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.

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.

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

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

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.

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.

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).

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.)

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. You will only need Part 1 of this notebook for today's experiment.

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.

The function you have arrived at is an example of a scaling law.

You may already be familiar with at least one form of the use of scaling laws. Aeronautical engineers construct small scale models of aircraft and test their designs in wind tunnels. If the model is aerodynamically stable, then the scaling nature of the physics involved says that the full size airplane will perform similarly.

In our case, as long as the underlying assumptions of the model remain valid, there is no reason that the functional relationship you have determined should not be valid for craters of all sizes (for impactors of similar density into material of similar density and granularity).

Towards the end of the period, your TA will lead the class into the hall and drop a larger stainless steel ball (from a significantly larger height). You will use your model to predict crater diameters for energies that are orders of magnitude larger than what you studied in Part 1.

• Your TA will provide an estimate of the ball's mass and the heights from which they will drop it. Use this information to calculate the kinetic energy of the ball upon impact.
• Predict the size of crater produced by such an impactor using your model.
• Provide your TA with your crater diameter estimates and uncertainties.

After the experiment, the TA will call the group together for a discussion of the class findings.

• How did your predictions (from the model) match the measured values?
• Was there much scatter in the predicted values from your classmates or were all groups roughly in agreement?

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.

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?
• Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the 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.

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.)

On December 24, 2021 a meteorite struck Mars near where the NASA InSight Lander is operating. Devices on the craft detected seismic waves that scientists at first thought were due to an earthquake (marsquake?), but satellite imagery from NASA's Mars Reconnaissance Orbiter later showed a new impact crater on the surface consist with the location of the source of the waves.

There are no questions to answer, but you might enjoy this summary of the recent study (published October 22, 2022): https://www.jpl.nasa.gov/news/nasas-insight-lander-detects-stunning-meteoroid-impact-on-mars.