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Formation of Impact Craters in Sand
Connor Barclay and Sharif Alkhass,
University of Kent
Abstract: The diameter of craters formed by spherical ball bearings of varying mass
dropped into sand at low velocities was studied, along with the effect of altering the
angle of the impactor. The relationship between the diameter of the crater formed and
the total energy of the projectile at impact was found to be similar to a scaled down
version of the equation for universal meteor craters. The relationship is shown to be a
power law where n is equal to 0.2462.
Introduction: The impact of asteroids, comets, and meteoroids onto the surfaces of
planets is a significant geologic process and for the purposes of this experiment the
focus is on studying the effect of cratering from meteors. Meteors are small chunks of
rock debris from our solar system that have been caught by the pull of a planet’s
gravity and are falling through the atmosphere; burning up in the process. They can
range anywhere from dust particles up to 10.0m in diameter, if they survive the
impact event they are known as meteorites [1]
.
Cratering formed by meteor impact events tend to be spherical excavated
holes with elevated rims around the edges since as the meteors are passing through an
atmosphere a significant amount of its body is burned away due to friction and on
most occasions is reduced to a spherical chunk of rock by the time it hits the surface
[2]
. When the planetary surface is hit with the impactor (meteor), a shock wave begins
to flow through the target, starting below the surface. This sends target material,
known as Ejecta [3]
, flowing upwards and away from the collision site. This material
then falls back down to the surface of the target, surrounding the hole from which it
was ejected. Layers of material that were initially below the surface of the target are
now exposed and transported up to lie around the impact site. Perhaps the most
famous cratering event to happen on Earth is Chicxulub Crater located in the Yucatan
Peninsula, Mexico (see Image 1). The crater is Earth’s largest impact occurrence at
180km in diameter and it is theorised by many palaeontologists that it caused the
Cretateous-Paleogene extinction of flightless dinosaurs [4]
.
Image 1.
Crater size overlaid on top of satellite image of Mexico, courtesy of:
http://www.esri.com/~/media/Images/Content/news/arcnews/spring14/p20p2-lg.jpg

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When choosing to research meteor impact craters the lunar surface of the moon is far
more preferable to study as Earth’s craters suffer from constantly active geological
environments such as plate tectonic shifts, tidal forces and interfering atmospheric
conditions. Because of this the moon has an ideal surface to investigate as 99% of the
top regolith has been untouched for nearly 3 billion years perfectly preserving meteor
impacts as early as the moon’s formation [5]
.
The size of an impact crater caused by a meteor is proportional to the total energy the
meteor has as it strikes the surface. Thus the greater energy the impactor retains, the
larger the meteor crater will be hence the equation relating the kinetic energy of a
meteor at impact and diameter of the resulting crater on earth is:
𝐷 = 𝑘 ∙
!!"!#$
!.!!"#$%&
!
!
(1)
Where D is the diameter of the crater, k is constant, E is the total energy of the
meteor at impact, 𝜌 is the density of the meteor and gplanet is the acting gravity on the
meteor. Equation (1) describes low velocity meteor collisions with a planetary
surface where k is to be found; the equation has been derived by using the function of
variables method [6]
. The function of variables derivation method relates variables
together by their units of measurement hence it follows that:
ℎ ∝ 𝐷 (2)
Where h is the height (depth) of the crater measured in meters and D is the diameter
of the crater also measured in meters. Since they are both measured in meters and to
the power of 1 they are as a result proportional to each other. Following on from
equation (2) we know that the volume of a hemi-spherical shape is proportional to the
Diameter3
.
𝑉 ∝ 𝐷!
(3)
The volume is recorded in meters3
hence in order for the diameter to proportional it
needs to be cubed. The following relationship between the meteors mass and the
diameter is shown to be:
𝑚 ∝ 𝜌. 𝐷!
(4)
Where m is mass of the meteor, 𝜌 is the density of the meteor. The proportionality
holds true when deriving using the function of variables method since:
𝑘𝑔 ∝
!"
!!
∙ 𝑚!
(5)
This equation shows that deriving equation (4) by looking at the variable’s units of
measurements, we can see that the crater diameter is proportional to the mass of the
meteor when it multiplied by the density. This is because the units of the product
𝜌 ∙ 𝐷!
reduce to just kg, the same as mass’ units.

