Crater formation math

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:

Code:

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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.

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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?

This is a math question as it is not the physics that confuses me but rather the mathematics.

Cheers
~Lance H

I'm thinking it's still a physics problem. If you are talking about vacuum impact or atmosphere entry speeds and the sorts of materials that impact and create craters, we see quite generally that the shape of the hurtling object is not very important.

Your ball bearings don't disintegrate on impact. It seems this would create an insufficient model for true atmosphere entry impact. The projectile remaining intact would rationally result in a bigger crater, maybe deeper and wider - with a projectile sitting in it. (I'm having trouble imagining it would bounce away.) Interestingly, a pointy projectile, that doesn't disintegrate, might produce a smaller crater with greater penetration. After all, "bunker buster" bombs do have different properties than regular bombs.

I wonder how different a perfect cube is when landing on 1) A flat surface, 2) An edge, or 3) A point.

No this is not the case as I am only investigating the kinetic energy aspect of the impacting object, thus any other factors if kept constant can be negated from this model. Please note it is only a model of a simplified version. Thus the problem does remain still. How can they state "This will be true as the scaling factor for any shape of crater. " for the above theory.

Chur