Dear Travis,
Thank you for question on meteors and atomic bombs. The kinetic
energy in the meteor can be expressed as:
K = ½ mv^2
However, as the object hurtles toward the Earth, it gains kinetic
energy as its potential energy is converted to kinetic. Therefore,
one must consider the total energy in the object:
E = ½ mv^2 + mgh
Just before impact, the height will decrease to zero (at impact) and
all of the energy will be kinetic energy. For simplicity, I will
neglect loss of energy to resistance due to the atmosphere and effects
of heating secondary to friction with the atmosphere during descent.
Of course, if one could measure the velocity of the object just prior
to impact, these effects would be inherently be taken into account
(the friction has already had its effect at that point), so one can
either calculate the equivalent energy based on the energy of the
object at a height above the impact site, or one can calculate the
equivalent energy more accurately by measuring the instantaneous
velocity (and mass) must prior to impact. The math is essentially the
same.
So, the total energy is given above, and is constant (conservation of
energy). With the units below, the energy calculated will be in
Joules. To convert to "atomic bomb units," also known as "equivalent
tons of TNT", we use the conversion
1 megaton (1 million metric tons) of TNT = 4.185 x 10^22 ergs =
4.185 x 10^15 Joules
or, one can say, there are
2.39 x 10^-16 megatons per Joule
A very complete physics unit conversion document can be found here:
http://scholar.google.com/url?sa=U&q=http://www.agu.org/reference/gephys/23_moskowitz.pdf
Table 4 on page 5 of the above document give multiple energy unit conversions.
The answer, then, is a matter of substitution of the conversion factor
into the energy equation:
Energy (in megatons TNT) = E * megatons / Joule
= E * ( 2.39 x 10^-16 megatons / Joule)
= (½ mv^2 + mgh) * ( 2.39 x 10^-16 megatons / Joule)
An *excellent* resource for both derivations and discussion of the
many ramifications of such an impact is the following article:
"Comet and Asteroid Threat Impact Analysis," by James A. Marusek.
http://scholar.google.com/url?sa=U&q=http://personals.galaxyinternet.net/tunga/TA.pdf
This document includes multiple estimates on the many variables
involved (e.g., object density, velocity, etc.)
Another good article can be found here:
http://scholar.google.com/url?sa=U&q=http://www.geocities.com/elidoro/collaborazione/whitepaper.pdf
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Here are definitions of all of the variables I've used above with the
units in brackets:
K: Kinetic energy [joules]
m: mass of object [kilograms]
v: velocity of object [meters/second]
E: Total energy of object [joules]
g: gravitational acceleration constant (9.8 m/s^2)
h: height of object from impact site [meters]
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I hope this was helpful. Feel free to request clarification.
Best,
-welte-ga |