Space Math: Craters are a Blast!
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Space Math: Craters are a Blast!

Topic:

Body:

Mission:

Science Education Standards:

Earth and Space Science -- Content Standard D

Motions and Forces

• Objects change their motion only when a net force is applied. Laws of motion are used to calculate precisely the effects of forces on the motion of objects. The magnitude of the change in motion can be calculated using the relationship F = ma, which is independent of the nature of the force. Whenever one object exerts force on another, a force equal in magnitude and opposite in direction is exerted on the first object.
• Gravitation is a universal force that each mass exerts on any other mass. The strength of the gravitational attractive force between two masses is proportional to the masses and inversely proportional to the square of the distance between them.

Short Description: Students measure crater diameters in a photo of the Moon, and determine the energy required to create them using a simple quadratic equation.

Source: Space Math (GSFC)

Have you ever wondered how much energy it takes to create a crater on the Moon. Physicists have worked on this problem for many years using simulations, and even measuring craters created during early hydrogen bomb tests in the 1950's and 1960's. One approximate result is a formula that looks like this:

E = 4.0 x 1015 D3 Joules.

where D is the crater diameter in kilometers.

As a reference point, nuclear bomb with a yield of one-megaton of TNT produces 4.0 x 1015 Joules of energy!

Problem 1 - To make the formula more 'real', convert the units of Joules into an equivalent number of one-megaton nuclear bombs.

Problem 2 - The photograph above was taken in 1965 by NASA's Ranger 9 spacecraft of the large crater Alphonsis. The width of the image above is 183 kilometers. With a millimeter ruler, determine the diameters, in kilometers, of a range of craters in the picture.

Problem 3 - Use the formula from Problem 1 to determine the energy needed to create the craters you identified.

Note: To get a better sense of scale, the table below gives some equivalent energies for famous historical events:

 Event Equivalent Energy (TNT) Cretaceous Impactor 100,000,000,000 megatons Valdiva Volcano, Chile 1960 178,000 megatons San Francisco Earthquake 1909 600 megatons Hurricane Katrina 2005 300 megatons Krakatoa Volcano 1883 200 megatons Tsunami 2004 100 megatons Mount St. Helens Volcano 1980 25 megatons

Problem 1 - To make the formula more 'real', convert the units of Joules into an equivalent number of one-megaton nuclear bombs.

Answer: E = 4.0 x 1015 D3 Joules x (1 megaton TNT/4.0 x 1015 Joules)

E = 1.0 D3 Megatons of TNT

Problem 2 - The photograph above was taken in 1965 by NASA's Ranger 9 spacecraft of the large crater Alphonsis. The width of the image above is 183 kilometers. With a millimeter ruler, determine the diameters, in kilometers, of a range of craters in the picture.

Answer: The width of the image is 92 mm, so the scale is 183/92 = 2.0 km/mm. See figure below for some typical examples: See column 3 in the table below for actual crater diameters.

Problem 3 - Use the formula from Problem 1 to determine the energy needed to create the craters you identified. Answer: See the table below, column 4. Crater A is called Alphonsis. Note: No single formula works for all possible scales and conditions. The impact energy formula only provides an estimate for lunar impact energy because it was originally designed to work for terrestrial impact craters created under Earth's gravity and bedrock conditions. Lunar gravity and bedrock conditions are somewhat different and lead to different energy estimates. The formula will not work for laboratory experiments such as dropping pebbles onto sand or flour. The formula is also likely to be inaccurate for very small craters less than 10 meters, or very large craters greatly exceeding the sizes created by nuclear weapons. (e.g. 1 kilometer).

 Crater Size (mm) Diameter (km) Energy (Megatons) A 50 100 1,000,000 B 20 40 64,000 C 5 10 1,000 D 3 6 216 E 1 2 8