Scaling of impact craters in unconsolidated granular materials
David R. Dowling and Thomas R. Dowling
Citation: American Journal of Physics 81, 875 (2013); doi: 10.1119/1.4817309
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NOTES AND DISCUSSIONS
A note on the gyromagnetic properties of the hydrogens
G. D. Severna)
Department of Physics, University of San Diego, San Diego, California 92110
J. P. Bolender
Department of Chemistry and Biochemistry, University of San Diego, San Diego, California 92110
(Received 5 June 2012; accepted 28 June 2013)
[http://dx.doi.org/10.1119/1.4813853]
How do NMR experiments work? This question is
addressed at the undergraduate level in both chemistry and
physics lectures and laboratories. In most descriptions,1 it is
taken for granted (without discussion) that hydrogen NMR is
proton NMR and how, for hydrogen, an external magnetic
field creates a two-state quantum system with an energy gap
simply and directly proportional to the value of the nuclear
magnetic moment. No mention is made of the magnetic
moment of the two electrons that make up the covalent bond
connecting the hydrogen atom to the (most often) carbon
backbone of the molecule. Nor is any mention made of the
complete cancellation of the electron-induced magnetic field
at the hydrogen nucleus; that is, the cancellation of the
hyperfine splitting (HFS) that is necessary for this technique
to yield such simple data. The chemist will examine this
point and say that the paired electrons are diamagnetic and
will not enlarge upon the meaning of this statement except to
say that the shift in resonance frequency is produced by the
distribution of electrons in proximity to this particular proton. This discussion leaves aside any mention of the quantum
physics that leads to the uncomplicated structure of the
NMR absorbance line. Finally, no mention is made of the
Pauli Exclusion Principle that demands that the entire wave
function be anti-symmetric with respect to exchange of the
electron particle identities. While the spatial wave function
is symmetric with respect to that interchange in order for the
electronic energies (electrostatic potential and the electron’s
kinetic energy) to possess a minimum leading to a chemical
bond, the spin wave function of the paired electrons must be
in an antisymmetric (singlet) state, which leads to the cancellation of the electron induced magnetic fields.
Essentially all discussions in texts and papers that describe
how these experiments work omit these considerations. We
believe a brief discussion of the implications of the Pauli
Exclusion Principle in connection with the physics of covalent bonds would clarify the context in which NMR experiments with the hydrogens can be simply understood,
especially with regard to the necessary disappearance of the
hyperfine splitting of the ground state of hydrogen. We begin
by reprising the typical explanation of NMR experiments.
Typical discussions of the NMR experiment as they are
presented to undergraduate physics and chemistry students
begin with the assumption that the hydrogen nucleus, by virtue of an externally applied magnetic field, is characterized
energetically as a two-state system corresponding to the nuclear spin arranged either “up” or “down” relative to the
external magnetic field. Thus, the energy is typically written
as
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Am. J. Phys. 81 (11), November 2013
http://aapt.org/ajp
E ¼ lp B0 ¼ cp hI B0 ;
(1)
where lp , I, and cp are the proton’s magnetic moment, nuclear spin, and gyromagnetic ratio, and B0 is the externally
applied magnetic field. Because I ¼ 1=2 for 11 H there are two
states separated by an energy gap that is resonant with an rf
photon of frequency x0 ¼ cp B0 . The resonance is achieved
experimentally either by scanning the value of the external
field B0 with precision coils while fixing the rf frequency or
by fixing B0 and scanning (or pulsing in Fourier Transform
NMR) the rf frequency. The energy of the absorbed photon
then furnishes a simple and direct measure of cp and lp —
this is the salient point of all descriptions of how NMR
experiments work. Many of these types of experiments
involve relaxation times as well; however, for the purposes
of this note, we will omit discussion of this detail. The main
point is that without the (unmentioned and unexplained) cancelation of the intrinsic magnetic field of the electrons, the
experiment would not work as described.
