Crater Investigation Theory

[Lance H]

Currently in my investigation I am trying to find the relationship between crater diameter and the kinetic energy of the ball bearing I’m dropping (changing the kinetic energy by dropping it from a greater height). I found a theory online that explains the relationship that should be seen but I am confusing about one part of it.

Anyway here is the theory behind it. I will underline and bold the part that confused me:

Theory

Here is an approach you may find helpful:- the formation of a crater is akin to digging a hole.
To start with let us consider the minimum potential energy change that occurs when a cubic hole, side-length s, is created in sand.
This will be the same as lifting a similar-sized amount of material onto nearby ground.

D = length of one side of the cube
d = density of the impact material
m = mass of sand 

Volume of hole = D^3 where D is the length of one side of the cube
Mass of material moved from hole
Mass = volume x density = D^3 * d

Weight of this material = mass * Acceleration due to gravity
Weight = g * D^3 * d

Potential energy gained = weight x height lifted; as the height lifted is equal to the length of one side of the cube so this is height through which the mass must be lifted:
E = g * D^3 * d * D
E = g * D^4 * d
As the density and the acceleration due to gravity are constants this can be re written as
E = kD^4 where k is a constant.

This will be true as the scaling factor for any shape of crater.

The last statement, about the scaling for any shaped crater, is the one that has me. How can this theory be true for any shape with any scaling factor? As most craters are spherical I don’t know how this above theory can be applied to that?

Cheers

~Lance H

[sweetnsimple786]

By "This" does it mean "E = kD^4 where k is a constant"?

I don't know if I'm understanding this correctly, but maybe you can apply it to other types of solids by following the process? Like for a sphere it's something like E=kr⁴?

[Lance H]

Yeah you are correct in your first assumption.

Secondly, unfortunately I cannot get it to prove to work out like this. This is because diameter of the crater and depth would need to be known in order to calculate the volume of the spherical cap (term for the volume of a partial sphere).

[sweetnsimple786]

So you'd encounter the same problem with the cube right? Is the equation assuming that the entire object is submerged in the ground, that it displaces its volume worth of earth?

I'm sorry. =/