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Finally to find the Diameter of the crater as a function of the impactor energy we use
the following equation:
𝐸!"!#$ = 𝑚𝑔!"#$%&ℎ =
!
!
𝑚𝑣!
(6)
Where Etotal is the total energy of the impactor, gplanet is the acceleration due to
gravity of the particular planetary body and v is the velocity of the impactor. To
relate the total energy to the crater diameter, sub in equations (2), (3) and (4) into
equation (6) to get:
𝐸!"!#$ ∝ 𝜌𝐷!
∙ 𝑔!"#$%& ∙ 𝐷 (7)
⟹ 𝐸!"!#$ ∝ 𝜌𝑔!"#$%& 𝐷!
(8)
Which then rearranges to yields equation (1), the diameter of a meteor crater given as
a function of the energy of the impactor using the function of variables derivation.
⟹ 𝐷 = 𝑘 ∙
!!"!#$
!!!"#$%&
!
!
(1)
⟹ 𝐷 ∝ 𝑘 ∙ 𝐸!
(9)
Equation (1) can be further simplified since gravity remains constant and the mass
and density of the meteor are proportional to the energy the equation can be further
reduced to equation (9). It is predicted that the diameter of a meteor crater is
proportional to the energy of the meteor raised to the power of 0.25 however due to
the macroscopic size of the meteor compared to ball bearings it is unlikely that
relationship will be applicable. The relationship between the kinetic energy of the ball
bearing and the crater diameter should be proportionate to a power relation, but with
an exponential with a comparable value from that deduced in equation (8).
This research will attempt to verify the relationship between the energy at impact of
multiple ball bearings dropped from a increasing heights into sand and flour; and the
diameter of the crater formed. With the constant of proportionality k to be found,
which will act as the scaling relationship for theoretical meteor craters.
Methodology:
Four ball bearings of varying size and different mass were used, as seen below in
image (2) a large basin of sand was prepared so that any wall effect where the rim of
the crater was not disturbed during impact. The sand was levelled into a flat surface
before the ball was released by running a ruler across the top of the sand. A stand
with a clamp attached was placed along side the basin edge so that the ball could be
released vertically over the sand, the clamp thusly allowed each ball to be dropped
simultaneously without gaining any spin as a consequence. The vertical drop height
was increased in increments of 0.1m above the sand; this was monitored by having a
ruler held fixed from on the top layer of sand that was held in place by another stand
with an attached clamp. After the ball impacted the sand, the diameter of the crater

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was measured using a vernier calliper for an increased accuracy and smaller error
margins. The diameter of the rim of the crater was measured.
Image 2: Experiment (1) setup Image 3: Measuring of crater diameter
After measuring the diameter of the crater, the sand was flattened again and the same
procedure was repeated. For each ball, a total of 3 repeat readings were taken to
quantify random errors and improve the accuracy. Measurements for the height of the
drop, the time taken to impact, the average velocity at impact and the average kinetic
energies were also measured. Throughout the experiment, the same type of sand was
used, and the whole experiment was conducted in the laboratory room over a three-
week period. A following study was carried out before the second section in the form
of an investigative experiment into understanding how such planetary impacts may
look on a more accurate representation of the lunar regolith (Latin for blanket-rock)
[7]
. This experiment takes into account that the moons surface density increases from
1.35gcm-1
beyond 0.3m so in the following experiment repeats the procedure in
experiment (1) however a top layer of plain powdered flour is used [8]
.
Image 4: Experiment (2) setup
The second section of experiments followed a similar setup as to experiment one with
the exception that since these sets of trials were to determine the effects of crater
diameter at varying impactor angles. To achieve this hollow plastic tubing of
1.0meter in length was structured at multiple angles and was held in place by two
clamps on stands at either end of the tubing. The end of the tubing was positioned so
that the tip rested on the top layer of sand then before the ball was dropped it was
made smooth by sliding a ruler across the surface.

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Results:
Ball No.
Mass
±0.01g Crater Diameter ±0.1mm
Average Diameter
±0.3mm
Average Kinetic Energy
±2.05mJ
Ball 1 58.90 76.8 70.4 74.5 73.9 127.41
Ball 2 32.58 56.0 62.8 62.0 60.3 89.48
Ball 3 4.46 41.1 45.9 41.8 42.9 10.39
Ball 4 19.44 66.1 67.0 60.1 64.4 54.24
Table 5: Mass of ball bearings and diameters of the resultant craters
Whilst the mass of the ball bearing may not be directly proportional to the diameter
of the crater formed, it is clear that there exists a correlation between them as seen in
equation (4). The experimental results as seen in table(5) reflect this, since when the
mass of the balls are increased the crater diameters expand with it; the repeat readings
indicate that this statement is accurate and takes into account any random errors that
may have arisen during practise.
The uncertainty provided for the ball mass and crater diameter as the precision of the
scales and vernier callipers respectively however the error margins deduced for the
average kinetic energy were found by multiplication of errors:
= (±𝒆𝒓𝒓𝒐𝒓 𝒊𝒏 𝒎𝒂𝒔𝒔 + ± 𝒆𝒓𝒓𝒐𝒓 𝒊𝒏 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 !
)
⟹ = (±0.01𝑔 + ±0.02𝑚𝑠!
+ ±0.02𝑚𝑠!
)
⟹ = ±0.05𝑚𝐽
Although the error margins for the kinetic energy of the ball at impact (the surface of
the sand) has been estimated as a negligible value of ±0.05𝑚𝐽, it assumed that no
energy is lost due to air resistance. In section 1’s experiments, the depths of the
craters were not measured; therefore the results do not reflect the total energy down
to the bottom of the crater. Since the ball ended up between 1 and 2 cm below the
surface of the sand, this might cause an error margin of around ±2.00𝑚𝐽 in our
results hence given a total error of ±2.05𝑚𝐽.
As seen in graph 1 on the next page, a trend line has been inserted for the results of
table 1 which illustrates the relationship between the kinetic energy of the projectile
against crater diameter when ball mass varies from a fixed height. The trend line
predicts that the Energy is related to the Diameter by a power, n of 0.2 to 2 𝑠. 𝑓. The
power law curve does not fit into the error bars of all the data points but this is due to
the fact that the error margins for the diameter of the crater were negligible since an
instrument with a high degree of accuracy was used.
The value of 0.2 is very close to the original 0.25 as stated in equation (1) and since
the experiment is dealing with a scaled down model of crater impacts the value itself
seems very reasonable.