This cancellation is easily forgotten, perhaps because it is
perfect, but this is the very reason we think it is remarkable
and noteworthy. However, the omission may unintentionally
convey to students that the magnetic field due to a 1s,
unpaired, electron is small and negligible. The student is in
for a shock to find out just how large a magnetic field is
required to produce the hyperfine splitting of the atomic
hydrogen ground state. A phenomenological estimate of the
magnetic field at the location of the nucleus arising from the
magnetic moment of the electron Be from the known value
of the hyperfine splitting, taking Ehf ¼ 2lp Be as these quantities are customarily defined, gives Be ¼ 33:5 T, a truly
enormous value. For perspective, to produce an equivalent
field strength using a solenoid with a one-inch diameter bore
and a turns ratio of n ¼ 100 turns/cm would require a current
of I ¼ B=ðnlo Þ 200,000 A. Not bad for the tiniest magnet
in the universe. However, thinking about the magnetic field
at the location of the nucleus in atomic hydrogen in this phenomenological way does not help us picture how the electron
creates this magnetic field, nor how a paired electron in a
molecular orbital can cancel it out.
How can a particle with such a small magnetic moment
cause such a large magnetic field? The quantum picture of
the wave function of the 1s electron, not to mention the
ground state molecular orbital of H2, makes this difficult to
understand; perhaps this too is a reason for the omission. It
isn’t because the electron in the 1s state gets close to the nucleus, it is because the electron can be inside the nucleus;
indeed, the magnetic field responsible for the hyperfine
C 2013 American Association of Physics Teachers
V
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splitting of the hydrogen atom ground state, conspicuous by
its absence in hydrogen NMR, is created by the small portion
of the 1s wave function that actually coincides with the volume of the proton.
There is a classical analog to help picture this result (as far
as classical pictures can help). Think of a uniformly magnetized sphere of any magnetic material—ferromagnetic materials will do for this picture. The magnetic field within the
sphere is uniform within the volume, and parallel to the magnetization vector M, the dipole moment per unit volume of
the material. Then, imagine a slightly smaller sphere but
with an equal but opposite magnetization vector M that we
add concentrically to the first one. The magnetic field of the
combined distribution of magnetized material vanishes over
the space of their mutual interior. Thus, a spherical shell of
uniformly magnetized material creates no magnetic field in
its interior. Each sphere in this example creates a magnetic
field at its center, but the magnetic material must exist at its
center in order to produce a magnetic field there. This is
where the proton exists in the case of the hydrogen atom.
This is the classical analog to the quantum picture—the overlap of the electron’s wave function with the tiny volume of
the proton is responsible for the creation of the magnetic
field that gives rise to the hyperfine structure of the ground
state of the hydrogen atom.
What, then, is the appropriate quantum picture? The
spherical symmetry of the ground state hydrogen wave function leads to the view that only the overlap of the wave function with the volume of the nucleus can create this enormous
magnetic field. This is the so-called Fermi contact interaction.2 But we stress that the conceptual difficulty, as recently
pointed out by Bucher and others,2,3 is not with the interaction between the spin vectors of the two particles (lp le ),
but rather with their locations—the splitting arises from the
nonzero probability of the electron penetrating the inside of
the proton. Indeed the term “overlap” or “penetration” seems
more apt than “contact.” The conceptual difficulty in picturing this interaction is no less difficult in the case of a hydrogen atom covalently bonded in a molecule. The Pauli
Exclusion Principle demands, in the case that Chemist’s call
r bonding, that the two electrons form a singlet spin state
(i.e., an S ¼ 0 state). Again, as far as the interactions
between the spin vectors are concerned there is no conceptual difficulty in the vanishing of the HFS, except that the
spatial part becomes even more difficult to picture semiclassically. How can spatially noncoincident spins cancel
out? How can electrons, which must spend a fair amount of
time in between the nuclei creating the electrostatic potential
minimum required to form the bond, cancel their contributions to the magnetic field at the location of the nuclei? No
simple semiclassical picture is tenable here in which we
think of the electron as a point particle with an assignable
probability of being in various locations relative to the centers of the nuclei. Instead, if we think formally of the demand
of the Pauli Exclusion Principle that the total spatial wavefunction must be symmetric with respect to exchange of particle identities, we see that each electron must be present
within each nucleus in precisely the same way, one spin up
and one spin down, so their effects can cancel out in the
same way that the HFS is created in the first place.
