Cratering (Part 1)

Crater formation is a complex process, and it isn't obvious that one would be able to learn much about it from a small tabletop experiment. However, by making a few assumptions about what happens to the kinetic energy of an impactor after it strikes a surface and by applying dimensional analysis, we can come up with a simple model for how crater diameter scales with the kinetic energy of the impactor.

In this lab, you will use small impactors (steel ball bearings) on sand to explore this model and to see if your model can be applied to estimate the kinetic energy responsible for creating some of the largest craters discovered on Earth.

Goals


The primary goals of this experiment are as follows:

  • to learn how to evaluate experimental results;
  • to learn how to answer the question “How many data points do I need?” by assessing data in real time;
  • to assess both statistical and systematic uncertainties in your experiment; and
  • to gain experience using Python in the Juypter notebook environment to perform calculations, plot data, and complete least-squares fits.

Modeling crater size as a function of kinetic energy


Craters are abundant throughout the solar system. Earth's moon and the surface of Mercury are both heavily cratered. On Earth, erosion effects tend to erase craters over geological time scales. Nevertheless, there exist numerous relatively young craters on Earth. The Chicxulub crater just off the Yucatan peninsula is one of the largest impact craters on Earth, and its creation is thought to be the cause of the mass extinction which wiped out the dinosaurs. Parts of the Nevada Test Site are covered in craters from nuclear weapons tests conducted mostly in the 1950s.

In a nutshell, craters are formed when the kinetic energy of the incoming object, $K=\frac{1}{2}mv^2$, where $m$ is the impactor's mass and $v$ is its velocity, is converted into some other form(s) of energy as the object comes to rest. The ways an impactor loses this kinetic energy include deformation (i.e. pushing the surface down and out of the way), ejection (i.e. pushing material up and out of the crater), heating (i.e. raising the temperature of the surface material or impactor), comminution (i.e. crushing the surface material into smaller bits), or generating seismic waves (i.e. turning the kinetic energy of the impactor into propagating wave energy of the surface material).

In certain cases, only one of these processes may dominate and it becomes easier to think about how a crater is formed. In such cases we can use a technique called dimensional analysis to create a model for how crater size depends on the impactor's kinetic energy.

For this experiment, we will consider two such models.

Why not just use kinematics?

In theory, we could use what we know about Newtonian physics to predict what would happen.  In fact, if we were to examine any individual grain of sand, we could use kinematic relationships to predict where it would travel after the impact.  

In practice, knowing the location and physical properties of millions of grains of sand is not feasible.  And in the event that it were possible, the resulting equations would almost certainly not have any analytical solution (i.e. some equation that would predict the exact outcome for any starting configuration).

To side-step this problem, we can instead predict bulk properties of a larger system (i.e. the size of the sand crater) through other sub-disciplines of physics, such as dimensional analysis or statistical mechanics.

Model 1: Ejection

For the first model, we will assume that the particles which constitute the material struck by the impactor are bound loosely enough that most of the energy of the impactor goes into ejecting material from the impact site.

Assume that a spherical crater is formed by ejecting material; the size of the crater is proportional to the amount of material which was ejected. If the material has a uniform density, then the total mass of the removed material, $M$, is proportional to the volume of the crater, $V$, which is in turn proportional to the crater diameter cubed, $d^3$: $M \propto V \propto d^{3}$. (See Fig. 1.)

Figure 1: Crater geometry

At a minimum, the impactor must provide enough energy to lift the volume of mass completely out of the crater. (See Fig. 2.) If the mass is lifted to a height $h$, the kinetic energy is converted completely to a gain in potential energy of the crater material $U$ as $K = U = M g h$, where $g$ is the acceleration due to gravity.

Reminder: We can use the same equation $U = mgh$ for the gravitational potential energy of any object near earth's surface.

Figure 2: Lifting the volume of mass out of crater.

Assuming that the crater is spherical, the depth of the crater is proportional to its diameter: $h \propto d$. Using this and the relationship $M \propto d^3$, we have $K = U = Mgh \propto d^4$. Therefore, our first model is that the crater diameter should scale as kinetic energy to the 1/4th power: $d \propto K^{1/4}$.

Model 1: $d \propto K^{1/4}$

Model 2: Deformation

For the second model, we will assume that most of the energy of the impactor goes into deforming the surface by pushing the material out of the way.

