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We can do some simple calculations to see what kind of destructive power a meteor or asteroid might have. The object doesn't have to be very big to cause an enormous amount of destruction. Consider an object that is roughly spherical, and one kilometer (1000 meters) in diameter. Its volume V is given in terms of its radius R as (π is the ratio of a circle's circumference to its diameter, roughly 3.14159): |
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| (1) | V = 4/3 π R3 | |||||||||||||||||||
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Since its radius is 500 meters, this gives a volume of approximately 500 million cubic meters. (Since 4/3 x π is roughly 4; multiply 4 x 500 x 500 x 500; this is 4 x (5 x 5 x 5) x (100 x 100 x 100); this is 4 x 125 x one million or 500 million.) Water has a density of a metric ton, or 1000 kilograms, per cubic meter. If the object's density is 3 times that of water (fairly typical of carbonaceous meteors, a bit light for stony), its mass will be 1.5 billion metric tons, or 1.5 trillion kilograms. (In the U.S., a trillion is equal to 1012, a one followed by twelve zeroes.)
The energy of motion (kinetic energy) Ek of an object of mass m, moving at velocity v, is given by: |
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| (2) | Ek = ½ m v2 | |||||||||||||||||||
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The velocity of a meteor striking the earth is roughly 25 km/sec. That's 25,000 meters/sec, or 25 x 103. Square that; get 625 x 106; put that value into (2); the result is 4.7 x 1020 Joules of energy. A Joule is a metric unit; it equals one kilogram-meter2/second2; it's approximately equal to three-fourths of a foot-pound, which is the energy required to lift one pound one foot in the earth's gravity.
Now when this meteor strikes the earth, all of its kinetic energy will be converted into impact energy — some into an air pressure wave, some into forming a crater. A megaton, the explosive equivalent of one million tons of TNT, is approximately equal to 4.2 x 1015 Joules of energy. So the impact energy of the meteor is about 105 megatons, or 100,000 megatons. Note that the largest nuclear device was about 50 megatons — this meteor has an energy of 2,000 times that! If an object is ten times smaller in linear dimension (100 meters in diameter, rather than one kilometer), its volume and mass will be 1,000 times smaller. So its impact energy will be 1,000 times smaller, or 100 megatons. If an object is ten times larger in linear dimension (10 kilometers in diameter, rather than one kilometer), its volume and mass will be 1,000 times larger. So its impact energy will be 1,000 times larger, or 100 million megatons. This is the approximate impact energy of the asteroid believed to have caused the dinosaurs' extinction. I want to reiterate that this is NOT 100 megatons — it is 100 MILLION megatons. Suppose we were to discover an asteroid that is approaching the earth, and we want to divert it. Suppose further we want to change its velocity by only ten centimeters per second. Over a period of ten years (which contain roughly 315 million seconds), the meteor will move from its original trajectory by a distance of 3.15 billion cm, which is 31.5 million meters, or 31,500 km — this is about five earth radii. So it will be diverted enough to miss the earth. The actual calculation is more complicated than this; you would have to determine the new orbit. You'd then want to ensure the earth's gravity didn't pull the meteor in as it got close. You'd also want to ensure the meteor didn't swing around and hit the earth on a subsequent orbit. And missing the earth by only five earth radii is a pretty close call. But this gives a rough idea of what's needed. So if we can change the meteor's velocity, and do it at least ten years before the impact, a change in speed of ten cm/sec will do the job. Suppose that we have a year in which to change the meteor's speed. Perhaps we will put some kind of engine on the meteor, and gradually push it. If we spend a year changing the object's velocity by ten centimeters per second, the acceleration needed is roughly (Divide ten cm/sec by the number of seconds in a year, about thirty million. Also, if the mass is in kilograms, we must use meters, not centimeters.) Now this isn't much, but we're trying to move a very massive object. The force F required is given by Newton's second law as the acceleration a multiplied by the mass m from (1): |
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| (3) | F = m a = (1.5 x 1012 ) x ( 10-9 ) | |||||||||||||||||||
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The result is 4500 Newtons. A Newton is a metric unit; it's defined to be a kg m/sec2; it's equivalent to about a quarter of a pound. So 4500 Newtons is about half a ton. This isn't much. We can exert half a ton of force — as long as we do it steadily, for a year — and move the object.
To see how much energy is required to move the meteor, use the expression for kinetic energy above. When we're on the meteor, it appears to be standing still. So calculate the kinetic energy required to change its velocity by ten cm/sec (which has to be converted to one-tenth m/sec first): |
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| (4) | ½ m v2 = ½ (1.5 x 1012 ) x (10-1)2 = 7.5 x 109 | |||||||||||||||||||
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The units are (again) Joules, and each Joule is about one-fourth of a foot-pound, so about two billion foot-pounds of energy will be required. To get an idea of how much energy this is, do the following. Near Colorado Springs, Colorado, there is a road to the top of Pikes Peak. The road starts in the nearby town of Manitou Springs, at roughly 7,000 feet elevation. The top of Pikes Peak is slightly more than 14,000 feet. Suppose we consider how much energy it takes to drive a car to the top of the mountain. We multiply the weight of the car (say it's about three thousand pounds) by the altitude change:
I have only calculated the energy required to change the car's elevation - since the road is about twenty miles long I should add the energy required to drive twenty miles along a flat course. But if you've ever driven this road you know the energy required to change the car's elevation is far greater. Likewise, if you've ever hiked up a mountain you know the effort required to go uphill is far greater than the effort to go a similar distance on a level course. The energy we need is about a hundred times this. So that means a hundred cars driving up the mountain use about two billion foot-pounds of energy. Another way is to recall it takes about 36 million Joules to put one pound of something into orbit. The energy we need is a bit more than 200 times that – so we certainly have the capability to generate enough energy. |
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