To gain some insight into the process scientists use to investigate physical phenomena. From formulating a question and developing a plausible model, all the way through to testing and evaluating that model. THe phenomena we will investigate is the formation of craters, such as those you see on the moon or which are created by explosions. In this process you will:

• Formulate A Question - Precisely define the question which you are investigating.
• Develop A Model - Use some very basic physics to develop a model which is both plausible and testable.
• Test The Model - Plan and conduct an experiment in the lab which tests whether or not the model describes the phenomena.
• Make A Prediction - Use your data and the model to make a prediction for conditions that go beyond what you tested in the lab.
• Test The Prediction - Conduct a second experiment to test the prediction.

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.

So the specific question that we will investigate is,

Almost all physical phenomena involve the concept of Energy at some level. In this case it takes energy to create a crater by either moving or removing the material which used to be where the crater is. Let us consider how this might work in the case of crater created when a meteor strikes the ground. This provides us with a simple starting point for developing a model as the energy carried by the meteor must go into creating the crater.

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.

We would like to design an experiment to determine if our model describes the functional relationship between the kinetic energy of an impactor and the size of the resulting crater. Devise an experiment that allows you to measure crater diameter (a measure of the size of the crater) as a function of impactor kinetic energy.

You have several different size steel ball bearings (impactors) and a container of fine sand of uniform grain size, along with some other pieces of equipment. The bearings can be dropped into the sand from different heights. By changing the size of the impactor (bearing) and the height from which it is dropped, you can vary the energy.

To help you decide which impactors to use and what heights to drop them from, it is helpful to begin by considering what are the largest and smallest energies which you can use? Obviously the largest impactor dropped from the largest height will provide the largest energy, and the smallest impactor dropped from the smallest height will produce the smallest energy. But these two end points may not produce useful craters. For example a very small impactor dropped from 1mm above the sand will probably not even produce a crater, while the largest impactor may bottom out in the container of sand when dropped from too great a height. Thus you need to conduct mini experiment to determine what are the smallest and largest energies which are practical to use.

Spend about 15 minutes making some initial observations using the apparatus. Drop different impactors from different heights and examine the craters which are produced. Record your observations along with the impactor sizes and heights used.

After about 15 minutes the instructors will lead a short discussion, asking each group to report on their results. Be prepared to report on the largest and smallest energies which you feel are useful for the experiment. Pay attention to the results obtained by the other groups, collaboration among scientists is an important part of how science progresses. We learn by both doing and hearing about the work of others.

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. Using what you learned from other groups (and using the hints above), develop a good measurement procedure.

To evaluate your experimental results you will plot your data using the program Logger Pro which is loaded on the lab computers. Using this program you can enter and generate an XY plot of your data. The software also has the ability to perform what is called a Curve Fit which determines whether or not your data follow the same functional form as the one predicted by your model. The software is easy to use and your instructors will help you with the details.

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.

Predict the size of crater produced by such an impactor using your model.