HOW FAST DOES IT FALL?
(C)1999 Alan M. Schwartz
It is my passion shared with a select few (who also manage to
avoid forensic psychiatric evaluation) to toss bottles bearing
messages into the Pacific Ocean. Succeeding at voyages to the
Far East requires catching the Alaska current flowing about 50
miles west of Point Conception, California. It sustains 3-5
knots south to Baja, Mexico and then sweeps west. We fly west
out of Santa Barbara in small planes and drop sealed champagne
bottles from 8500 feet. What is our bottles' impact velocity?
Given an 8.6 cm diameter spherical bottle of mass 950 grams,
http://www.processassociates.com/process/separate/termvel.htm
gives a terminal velocity (weight=drag) of 172 mph and a Reynolds
number of 4.5x10^5 (turbulent). This is pleasant but irrelevant.
Our bottle has a density of 0.85 gm/cm^3 and a sphere at the same
diameter and mass is much denser, volume=[(4/3)(pi)r^3]. WHAMMO!
It does lend a reference point to scale further calculations.
The weight of an object is its mass times the acceleration of
gravity, mg. The drag of an object is (1/2)[(rho)(Cd)Av^2] where
m =0.95 kg, loaded and sealed champagne bottle
g =9.8 m/sec^2
rho=density of air, 1.213 kg/m^3 (68 degrees F and moist - ocean)
Cd =coefficient of drag (measured fudge factor)
A =projected area, (0.01958 m^2 horizontal, 0.005811 punt end)
v =velocity, m/sec
Plug in the Internet answer and find Cd(sphere)=0.446. This is
reasonable for the given Reynolds number. If the sphere works,
then Cd(cylinder)=1.1 to give a horizontal champagne bottle
terminal velocity of 70 mph ("Fluid Mechanics," 4th Ed., Binder
and Raymond, 1962). Alas, the bottle does not fall this way.
Punt end (heavy end) down seems plausible. Hydrodynamics texts
are punctilious in their avoidance of the Cd of a cylinder with
long axis parallel to fluid flow (but go orgasmic if its long
axis is perpendicular to flow). What dysfunctional cognitive
impairment would fire a flat-nosed projectile? Visit your
neighborhood university library and skim hydrodynamics texts
(bunched together with Library of Congress cataloging) looking
for one displaying military data. "Elementary Fluid Mechanics,"
2nd Ed., Vennard, 1951 has drag coefficients for flat-, round-,
and pointy-nosed projectiles to Mach 4. Given a Cd of 0.64 a
champagne bottle falling punt end down comes in at 144 mph.
Though the bottle could fall this way it is a metastable
configuration - a saddle point. Any minor perturbation will
launch it toward its equilibrium orientation.
A loaded and sealed bottle floats horizontally. Drag force is
proportional to projected area. Drop a champagne bottle
horizontally and its narrow neck intercepts less wind force than
its fat bottom. The bottle rotates until its narrow top is
bottommost, mostly. With pointy or rounded nose, a subsonic
projectile with a flat bottom has Cd=0.18. A pimple up front
makes a big difference! (At supersonic velocities pointy noses
have respectably less drag. Marginally improve things further by
reshaping the cylindrical wall perpendicular to the flat backside
as a slight boat tail taper.)
Do not be standing under one of our bottles looking into the sky
with your mouth open. Our baby will be screaming toward the big
blue marble at 270 mph (though it probably orbits about its
centerline as the wind buffets it and loses a few mph). A
spreadeagled human falls at about 130 mph. A plastic shopping
bag with its corners cut off and handles slit acting as a four-
line parachute would slow the bottle to about 40 mph.
What happens when a stoppered champagne bottle hits water stopper
first at 270 mph? At least some of them survive because we get
messages back from the Philippines and Malaysia among other
Pacific Rim backwaters. One could plug in a new density for the
drag equation (1020 kg/m^3 for salt water), correct weight for
buoyancy, turn the crank, and get a silly answer (the bottle
floats). Better might be Isaac Newton's blunt penetration
equation alleging that an object of density d1 and length L
penetrating into a medium of density d2 propagates a distance of
(d1/d2)L (it displaces its own mass). Go ahead, work it out.
The sealed bottle is 31.5 centimeters long. Figure out the time
of deceleration and then the deceleration itself. Them's a whole
lotta gees, eh?
Champagne bottles survive tremendous grief. The ones used to
launch ships are deeply multiply scratched to weaken them. There
is at least one movie of a ship's christening where the scratches
were omitted. The lady whams it and it bounces. She does it
again and again. The gent behind her finally grabs the thing and
WHAMMO! It bounces. Despite evidence that it might not be
necessary, from now on we will use parachutes.