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Am. J. Phys., Vol. 81, No. 11, November 2013
Chemists get at this sort of quantum fuzziness with respect
to the location of the electron in a molecule with the term
“delocalization.” Delocalization refers to the electrons within
the chemical bonds of a molecule as being spread out or
shared over the entire space of the molecular bond or molecule; that is, the electron is not associated solely with just
one nucleus. The most common example of delocalization is
the benzene molecule. In this molecule, the six p electrons
are all equally shared with the six carbon atoms that make up
the ring. Additionally, delocalization also occurs in r bonds,
where the electrons are found to encompass all locations
around the two bonded atoms, and in atoms, where the electrons are expressed as being in atomic orbitals. The concepts
of delocalization are most often exhibited in electron density
plots, which represent the volume where there is a 95% probability of finding the electrons (and which includes the
nuclei). These probability densities reflect the abilities of
various atoms to more strongly attract electrons within a
covalent bond (electronegativity), which indicates to a student that the electrons are more often “found” at or near a
particular nucleus.
Consider hydrogen in a covalent bond, say, with itself. In
the most simplistic descriptions, the pair of electrons are
“equally shared” by the two nuclei and students assume this
means the electrons are found “between” the two nuclei.
However, the probability distributions indicate larger electron
densities nearer the two nuclei. The Pauli Exclusion Principle
demands not only that the shared electrons have opposite
spins, consistent with the singlet spin state of the molecular
orbital, but also that the entire wave function be antisymmetric
with respect to the exchange of electron particle labels. This
implies that the spatial wave function be symmetric with
respect to interchange of particle labels, consistent with a nonvanishing wave function in between the two nuclei. While this
is consistent in some sense with student conceptions of possible locations of the electrons forming the bond, another implication is typically missed. Both shared electrons are always
within the volume of each proton, one spin up and one spin
down, perfectly canceling the very large magnetic field.
Wherever one of the 1s electrons can be, the other one is also
there in the opposite spin state, equally as often—which is
always—because the hyperfine splitting is always missing for
a covalently-bonded hydrogen nucleus (even when not
observed). This is the situation for all NMR experiments. We
think that it is too important, too interesting, and too curious—however difficult it is to picture—to leave out of the
story for either physics or chemistry students.
This work was supported by NSF Grant No. CBET0903832.
a)
Electronic mail: [email protected]
See, for example, A. C. Melissinos and J. Napolitano, Experiments in Modern Physics, 2nd. ed. (Academic Press, San Diego, 2003), Chap. 7, and D.
A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach
(University Science Books, Sausilito, CA, 1997) Chap. 14. These are excellent texts which we use in our courses. We allege no errors in them about
NMR experiments.
2
M. Bucher, “The electron inside the nucleus: An almost classical derivation
of the isotropic hyperfine interaction,” Eur. J. Phys. 21, 19–22 (2000).
3
D. J. Griffiths, “Hyperfine splitting in the ground state of hydrogen,” Am. J.
Phys. 50, 698–703 (1982).
1
Notes and Discussions
874
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Scaling of impact craters in unconsolidated granular materials
David R. Dowlinga)
Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109
Thomas R. Dowling
Skyline High School, Ann Arbor, Michigan 48103
(Received 24 February 2013; accepted 17 July 2013)
A simple experiment suitable for introductory physical science laboratory instruction involves
forming impact craters by dropping spherical objects into a bed of unconsolidated granular
material. The experiment is straightforward, and laboratory results for different impact energies,
spheres, and granular materials may be collapsed to a single power-law using parametric scaling
determined from dimensional analysis. The resulting power law is extrapolated over more than
16 orders of magnitude to produce an estimate of the impact energy that formed the 1.2-km-diameter
Barringer Meteor Crater in northern Arizona. VC 2013 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4817309]
I. INTRODUCTION
A general scaling law for the size of an impact crater made
by dropping a sphere into a deep bed of unconsolidated granular material may be determined from a combination of
dimensional analysis and simple experimentation. This paper
presents this combination and recommends it for instructional
purposes at the high school and undergraduate levels. It is a
follow-on study that complements prior reports on laboratory
crater-formation experiments.1,2 The new contributions here
are threefold, including a complete description of the parametric scaling of crater size, extensions of the prior experimental results to spheres of differing composition and to
different granular materials, and an extrapolation of simple
laboratory crater-formation results to obtain an independent
estimate for the impact energy that produced the 1.2-km-diameter Barringer Meteor Crater in northern Arizona
Dimensional analysis is a broadly applicable technique for
developing scaling laws, interpreting experimental data, and
simplifying the description of a wide variety of physical phenomena. It is routinely taught in college-level fluid mechanics courses3–5 and is naturally applicable in any branch of
physics. The fundamental basis for dimensional analysis can
be summed up as follows: the natural world functions without any knowledge of humankind’s units of measurement,
and therefore all meaningful physical laws can be stated in
dimensionless form. This simple concept constrains the possible combinations of boundary- and initial-condition parameters, material constants, and fundamental constants that
may appear in correct physical laws. Such constraints may
produce parametric simplifications that are not obvious.