Assume that a spherical crater is formed by pushing surface material out of the way; the size of the crater is proportional to the amount of material which was pushed away.

Since the material only needs to be pushed out of the way (and not raised up to some height), the energy required is simply proportional to the volume which needs to be moved: $K \propto V \propto d^3$. Therefore, our second model is that the crater diameter should scale as kinetic energy to the 1/3rd power: $d \propto K^{1/3}$.

Model 2: $d \propto K^{1/3}$

Part 1: Making craters


We have two potential models which are quite similar. We therefore would like to design an experiment to determine which model better describes the data. Devise an experiment that allows you to measure crater diameter as a function of impactor kinetic energy. Since your ultimate goal is to distinguish between these two similar models, you will need to think about how to achieve sufficient precision and how to collect enough data to make a conclusive statement at the end of the project.

Begin by opening up this project's lab notebook template, and sharing the lab notebook with everyone int he group.

Initial observations

You have a number of different size steel ball bearings (impactors) and a container of fine sand of uniform grain size, along with some other pieces of equipment. Spend about 10 minutes making some initial observations using the setup, with a focus on testing possible procedures for releasing the ball bearing and measuring craters.

After this period, your TA will have a short discussion with the group to discuss what you've discovered.

Hints

Some key points to keep in mind as you consider how to go about designing and conducting your experiment are as follows:

  • How can you determine the kinetic energy of the impactors (ball bearings)?
  • How can you measure the diameter of the craters formed in the sand? (As a standard for defining the edge of the crater, use the highest point of the outermost ring. Note that for larger craters, the outermost ring of the crater may be relatively flat. In these cases, use the middle of the outermost ring. See Fig. 3.)
Figure 3: Determining the diameter of a crater with a ridge ring.
  • How will you consistently release the impactors?
  • What range of kinetic energies are necessary to test the model? (Since the model predicts a power law relationship between size and energy, you should cover at least 2 decades of energy.)
Decades? What do you mean?

In physics, a decade is often used to denote that something varies by a factor of 10 to some power. 

For example:

  • If you investigate lengths between $1\text{ cm}$ and $10\text{ cm}$, that would be one decade ($10^1$).
  • If you investigate lengths between $1\text{ cm}$ and $1\text{ m}$, that would be two decades ($10^2$).
  • Between $1\text{ cm}$ and $1\text{ km}$ would be a range of five decades ($10^5$).

In our case, we'd like the ratio between your smallest and largest energy to be at least a factor of 100.

Taking and visualizing data

After the discussion, you will continue taking data. In order to visualize this data, we again provide a Google Colaboratory notebook to make calculations and to plot as you go.

In order to either support or rule out the models under consideration, you will need to pay careful attention to the uncertainties in your measurements. For the purposes of this experiment, you will average repeated measurements and use the standard deviation of the mean (standard error) as an estimate of the uncertainty in your data. We provide functions for these calculations in the notebook above.

You will use the majority of the period today to collect data. While we do not specify what data to take or how much, your group will need to decide when you have enough. Use the plots you create (and feedback from your TA) to decide when you are finished.

Post-lab assignment


After the lab is over, each student in your group will write up their own conclusions and submit them to their TA via Canvas. Your individual conclusions are due no later than 48 hours before the start of your next lab. Your conclusions will be graded for completeness and quality according to the rubric on the PHYS141 main wiki page.

Even though you worked as part of a group in the lab, and submitted one group notebook, your individual conclusions must be your own work.

Your conclusions should not require more than one or two pages of text, though the final document may be longer if you include plots of data. The focus of your writeup should be on the final conclusions which you are able to draw based on your work in the lab. Assume that the reader, i.e. your TA, knows what the lab is about and has access to your groups notebook. As such you do NOT need to write about the following:

  • What you did in the lab.
  • Background and motivation for the experiment.
  • Theory.
  • Details of apparatus used.

You are expected to clearly articulate your conclusions and discuss how your data support those conclusions.

Conclusions

Take a moment to think a bit about the learning objectives for this lab course. These were listed on the lab homepage, but as a reminder we provide them again here.

For the introductory physics laboratories here at the University of Chicago, we have adopted a set of learning objectives. By the end of this course, you should be able to do the following:

  • collect data and revise an experimental procedure iteratively and reflectively;
  • evaluate the process and outcomes of an experiment quantitatively and qualitatively;
  • extend the scope of an investigation whether or not results come out as expected;
  • communicate the process and outcomes of an experiment; and
  • conduct an experiment collaboratively and ethically.

