Low velocity impact craters in sand

[Liverpool8]

Hi, i am working on a high school project on how various factors (mass, height and sand particle size) affect crater diameter.

I am struggling to find a solid hypothesis/ theory on this question. I have found out that the cube of the diameter is proportional to the KE of the ball, but i am struggling to find justification of this with physical theory.

For example why is the energy of the ball proportional to the volume of the material excavated on impact?

 

[mgb_phys]

Think where the energy of the ball goes.
It must lift the material it hits.
The mass of material lifted is proportional to what?