Dimensional analysis has been used for such diverse tasks as
proving the Pythagorean Theorem6 and determining the yield
of the first atomic blast.7
For the present purposes, dimensional analysis provides a
means for organizing the experimental crater data so that students can directly test hypotheses about which characteristics
of the sphere and the granular material matter, and whether a
power-law scaling should apply between the rim-to-rim crater diameter and the kinetic energy lost by the sphere during
the impact. In particular, dimensional analysis combined
with the experimental results shows that only the impact
energy matters; the size of the falling sphere has no separate
importance.
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Am. J. Phys., Vol. 81, No. 11, November 2013
The experiments reported here are nearly identical with
those already reported1 but with a less refined experimental
technique. Notwithstanding, when data are plotted using
logarithmic axes, our final experimental findings remain
robust even when generated from low-precision instrumentation. Thus, the crater-formation experiment can be used in
undergraduate and high-school-level science investigations
(in fact, the experimental measurements provided here were
collected by the second author for a high-school science-fair
project).
II. SIMPLE DIMENSIONAL ANALYSIS OF IMPACT
CRATER FORMATION
The goal here is to use dimensional analysis to determine
a scaling law for D, the rim-to-rim diameter of the crater
formed from the impact of a vertically falling sphere on the
flat horizontal surface of a deep bed of granular material
(see Fig. 1). When the impact energy is low enough, heat
transfer, chemical reactions, vaporization, fragmentation,
and melting (i.e., alteration of matter) of the sphere or the
impacted material will be unimportant. Thus, the dependent
parameter D is anticipated to depend on four independent parameters: E, the energy lost by the sphere during the impact;
d, the sphere diameter; q, the density of the impacted material; and g, the gravitational field strength at the surface of
Earth. This listing of parameters explicitly excludes a
strength modulus or fracture toughness for the granular material. Such exclusion implies the granular material is unconsolidated and grains may separate without expenditure of
energy.
Based on the Buckingham pi-theorem,8 these five parameters (D, E, d, q, g) may be reduced to 5 – 3 ¼ 2 dimensionless
groups (called pi-groups) because these parameters embody
3 independent dimensions (mass, length, and time). In this
case, the two dimensionless groups may be found by inspection. The first group traditionally includes the dependent parameter D. Here, D is readily combined with d to form the
first dimensionless group: D/d. Only two (E and q) of the
remaining parameters contain units of mass, so these must
appear as a ratio E/q, which has units of length5/time2. The
units of time may be eliminated from this ratio via division
by g, giving the quantity E/qg, which has units of length4.
One further division by d4 then yields the second
Notes and Discussions
875
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Table I. Characteristics of the spheres used in the impact experiments.
Diameter (60.5 mm)
Fig. 1. Cross-sectional view of an impact crater of diameter D formed by a
falling sphere of diameter d that loses kinetic energy E during the impact.
Here, g is the gravitational field strength and q is the granular material
density.
dimensionless group: E/qgd4. These two dimensionless
groups must form the scaling law for D, given by
D=d ¼ f ðE=qgd 4 Þ;
(1)
where f ðxÞ is an unknown function that can be determined
from crater-formation experiments. Interestingly, if the
unknown function has the form f ðxÞ / x1=4 , then Eq. (1)
implies that the crater diameter D is independent of the
sphere diameter d. Alternative crater scaling approaches that
include additional properties of the impacted material and
energy source are available elswhere.9,10
From an instructional standpoint, the student effort needed
to reach Eq. (1) may be modest. When given the parameter
list, an undergraduate student who has successfully worked
several dimensional analysis homework problems can be
expected to correctly reach Eq. (1) in less than 10 min. The
effort to reach Eq. (1) (or its equivalent in more complicated
situations) can be increased by asking students to develop
parameter listings from scratch or from a partial listing provided by the instructor. When students generate their own
versions of Eq. (1), the experimental results can be used to
identify spurious parameters. Alternatively, high school science students could be given Eq. (1) as a hypothesis and
asked to experimentally determine its validity and/or the
unknown function f.
10.
13.
15.
25.
25.
34.
43.