Put succinctly, the goal is to understand how we know, not what we know.*

* These goals were first outlined by the Physics Education Research Lab at Cornell University for labs at all levels, but especially for introductory labs. You can read more about the philosophy behind these learning goals  here.) </WRAP>

Drawing scientifically appropriate and meaningful conclusions is not easy. It is a skill which is learned and which you develop over time. Here are some thoughts to guide you in drawing conclusions from your lab work.

The conclusion is your interpretation and discussion of your data.

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • At the end of the lab period were there things you learned about what you did and how you did it that you could use to improve your experiment and obtain more precise results?

In about a few paragraphs, draw conclusions from the pendulum data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

Remember… your goal is not to discover some “correct” answer. In fact, approaching any experiment with that mind set is the wrong thing to do. You must always strive to reach conclusions which are supported by your data, regardless of what you think the “right” answer should be. Never should you state a conclusion which is contradicted by the data. Stating that the results of your experiment are inconclusive, or do not agree with theoretical predictions is completely acceptable if that is what your data indicate. Trying to shoehorn your data into agree with some preconceived expectation when you cannot support that claim is actually considered to be fraudulent, don't do that.

In the next lab, we will return to the pendulum to take more measurements with the goal of increasing the precision of your measurements based on what you learned in this lab period. This is a common theme in experimental physics; making measurements, analyzing your data, gaining experience and learning how to improve your experimental technique, going back into the lab to refine your experiment and improve your results… We assure you that no one plans and executes an experiment all in one go.

REMINDER: Your post-lab assignment is due 48 hours before your next meeting. Submit a single PDF on Canvas.



Preview of Next Weeks Lab

The remainder of this wiki page is just a preview of how your measurements from today will be used in next weeks lab.

You do not have to do any of the following this week.

The power of Scaling Laws

The models which we developed using dimensional analysis are examples of Scaling Laws. The functional form of these two models is of a power law relationship between crater diameter and impactor energy. In principle, if the underlying physics remains the same, this relationship should hold true over a wide range of energies. In other words the relationship between the variables should scale with energy.

You are already likely familiar with the use of scaling laws in real world applications. For example when designing aircraft, engineers will build scale models to test in wind tunnels with the expectation that a design which works with a hand held model in a wind tunnel will also work for the full size plane.

Likewise the scaling law which you have developed in the lab should, in principle, work for much larger energies so long as the physics does not change. Over how large a range of energies would you expect your relationship to be accurate? Scaling laws such as the one you have developed have been used to estimate the energy of asteroids which created craters on the moon. Similar scaling laws have been used to estimate the yield of thermonuclear weapons tested on earth.

The second part of this lab, which will be performed next week, will involve testing and refining your scaling law at higher energies, all the way up to the creation of terrestrial craters. Part of this lab will involve using your scaling law to predict the size of the crater produced by steel balls dropped from the second and third floors of KPTC into a large container of sand on the first floor. You will then use this data to refine your model and then derive an estimate of the yield of the nuclear device which created the Sedan crater in Nevada as well as the Chicxulub crater which is thought to have caused the extinction of the dinosaurs.

Sedan Crater

Below is a Google Maps image of a portion of the Nevada Test Site where over 1000 nuclear weapons tests were conducted. You can see numerous craters formed from both above ground and below ground detonations of nuclear weapons which occurred in the 1950s. On the left side of the image is an impressive crater known as the Sedan Crater which was produced as part of Operation Plowshare to test the feasibility of using nuclear weapons for civilian construction purposes. The crater was produced by the detonation of a $10^{4}$ kiloton ($4.40 \times 10^{12}\text{ J}$) thermonuclear explosion.

Figure 4: The Nevada Test Site with the Sedan Crater marked on the far left. (Image via Google Maps.)

Chicxulub crater

Another famous crater is Chicxulub, the crater formed by the asteroid that struck the earth around 65 million years ago causing the mass extinction of the dinosaurs (and many other species). Unlike your experiments where the impactor was dropped directly down onto the surface, it is estimated that the asteroid hit the earth at an angle of between 45$^{\circ}$ and 60$^{\circ}$ from horizontal (source: Wikipedia).

Figure 5: Location of the Chicxulub Crater on the Yucatan Peninsula in Mexico.