University of Michigan - Department of Astronomy

Version: Intro


Watch out! The sky is falling, the sky is falling!
-- Chicken Little



Answer these questions:


One look at the surface of the Moon should convince you that "empty space" is not so empty after all. There is actually a wide range of objects floating between the planets, from tiny particles to asteroids that can be a hundred miles across, debris left behind when the planets were formed. These objects can be perturbed from their orbits (by a close passage by a planet, a passing star, any number of things) and onto paths that cross ours -- or any other planet or moon. When that happens, a collision occurs and an impact crater is formed.

The size and shape of the crater depend on the impactor: its size, shape, speed, and the angle is hits the ground with. Specifically, the size of the crater depends on the energy of the impactor. However, the relationship is not linear, but rather is a power law:

eqn. 1

where D is the diameter, E is the energy of the impactor when it hits the ground, n is the power, and k is a constant. In part one of this lab, you will use a model to determine what k and n are.

The energy when the impactor strikes the ground is all kinetic,

eqn 2Ek = 1/2 *m * v^2

where m is the mass of the impactor, and v is the speed it's going when it hits the sand. Unfortunatly, v is inconvenient to measure in our classroom. Fortunatly, energy is conserved, so we can give the impactor a known energy and know it will hit the sand with that amount of energy. The total energy of a falling object is the sum of the kinetic and potential energy. If you drop the impactor so it starts with v = 0, the total energy is just potential (called gravitational potential energy):

eqn. 3

where m is still the mass, g is the acceleration due to gravity = 9.81m/s2 at the surface of the Earth, and h is the height above the ground.

To demonstrait cratering, you'll have a box of sand. There should be a thick base of white sand which you'll add a thin regolith of colored sand to. Get a box and some colored sand from your instructor (note if the sand in the box isn't white, you should get colored sand with a good contrast). Be sure to make the layer of colored sand very thin, since you want to see the pattern of ejecta when you make the crater.

Part 1: the Power Law

In this part you'll be figuring out the values of k and n for your model.

Find the mass of your ball bearing and fill in its value on the worksheet (note you can do this at any time during data collection) To use the scale, turn it on using the power button on the far right side, place the tray on it with the arrow pointing to the back, and push "TARE". When it reads "0.000 kg" (zero kg) place the ball bearing in the tray and take the reading when the numbers stabalize. If the display has the wrong unit, push the unit button until "0.000 kg" is displayed.

Smooth the sand and sprinkle a thin layer of colored sand on top. Do not knock the boxes on the floor to smooth the sand! Drop the ball bearing from at least 5 cm into the sand and measure the diameter of the crater from rim to rim. If the crater isn't round (especially if it lands too close to the side of the box) measure the longest axis. Enter the height and diameter in table 1. Pay attention to the shape of the crater and the pattern made by the ejecta.

Retrieve the ball bearing and drop it again from another height into another part of the container. Record the height and crater diameter in table 1. When the surface becomes too lumpy or the regolith is too well covered, smooth the sand and sprinkle more colored sand on it. Repeat until you have the two columns of table 1 filled.

Get the broom and dustpan from your GSI and clean up any spilled sand (the custodians will not clean up messes made by lab equipment!)

Recall from the introduction that the impact energy is equal to the energy you gave the ball bearing. Write the equation you'll use to calculate the impact energy below table 1 and show the calculation for the first row (watch your units!) Check it with your GSI, then fill in the rest of the column.

Since the relationship between D and E is a power law, a graph of D vs. E would be curvy, and it would be very hard to figure out what n and k are. However, n is the slope of a log-log graph (see your GSI if this doesn't make sense to you). Calculate logD and logE and fill in those values in table 1. Do the graph of logD vs. logE. Be sure to include the graph with your lab.

Draw a "best-fit" line through the data points on the graph. Remember, a best-fit line is a straight line that comes close to, but probably not through, as many points as possible. Mark the two places (not data points) on your line and find the slope. Record this as n on the worksheet. If you use a computer program or graphing calculator to find the slope, include the equation of the best-fit line on your graph.

To find k, you will use one of your data points. Pick a data point that is close to your best-fit line (one that you think is a good value), and plug D, E, and your value for n into the equation D = k En. Now you can solve that equation for the value of k. For example, suppose you dropped a 0.03kg ball bearing from a height of 2m. This give you an impact energy of E = mgh = (0.03kg) * (9.81m/s^2) * (2m) = 0.59Joules. And suppose that when you dropped the ball bearing, you measured the crater diameter to be 5cm = 0.05m, and also your graph had given you a slope = n = .25. Then,

D = 0.05m, E = 0.59Joules, and n = 0.25,

plugged into the equation D = k En

k = D / (En) = 0.05 / (0.59^0.25) = 0.057 .

Now, go and put all this knowledge to work in the Activity #1 Questions!

Activity 2: History of Cratering

Craters can be used to find out information about conditions on the planet or moon. An active planet will have few craters because tectonics and vulcanism recycle the planet's surface. On a planet with an atmosphere or craters can be worn away due to wind or water erosion. A geologically dead planet with no atmosphere has no way to remove craters, except through more cratering.

Smooth the surface of the sand and apply a thin layer of colored sand. Try throwing a ball bearing sideways into the sand and see what shape it makes. Get some of the other objects and throw them in. Make a bunch of craters without wiping the sand clean, to see how they pile up on top of each other (be careful not to mess up your craters pulling the objects back out!) Look carefully at the patterns in the sand and in the colored sand on top of the white sand -- around the crater you should see a crater rim and a little further away, rays of ejecta thrown out by the objects. See what you have to experiment with in order to answer the following Activity #2 Questions!

Updated 09-Jul-2010 by SAM

Copyright Regents of the University of Michigan.