Root Density Law equation - maximum impact depth I'm reading the article "Depleted-Uranium Weapons:the Whys and Wherefores" by André Gsponer. In it he gives an equation and calls it "Root Density Law", for the maximum penetration length $L_{max}$ of a cylinder with density $\rho_1$:
$$L_{max} = \sqrt{\frac{\rho_1}{\rho2}}  L$$
where $L$ is the length of the cylinder and $\rho_2$ is the density of the impact ground.
He also claims:

A very simple result that can be derived as an exercise by students of a final-year high-schoolphysics class. 

I can't find this equation anywhere does anyone know any literature?
 A: Okay after some lucky digging I was able to find something in "Hazell, Paul J - Armour Materials, Theory, and Design-CRC Press (2015)", page 120.
First this equation only holds in extreme high impact velocities, where matter behaves as fluid. the regime is called Hydrodynamic Penetration. (For tugsten this is 3000 m/s).
A jet is flying with speed $v$ and has density $\rho_j$.  The target has density $\rho_t$. We change our cordinate system to be in the tip of the jet (also called stagnation point):

Fig: Jet impacting target hydrodynamically. From "Hazell, Paul J - Armour Materials, Theory, and Design-CRC Press (2015) figure 4.15
From this new coordinate system, the target is approaching the jet with some speed - let's donate that $u$.
The tail of the jet, with respect to the new coordinate system is then  $v-u$. 
We assume our material is incomprehensible. By Bernoulli's equation the pressure at the stagnation point should be equal so:
$$ \frac12 \rho_j (v-u)^2 = \frac12 \rho_t u^2 $$
We'll need $v-u $ later, so isolate that in the equation above. We get:
$$  (v-u) = \sqrt{\frac{\rho_t}{\rho_j}} u $$
Let's denote the length of the jet by $L$l The book claims the penetration speed is constant, so the time of impact is given by:
$$t = \frac{L}{v-u}$$
The penetration $L_{max}$ depth is given by:
$$ L_{max} = u t$$
These two equations can be written to:
$$L_{max} = \frac{L u}{v-u}$$
Now just insert the expression for $(v-u)$ we found earlier:
$$L_{max} = L \sqrt{\frac{\rho_j}{\rho_t}}$$
Which was what we wanted.
