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.

 

[kerimek]

Hello! It's great to see that you are taking on this project and trying to prove the relationship between crater depth and height. It is indeed an elementary concept, but it is important to have a solid understanding of the underlying principles.

To mathematically relate the crater depth to the increase in energy, you can use the conservation of energy principle. This principle states that energy cannot be created or destroyed, only transferred from one form to another. In the case of an object dropped from a certain height, the initial energy is in the form of potential energy (Ep=MGH) and as it falls, it is converted into kinetic energy (Ke=1/2MV2).

So, to relate the crater depth to the energy increase, you can use the equation for kinetic energy (Ke=1/2MV2). As the object falls from a taller height, it will have a higher velocity at the point of impact, resulting in a greater amount of kinetic energy. This increased kinetic energy will then be transferred to the sand upon impact, creating a deeper crater.

To factor in the velocity of impact, you can use the equation Vf2=Vi2+2AΔX, where Vf is the final velocity, Vi is the initial velocity, A is the acceleration (in this case, due to gravity), and ΔX is the distance traveled. Using this equation, you can calculate the velocity of impact for each height and use that to determine the corresponding change in crater depth.

As for your question about a constant increase in crater depth with each drop, it is important to note that there are other factors that can affect the size of the crater, such as the density and composition of the sand, the angle of impact, and air resistance. So, while you may see a general trend of increasing crater depth with increasing height, it may not be a perfectly linear relationship.

I hope this helps guide you in your project. Remember to always double check your calculations and consider all factors that may affect your results. Good luck!