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.

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.

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.

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.

*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.)

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

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

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.

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.

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.)

- 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.)

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.

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.

Once you have enough data, your TA will guide you through some exercises to understand how to perform a least-squares fit to compare a full data set to a model. After that, you will use your model to make predictions for a larger crater that the class will create (and measure) at the end of the period.

When you make a comparison between a measured number $a \pm \delta a$ and prediction $b$, you can perform a $t^{\prime}$ test to see whether your measured value agrees with that prediction as

$t^{\prime} = \dfrac{a-b}{\delta a}$, |

where, as discussed in past weeks…

- …$\lvert t' \rvert \leq 1$ is
*agreement*, - …$1 < \lvert t' \rvert < 3$ is
*inconclusive*, and - …$\lvert t' \rvert \geq 3$ is
*disagreement*.

Some models, however, have *free parameters* (also called *unknown* or *undetermined parameters*). For example, our two potential cratering models predict that the cratering diameter increases with kinetic energy as either $D = AK^{1/3}$ or $D = BK^{1/4}$, but they don't specify exactly what value the coefficients $A$ or $B$ should have.

How can we make quantitative comparisons in this case?

*Fitting* data to a model is a way to determine the best values for free parameters. In Part 1, you adjusted the values of $A$ and $B$ in your models until you had what looked to your eyes like the best match to the data. In this part, we will use a more rigorous method for determining the “best” values.

Suppose that you have $N$ data points, where the $i$th data point is $(x_i, y_i \pm \delta y_i)$. You want to see if your data are well-described by a particular function, $f(x)$. Let's suppose, just as an example, that you wanted to test a linear function $f(x) = mx$, where $m$ is the slope of the line. This slope is an example of a *free* parameter in the model, and we want the fit to tell us what the “best” value is for $m$.

First, let's define a function called the **residual**. The residual of the $i$th point is $\chi_i$ and is defined as

$\chi_i \equiv \dfrac{f(x_i) - y_i}{\delta y_i}$. |

This form should immediately look familiar: it is the $t^{\prime}$ comparison between the measured value $y_i \pm \delta y_i$ and the value predicted by the function at the same point, $f(x_i)$.

A “good” fit will be one where the value of the free parameters in the fit function make this residual small (meaning that the |$t^{\prime}$| at this point is small, and therefore *in agreement*). But perhaps values which are good for this point aren't quite as good for the next point… or the point after that. So, rather than focus on a single residual, we want to instead pick values for the free parameters which minimize the *sum* of all the squares of the residuals,

$\chi^2 \equiv \sum_i^N \dfrac{(f(x_i) - y_i)^2}{\delta y_i^2}$. |

(We look at the *squares* of the residuals because we want each contribution to the sum to be positive. If a data point is a little bit higher than what is predicted by the fit function, that should count the same as if the data point is a little low. Squaring accomplishes this.)

It is possible to minimize a function by hand by using calculus, but for this class, we will instead rely on a computer algorithm to find the value for each fit parameter in a model which minimizes the value of $\chi^2$. Such a fit is called a **least-squares fit** because it finds the fit values that give the *least* value of the sum of the *squares* of the residuals.

In such a fit, you supply the data points and the model function, and the fit returns the best values for each fit parameter and the minimum $\chi^2$ value that results when you use those values.

You can now apply the same formalism from above to your question: what are the best values for $A$ and $B$ in the 1/3- and 1/4-power law models?

To do this, return to the Google Colab notebook above, and start working on Part 2. There you will need to enter your data – energy and corresponding average crater diameter with uncertainties – and the code will guide you through the least-squares fits.

**Work through the notebook slowly!** Try to understand what is happening at each step, and talk to your TA if you don't know what you are looking at.

When the fits are done, you will have values for the best fit parameters and plots showing what the fits look like. The plots should look similar to what you found in Part 1 by-eye, but now we have a more quantitative justification for what is the “best” fit.

In addition to being the thing which is minimized, we can use the final $\chi^2$ value to determine whether our model overall is in agreement with the data or not. (It is still possible for the “best” fit to be a “bad” fit, for example.) For this reason, $\chi^2$ is sometimes referred to as the “goodness of fit” parameter“.

First, note that $\chi^2$ can grow arbitrarily large; if we increase the number of data points used in the fit, we increase the value of $\chi^2$. Therefore, it will help us to look not at $\chi^2$ itself, but at quantity called the **reduced chi-squared**, $\chi^2_{red}$. If we have $N$ data points and $k$ free parameters in the fit, then the number of **degrees of freedom** is $\nu = N - k$ and the reduced chi-square is

$\chi^2_{red} = \chi^2/\nu$. |

The reduced chi-square is sort of like the *average* chi-square per data point, or equivalently the *average residual*.

