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.

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.

(left) Location of the Chicxulub Crater on the Yucatan Peninsula in Mexico. (right) The Nevada Test Site with the Sedan Crater marked on the far left. (Image via Google Maps.)

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 in the 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 15 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.

Equipment

Each station has the following equipment:

  • a box of loose, uniformly-sized sand;
  • a collection of small stainless steel ball bearings of different sizes;
  • a metal lunch tray and a shield, which together should be used to minimize and catch sand splatter when dropping
  • a stir stick, which can be used to agitate and mix up the sand; and
  • a light, which can be angled across a crater in order to produce more light contrast (making the crater easier to see while making measurements).
Base equipment available for each station

In a common area in each room, there is also the following equipment:

  • rods and clamps;
  • digital mass balances and triple beam (analog) mass balances;
  • meter sticks, two-meter sticks, rulers and calipers; and
  • drafting compasses (which can widened/narrowed to measure or mark circle diameters).

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


Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

Conclusions (Partial)

In about one or two paragraphs, draw conclusions from today's experiment. 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. (See the last experiment for suggested questions you should ask yourself as you consider the conclusion.)

NOTE: We're leaving this week's experiment right in the middle – after you've taken data, but before we've fully explored the model. So, your conclusions are expected to be partial and incomplete. Maybe you will have to rely more on qualitative rather than quantitative data, or you will have to speak more towards what you plan to do next time than on what you've already done. That's all okay!

Questions

Consider the following questions:

  1. Think about the models we are testing…
    1. What assumptions go into those models? (About the type of impactor, about the sand, about the direction of impact, etc.)
    2. We test between “ejection” and “deformation” impacts. What other ways might energy be dissipated in an impact that we are not considering here?
    3. We are dropping spherical balls and creating spherical craters. What do you think would happen if you dropped objects that were not spherical? (There is no right or wrong answer here, but justify your answer with an explanation of why.)
  2. Think about real craters produced on Earth (or the Moon or other planets)…
    1. Could you apply the model (with the best fit parameters that you determined at the end of the lab) directly to such craters? (Why or why not?)
    2. What might be different about these real craters from the ones we created today in lab? (Consider both differences in assumptions and differences in physical properties.)

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.