Mass (60.5 g)
Description
8
14
4
8
64
21
45
Steel
Steel
Hard plastic
Elastic plastic
Steel
Elastic plastic
Golf ball
surface is returned to the granular material bed by manual
agitation of the mixing bowl.
IV. RESULTS AND DISCUSSION
To illustrate the experimental results, an image of a laboratory crater is shown in Fig. 2. This particular laboratory
crater has a diameter of D ¼ 5.7 cm and was formed by
dropping the 14-g steel sphere from a height of 116 cm
(E ¼ 0.16 J) into playground sand. For comparison, an aerial
picture of the Barringer Meteor Crater (Arizona, USA) is
also shown in Fig. 2.11 This impressive natural crater has a
diameter of 1.2 km and was formed by a meteor impact
involving at least one hundred times more energy than the
Hiroshima atomic blast (around 60 TJ).
Scaled results from experiments involving all seven spheres,
all four heights, and both granular materials are shown in Fig.
3. This figure shows a log-log plot of the (dependent) dimensionless group D/d as a function of the (independent) dimensionless group E/qgd4. The solid upward-sloping line has a
slope of 1/4 and was visually fit to the measurements. The
III. APPARATUS AND EXPERIMENTS
The experiments conducted for this study are described
elsewhere,1 so the description here will be brief. The basic
experimental apparatus includes seven different spheres (see
Table I) and two granular materials. The granular materials
are playground sand with density 1.60 6 0.07 g/cm3 and
granulated sugar with density 1.00 6 0.07 g/cm3. The impact
energy E for any crater-forming experiment comes from converting gravitational potential energy into kinetic energy that
is lost on impact. Thus, a sphere of mass ms dropped from a
height h above the undisturbed granular material surface will
have an impact energy E ¼ msgh. In these experiments the
spheres are dropped by hand from four nominal heights
(24 cm, 56 cm, 116 cm, and 257 cm; heights of repeated trials
fall within 62 cm of these nominal values) into the central
50% of a 32-cm-diameter semi-hemispherical mixing bowl
filled to a central depth of approximately 8 cm with granular
material. Crater diameter measurements varied between
2.9 cm and 9.8 cm (610%), obtained by viewing the crater
through a clear plastic ruler. After an experiment, the sphere
is removed from the crater and a (nominally) flat horizontal
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Am. J. Phys., Vol. 81, No. 11, November 2013
Fig. 2. Impact crater images. An experimental crater in playground sand
with a diameter of 5.7 cm (top) and the Barringer Meteor Crater in northern
Arizona with a diameter of 1.2 km (bottom, credit: U.S. Geological Survey/
photo by D. Roddy).11
Notes and Discussions
876
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dashed line segment with the same slope is the data-fit line
from the prior report1 assuming the white silica sand used there
has the same density as playground sand. For clarity, uncertainty error bars are only provided for the measurement having
the largest D/d value as a visual example. The horizontal line
segment in the upper right of Fig. 3 represents scaled results
from the Barringer Crater using the density of Coconino sand
stone from the impact site (2.26 g/cm3),12 a crater diameter of
1.2 km,13 an equivalent sphere diameter of 40 m,13 and the
smallest (2.5 mega-tons of TNT)13 and largest (20 mega-tons
of TNT)14 readily available estimates of the impact energy that
formed the Barringer Crater.
In spite of the scatter in the data, Fig. 3 suggests a few interesting conclusions. First, the data follow a 1/4-power law and
the fit-line from the prior laboratory crater formation results1
falls within the scatter of the current measurements. The prior
and current fit lines would coincide if the white silica sand
used in the prior study is 25–30% less dense than the current
playground sand. Second, when anti-logs are taken, the fit line
determines the unknown function f in Eq. (1) to be
D=d ¼ 0:97ðE=qgd4 Þ1=4 ;
(2)
D ¼ 0:97ðE=qgÞ1=4 ;
(3)
or
thus completing this combined theory-experiment exercise.
Third, the dimensional-analysis- derived scaling of E with q
shown in Eq. (2) appears to be correct because there is no
discernable separation of the playground-sand (darker symbols) and granulated-sugar (lighter symbols) crater data.
Fourth, the experimental and dimensional analysis results together show that the size of the crater does not depend on the
size of the impacting mass—the parameter d disappears from
Eq. (2).