Conceptually, what does the reduced chi-square represent and how can we use this value to determine if our model is in agreement with the data or not? Suppose you have one point that is very close to the fit line so that its distance away is less than the size of its uncertainty; for such a point, $\chi_i = (f(x_i)-y_i)/\delta y_i < 1$. Now suppose another point is far away from the line, so that its distance away is greater than its uncertainty; therefore, $\chi_i = (f(x_i)-y_i)/\delta y_i > 1$. If we have a “good” fit, then we'd expect to have some close points ($\chi_i <1$), some far points ($\chi_i >1$), and some medium points ($\chi_i =1$), so we would expect our average residual (i.e. our reduced chi-square) to be about 1.

Let's look at a few scenarios:

- $\chi^2_{red} \approx 1$: the scatter of the data around the fit line is about what you would expect based on the size of the uncertainties. The data and the fit
*agree*. - $\chi^2_{red} \gg 1$: the scatter of the data around the fit line is greater than you'd expect based on the size of the uncertainties. Either the model does not agree with the data, or the uncertainties on the data points are too small (possibly because there are unaccounted for uncertainties or because of a systematic bias).
- $\chi^2_{red} \ll 1$; the scatter of the data around the fit line is smaller than you'd expect based on the size of the uncertainties. The uncertainties are likely
*over-estimated*or more data is needed to test the model.

Unlike the $t^{\prime}$ test, these are not hard rules about agreement or disagreement. But it can be helpful as part of the discussion about the quality of your fits.

Why do we divide by the number of *degrees of freedom*, $\nu = N - k$, instead of just the number of points, $N$?

Each time we add another free parameter to the model, we “constrain” the model more. Think, for example, about what happens when you have two data points and you try to fit them to a line $f(x) = mx + b$. We have two data points ($N = 2$) and two fit parameters ($k = 2$), so we have zero degrees of freedom ($\nu = N - k = 0$). The line will go *exactly* through both points and the chi-squared value will be zero, $\chi^2 = 0$. We have effectively “used up” two data points worth of information to do the fit, so we have no “freedom” left to let the fit wiggle around the data points.

Now consider doing the same fit with three or more data points. The line is now no longer guaranteed to go through each point exactly, and so $\chi^2$ value will no longer be zero.

By dividing $\chi^2$ by the number of degrees of freedom instead of just by $N$, we better account for the information lost (used to constrain the model).

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). Craters on the moon for example should likely follow the same scaling law which applies to your sand craters.

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?

When you TA says it is your turn, you will be allowed to go over to a large box of sand and drop a significantly larger stainless steel ball into it to see how well your model predicts crater diameters for energies that are several orders of magnitude larger than what you studied in Part 1.

- Measure the mass of the ball and determine the kinetic energy it will have after being dropped from a height of 1 meter.
- Predict the size of crater produced by such an impactor using your model.
- Perform the experiment. Estimate both the crater diameter and uncertainty. You may repeat the drop if necessary.
- Provide your TA with your final crater diameter measurement (in secret).

After everyone has has dropped the mass and made their measurements, the TA will call the group together for a discussion of the class findings.

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

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.*)

Even though the crater was produced by an explosion at the surface near the surface instead of an impactor from space, we want to see if your model holds for this crater.

- Rearrange your scaling law equation so that it becomes kinetic energy as a function of diameter.
- The Sedan Crater has a diameter of 390 m. Using your rearranged scaling law, what would you predict for the energy yield of the nuclear weapon that produced it? Is the value of the same
*order of magnitude*as the known value ($4.4 \times 10^{12}$ J)? **(For PHYS 131/141 only)**: Using propagation of uncertainties, determine an uncertainty on this energy.**(For PHYS 131/141 only)**: You may find that your model predicts a value which is the right order of magnitude, but which does not actually agree within uncertainties. Can you think of some reasons why the model may not be correct? (As a hint, think about the dominant process in your model… areas of these craters have been found to be compacted and some of the sand was heated and turned into glass. If some of the energy of the blast went into these processes, what does that mean for your prediction? Will it be too high or too low?)

In the Introduction to Experimental Physics lab about making your own ruler, we introduced the uncertainty propagation formulas. A more complete treatment (with examples) is available on its own uncertainties page.

We'll review the most important points here.

**General formula**
Suppose you have a function $f(x,y,... z)$. No matter how complicated this function (or how many variables you have with meaningful uncertainties), you can always start with the following general uncertainty formula:

That said, there are several very common cases we encounter in the labs, so let look at these specific examples.

**Sums and differences**

If $f(x,y,z) = x - y + \dots +z$, then the general formula becomes

$\Delta f = \sqrt{(\Delta x)^2 + (\Delta y)^2 + \dots + (\Delta z)^2}$. |

**Products and quotients**

If $f(x,y,z) = \dfrac{xy}{z}$, then the general formula becomes

$\dfrac{\delta f}{f} = \sqrt{\left( \dfrac{\delta x}{x}\right)^2 + \left( \dfrac{\delta y}{y}\right)^2 + \left( \dfrac{\delta z}{z}\right)^2}$. |

**Powers**

If f(x) = x^n$, then the general formula becomes

$\dfrac{\delta f}{f} = n \dfrac{\delta x}{x}$. |

**Reciprocals**

If $f(x) = 1/x$, then the general formula becomes

$\dfrac{\delta f}{f} = \dfrac{\delta x}{x}$. |

(Note that reciprocals are just a special case of products or quotients.)

