[meteorite-list] Terminal velocity of small falling objects

From: Piper R.W. Hollier <piper_at_meteoritecentral.com>
Date: Thu Apr 22 10:08:25 2004
Message-ID: <5.1.0.14.2.20020829001421.0224ac20_at_pop.xs4all.nl>

Hello Tom, Tom, Shaun, Dave, Ron, and list,

> As I said before, simple ballistics made it impossible, the girls foot would
> have been damaged and in need of a hospital.

Ron, I DON'T agree with this statement made by an unnamed contributor, nor
with several other similar statements. My disagreement is based not on
"simple" ballistics, but on "mathematical" ballistics, that is, taking
known formulas and plugging in reasonable input values and seeing what
comes out.

I will make an amateur attempt here to contribute something about the
physics of small meteorite falls. Like many other phenomena where most
humans have little or no first-hand experience, some people may develop
intuitive notions about the subject which are rather inconsistent with the
laws of physics. The sensationalist bent of the media doesn't help much. I
am a firm believer in "doing the math". Calculations based on known
formulae are arguably not as good as making experimental measurements, but
are still considerably better than seat-of-the-pants guesswork. Somebody
please correct me if my logic or my math are off track in the following
analysis.

An object dropped and allowed to fall through the lower atmosphere does not
continue to accelerate indefinitely. There will instead be an upper
boundary on the velocity at which it falls, called the "terminal velocity".
At the terminal velocity, the downward force cause by gravitational
attraction between the object and the earth (the "weight" of the object) is
exactly balanced by the upward force of the aerodynamic drag of the
object's passage through the atmosphere. When these two forces are in
balance there will be no further acceleration and the object will maintain
an essentially constant downward velocity.

For a sky diver the terminal velocity is about 53 meters per second, or 120
mph. There is only significant acceleration during the first 10 seconds or
so of fall. After that, the onrushing air pushes the person up just as hard
as the earth pulls him or her down, and the fall velocity levels off.
Animals smaller than sky divers have a lower terminal velocity -- in the
case of small insects it can be much less than 1 mph. This is part of the
reason why bugs, lizards, tree frogs, and even small mammals can fall out
of tall trees, hit the ground, and usually simply walk away unharmed, while
a human would be seriously injured by a fall from the same height. Terminal
velocity is also the reason that a small meteorite can fall "for miles",
hit a person, animal, car, or building and do little or no damage. Small
meteorites will in most cases have lost all of their cosmic velocity at a
considerable distance above the ground, and the fall velocity upon reaching
the ground is the aerodynamic terminal velocity.

How small does a meteorite need to be to not be dangerous?

The magnitude of the terminal velocity depends on a number of things:

1. The density and viscosity of air. The exact values of these at sea level
depend on barometric pressure, temperature, and humidity, but for the
purposes of rough calculations can be assumed to be about 1.222 kg per
cubic meter density and 1.73 x 10^-5 newton-seconds per square meter
viscosity on a typical pleasant afternoon in Northallerton.

2. The density of the falling object. This is typically 3.25 to 3.90 grams
per cubic centimeter for ordinary chondrites and about 8 grams per cubic
centimeter for irons.

3. The size of the falling object. Use a ruler or make an estimate from a
photo. The meteorite in the Northallerton photo looks like it is about one
inch (2.5 cm) in diameter.

4. The shape of the falling object.

5. The rigidity of the falling object.

If one makes the simplifying assumptions that the object is spherical and
rigid, the calculation of terminal velocity is rather straightforward.
These are not unreasonable approximations when making rough calculations
for falling meteorites.

There is a convenient web page where one can simply plug in the relevant
values and have a computer do the calculation:

http://www.processassociates.com/process/separate/termvel.htm

If we assume a small meteorite with a typical chondritic density of 3.65
grams per cubic centimeter and a diameter of 2.5 cm (roughly one inch), we
come up with a terminal velocity of about 46.8 meters per second, or
roughly 105 mph. This meteorite would weigh about 29.8 grams, or roughly
one ounce. The question of how much damage a hard one-ounce object
traveling at 105 mph might do upon striking a human being is left to the
reader's own judgment and intuition. Personally I think it would hurt a lot
but would not necessarily require a trip to the emergency room, much less
the morgue. This is far from a "speeding bullet" velocity. It is not a lot
faster than a fast ball pitch and the object is a lot lighter than a
baseball -- batters routinely survive getting hit with a fast ball.

An iron meteorite of the same diameter with a density of 7.9 grams per
cubic centimeter would weigh about 65 grams, or about 2.3 ounces. The
terminal velocity would be about 69 meters per second, or roughly 155 mph.
I would not want to be in the way, but getting hit by even this falling
object would probably cause a lot less damage than a gunshot at close
range. Short of a hit on the top of the head, this is probably a survivable
encounter in most cases.

Somewhat counter-intuitively, it doesn't really matter whether these
meteorites had been falling through the air "from miles up" or only a few
hundred meters. Once the velocity approaches terminal velocity, which
happens in just a few seconds for small objects, there is no significant
further acceleration.

The 3 gram chondrite fragment which hit the boy at Mbale would have been
falling at approximately 33 meters per second, or about 73 mph. Once again,
I think it would have been painful if it had hit him directly, but accounts
stating that his life was saved because the meteorite was slowed down by
hitting a tree before striking him seem sensationalistic to me. Falling
stones that weigh only three grams are just not a big danger, regardless
of what height they fall from.

The Sylacauga, Alabama stone which fell through a house in 1954, hitting a
woman and severely bruising her, weighed a very substantial 3.9 kg (8.6
pounds). This is another situation entirely compared to small stones
weighing an ounce or less. Calculation of the fall velocity is left as an
exercise for the reader. If several layers of house construction hadn't
slowed the rock down, she would almost certainly have qualified for a very
unique epitaph on her gravestone.

Best wishes to all,

Piper

PS -- BTW, I don't doubt in the least that it is dangerous, irresponsible,
and idiotic to amuse oneself by shooting a gun into the air in populated areas.
Received on Sat 14 Sep 2002 03:59:45 PM PDT