Finally, Eq. (3) can be solved to give the energy in terms
of the physical characteristics of the impact site as
E ¼ qgðD=0:97Þ4 :
(4)
This equation provides a novel means to estimate the
Barringer-Crater impact energy. The energy determined
from this extrapolation (E ¼ 5:2 1016 J, or 12 mega-tons of
TNT) falls within the range of Barringer-Crater impact
energy reference values. Given that the experimental impact
energies ranged from 0.009 to 1.6 J, the quantitative agreement of this extrapolation to an impact energy more than 16
orders-of-magnitude larger is remarkable. There are, of
course, differences between the laboratory crater-forming
experiments and the physical processes that formed the Barringer Crater. For example, the meteor is believed to have
struck the ground at a 45 angle13 (not perpendicular), and
the tremendous energy released on impact likely caused
additional phenomena not present in the laboratory experiments including cracking, fragmentation, deformation, and
melting of the meteor and the Coconino sandstone. Moreover, the present extrapolation is only valid for impacts in
unconsolidated granular material where grains separate without energy expenditure, a requirement that may not be valid
in the case the Barringer Crater. However, the coincidence
of the extrapolated E value with the range of BarringerCrater reference values suggests that the importance of these
additional phenomena for crater size determination is small
compared to the physical process of granular material displacement produced by deceleration of a falling object, a
conclusion that conforms to observations from field work at
the impact site.13
V. CONCLUSION
Simple laboratory crater-formation experiments can be
combined with dimensional analysis into a rich instructional
opportunity that should appeal to high school and undergraduate physical science students. The dimensional analysis component allows parametric dependencies beyond
impact energy to be investigated, and the successful
Fig. 3. Experimental crater-diameter results in the dimensionless form of Eq. (1). The solid line has a slope of 1/4. Representative error bars are shown for the
measurement with the largest value of D/d. Reference results for the Barringer Meteor Crater (more than 16 orders of magnitude greater impact energy) are
shown at the upper right.
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Am. J. Phys., Vol. 81, No. 11, November 2013
Notes and Discussions
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extrapolation of the laboratory results to the Barringer Crater is the type of result that intrigues and delights scientists
of all ages.
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
Joseph C. Amato and Roger E. Williams, “Crater formation in the laboratory: An introductory experiment in error analysis,” Am. J. Phys. 66, 141–
143 (1998).
2
S. Kasas, G. Dumas, and G. Dietler, “Impact cratering study performed in
the laboratory without a fast recording camera,” Am. J. Phys. 68, 771–773
(2000).
3
Robert W. Fox, Philip J. Pritchard, and Alan T. MacDonald, Introduction to Fluid Mechanics, 7th ed. (John Wiley, New York, 2009), Chap.
7.
4
Bruce R. Munson et al., Fundamentals of Fluid Mechanics, 7th ed. (John
Wiley, New York, 2013), Chap. 7.
5
Pijush K. Kundu, Ira M. Cohen, and David R. Dowling, Fluid Mechanics,
5th ed. (Academic Press, Boston, 2012), pp. 21–30.
1
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Am. J. Phys., Vol. 81, No. 11, November 2013
6
Grigory I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics (translated from Russian by Norman Stein, Consultants Bureau,
New York, 1979), p. 23.
7
Geoffrey I. Taylor, “The formation of a blast wave by a very intense
explosion. Parts I and II,” Proc. Roy. Soc. London Ser. A 201, 159–174
and 175–186 (1950).
8
Edgar Buckingham, “On physically similar systems; Illustrations of the
use of dimensional analysis,” Phys. Rev. 4, 345–376 (1914).
9
Donald E. Gault et al., “Some comparisons of impact craters on Mercury and
the Moon,” J. Geophys. Res. 80, 2444–2460, doi:10.1029/JB080i017p02444
(1975).
10
Albert J. Chabai, “On scaling dimensions of craters produced by buried
explosives,” J.
Geophys. Res. 70, 5075–5098, doi:10.1029/
JZ070i020p05075 (1965).
11
See pictures at <http://www.solarviews.com/cap/earth/meteor.htm/> for
example.
12
Eugene M. Shoemaker et al., “Hypervelocity impact of steel into Coconino sandstone,” Am. J. Sci. 261, 668–682 (1963).
13
H. J. Melosh and G. S. Collins, “Meteor crater formed by low velocity
impact,” Nature 434, 157 (2005).
14
Meteor Crater Visitor Center website <http://www.meteorcrater.com/>.
Notes and Discussions
878
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