**An example**

Suppose that the function you are trying to propagate uncertainties for is $E = (1/2)mv^2$.

This can be rewritten as $E = (1/2) \times m \times v^2$. There is no uncertainty on $(1/2)$, so using the rule for powers and the rule for products, we have a total uncertainty of

$\dfrac{\delta E}{E} = \sqrt{\left( \dfrac{\delta m}{m}\right)^2 + \left(2 \dfrac{\delta v}{v}\right)^2}$. |

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.

- The Chicxulub Crater has a diameter of about 100-150 km. Using your rearranged scaling law, what would you predict for the energy of the asteroid that produced it?
**(For PHYS 131/141 only)**: Expert estimates – using scaling laws like yours as well as other evidence – suggest the kinetic energy of the asteroid at the time of impact was between $3 \times 10^{23}$ and $6 \times 10^{25}$ J (source: Wikipedia). Why might your prediction not agree with the expert's estimate? Is your prediction too high or too low, and is that consistent with your reasons for what may be causing disagreement?

Answer the questions/prompts below in a **new** document (not your lab notebook) and submit that as a PDF to the **Lab 2 - Report** 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 assignment is due 48 hours before your next day in lab.*

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.)

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.*)

Even though the crater was produced by an explosion at the surface near the surface instead of an impactor from space, we want to see if your model holds for this crater.

- Rearrange your scaling law equation so that it becomes kinetic energy as a function of diameter.
- The Sedan Crater has a diameter of 390 m. Using your rearranged scaling law, what would you predict for the energy yield of the nuclear weapon that produced it? Is the value of the same
*order of magnitude*as the known value ($4.4 \times 10^{12}$ J)? **(For PHYS 131 only)**: Using propagation of uncertainties, determine an uncertainty on this energy.**(For PHYS 131 only)**: You may find that your model predicts a value which is the right order of magnitude, but which does not actually agree within uncertainties. Can you think of some reasons why the model may not be correct? (As a hint, think about the dominant process in your model… areas of these craters have been found to be compacted and some of the sand was heated and turned into glass. If some of the energy of the blast went into these processes, what does that mean for your prediction? Will it be too high or too low?)

In the Introduction to Experimental Physics lab about making your own ruler, we introduced the uncertainty propagation formulas. A more complete treatment (with examples) is available on its own uncertainties page.

We'll review the most important points here.

**General formula**
Suppose you have a function $f(x,y,... z)$. No matter how complicated this function (or how many variables you have with meaningful uncertainties), you can always start with the following general uncertainty formula:

That said, there are several very common cases we encounter in the labs, so let look at these specific examples.

**Sums and differences**

If $f(x,y,z) = x - y + \dots +z$, then the general formula becomes

$\Delta f = \sqrt{(\Delta x)^2 + (\Delta y)^2 + \dots + (\Delta z)^2}$. |

**Products and quotients**

If $f(x,y,z) = \dfrac{xy}{z}$, then the general formula becomes

$\dfrac{\delta f}{f} = \sqrt{\left( \dfrac{\delta x}{x}\right)^2 + \left( \dfrac{\delta y}{y}\right)^2 + \left( \dfrac{\delta z}{z}\right)^2}$. |

**Powers**

If f(x) = x^n$, then the general formula becomes

$\dfrac{\delta f}{f} = n \dfrac{\delta x}{x}$. |

**Reciprocals**

If $f(x) = 1/x$, then the general formula becomes

$\dfrac{\delta f}{f} = \dfrac{\delta x}{x}$. |

(Note that reciprocals are just a special case of products or quotients.)

**An example**

Suppose that the function you are trying to propagate uncertainties for is $E = (1/2)mv^2$.

This can be rewritten as $E = (1/2) \times m \times v^2$. There is no uncertainty on $(1/2)$, so using the rule for powers and the rule for products, we have a total uncertainty of

$\dfrac{\delta E}{E} = \sqrt{\left( \dfrac{\delta m}{m}\right)^2 + \left(2 \dfrac{\delta v}{v}\right)^2}$. |

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.

- The Chicxulub Crater has a diameter of about 100-150 km. Using your rearranged scaling law, what would you predict for the energy of the asteroid that produced it?
**(For PHYS 131 only)**: Expert estimates – using scaling laws like yours as well as other evidence – suggest the kinetic energy of the asteroid at the time of impact was between $3 \times 10^{23}$ and $6 \times 10^{25}$ J (source: Wikipedia). Why might your prediction not agree with the expert's estimate? Is your prediction too high or too low, and is that consistent with your reasons for what may be causing disagreement?

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