Crater Depth and Height Relationship

[IB tired]

Homework Statement

Hello! I am an honors physics student and i have been assigned a project that has puzzled me. I have a container full of sand, a couple meter sticks, a tennis ball, and a stopwatch. What I need to do is prove something through the use of these items. What I am going to try to do is prove that an object dropped from a taller height will create a bigger impact crater. This is a very elementary concept, I know. I just don't know how to mathematically relate the crater depth to the increase in energy. I see that with the height increase, the total energy increases. But how do I directly relate the crater depth to the energy increase? I can find the velocity before the impact, how do I factor in the velocity of the impact?

Homework Equations

Ke=1/2MV²
Ep=MGH
Vf²=Vi²+2AΔX

The Attempt at a Solution

I am planning on conducting 25 drops: 5 from .5 meters, 5 from 1 meter, 5 from 1.5 meters, 5 from 2 meters, and 5 from 2.5 meters. With each height increase, I am expecting an increase in crater depth. I am just confused on how I am going to prove mathematically that the height increases crater size. I know that the combined energy before the impact will increase because Kenetic energy=1/2Mass * velocity² (Ke=1/2MV² ) and potential energy=Mass * Gravity * Height (Ep=MGH). The total energy would be the kenetic plus potential energy. As the height increases, the energy increases. Should there be a constant increase between the crater depth with each drop?

 

[kuruman]

I just don't know how to mathematically relate the crater depth to the increase in energy.

Since you are doing a physics project, you need to make some measurements first, then try to make sense of what you got. If you measure the crater depth after each drop you can prove (or disprove) whether the crater is deeper when the ball is released from a greater height. That's the easy part. More interesting is finding out the relationship between the independent variable (height of release) and the dependent variable (crater depth). For this, you need to do some plotting and thinking. The mathematical relation, if there is one, should emerge from the plot.