downward phase:
y(t)=\frac{m}{k_{y}}ln\gamma(t), \gamma(t)=\frac{\sqrt{1+\frac{k_{y}}{mg}v_{y,0}^{2}}}{Cosh(\sqrt{\frac{k_{y}g}{m}}t)}
Hence, the total time passed when the ground is reached again,T, is:
T=t_{1}+\sqrt{\frac{m}{k_{y}g}}Cosh^{-1}(\sqrt{1+\frac{k_{y}}{mg}v_{y,0}^{2}})
It's worthless, though, since the expression for the frictional force is probably dependent on the choice of coordinates..
