Low velocity impact craters in sand

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The discussion focuses on how mass, height, and sand particle size influence the diameter of impact craters. The participant seeks a solid hypothesis linking kinetic energy to crater size, noting that the cube of the diameter relates to the ball's kinetic energy. They question the physical justification for why the energy expended during impact correlates with the volume of material excavated. Additionally, they ponder the relationship between the mass of the lifted material and the factors affecting crater formation. Understanding these dynamics is crucial for the high school project on low velocity impact craters in sand.
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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?
 
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Think where the energy of the ball goes.
It must lift the material it hits.
The mass of material lifted is proportional to what?
 

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