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Graph 1: Crater Diameter as a function of the kinetic energy of the ball bearings with a fitted power
law curve
Graph 3: Crater Diameter as a function of the Kinetic Energy of the Ball Bearings with a fitted power
law curve
In this graph the mass and drop heights are dependent variables to be measured, as
both are proportional to energy in the equation:
𝐸!"#$%#&'( = 𝑚. 𝑔. ℎ (10)
As you can see in graph (3) the power law curve fits much closer to the data points as
more variables have been taken into account thus providing a more precise result of:
⟹ 0.2462 ≈ 0.25 𝑡𝑜 2 𝑠. 𝑓
y
=
k.x0.1982
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
Average
Crater
Diameter
±0.3mm
Average
Kinetic
Energy
±2.05mJ
Relationship Between Kinetic Energy vs Crater Diameter When Impactor
Mass Varies with A Constant Drop Height
y
=
k.x0.2462
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
0
20
40
60
80
100
120
140
160
180
Average
Crater
Diameter
±0.3mm
Average
Kinetic
Energy
±2.05mJ
Relationship Between Kinetic Energy vs Crater Diameter When Both Impactor
Mass And Drop Height Are Variables To Be Changed
Ball
1
Ball
2
Ball
3
Ball
4

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Angle 𝜽 ±5.0
Minor Axis
±0.1mm
Major Axis
±0.1mm
Trail Length
±0.1cm
Average
𝚫𝑻 ±0.01s
Average
Velocity
±0.01m/s
Energy
±2.05mJ
30.0 47.7 48.7 77.6 76.8 42.50 40.00 0.69 1.43 19.84
40.0 53.4 56.2 85.2 91.1 33.00 34.00 0.67 1.11 11.95
50.0 57.1 66.8 93.9 72.8 32.90 27.00 0.62 1.56 23.61
60.0 56.7 66.4 77.7 70.1 20.50 19.50 0.53 1.79 31.08
70.0 65.8 72.4 77.3 77.8 0.00 0.00 0.59 1.30 16.39
80.0 59.7 62.2 63.9 62.3 0.00 0.00 0.38 2.62 66.58
Table 8: Showing the Relationship between the crater diameters and the Kinetic Energy when the
angle of impact is changed in increments of 10.0 ±0.1degrees using ball 4
The results obtained in table (8) describe what was found during the series of
experiments carried in the second section where the changing impact angle variable
was introduced. What it is noticeable first during the experiments is that the crater
formed as a result is no longer circular but elliptical in shape, see lab book for a
sketch a of the new crater shape. This means that two different diameters are formed
as a direct consequence of this, in which the minor axis (smallest width) is the
diameter in the plane perpendicular to the direction the ball travels and the major axis
(largest diameter) is formed in the direction of travel.
Since the direction the ball travels prior to impact is along an axis and the surface of
the sand is flattened, the eccentricity remains constant and can be ignored when
determining the real diameter and because of this the minor axis can be thought of as
the circular diameter of the crater since the resultant direction of travel for the
impactor is constant at 90.0 degrees to this axis [9]
. As you can see from table (8) the
crater diameters and the kinetic energy at impact both remain proportional to each
other however the energy also is indirectly proportional to the impact angle. The
image below explains that as the impact angle steepens from 0.0-90.0 degrees, the
kinetic energy gained is increased due to the fact that as the ball is dropped closer to
90.0 degrees more resultant gravitational potential energy it contains as ultimately the
height of the drop is increased.
Y
X
Image 5: showing the increase in impact angle towards the y-axis in increments

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Graph 4: Showing the resultant average diameter of the minor axis as the angle of the ball bearings
impact is changed
The major axis is clearly seen to be of a larger magnitude diameter compared to the
minor axis due to the fact that the impactor retains as great enough x-component of
the velocity at impact. This was seen to occur when at angles less than 70.0 degrees,
see table (8), to remain in motion after excavating a crater, which caused the rim to
extend in the direction of travel as the force of the ball pushed the sand out of the way
until the friction caused from travelling across the sand caused the ball to stop. What
was also seen during the experiments is that as the angle of impact steepens, the trail
length steadily receded towards the crater until around at around the 70.0 degree
margin the impact becomes too sheer for the ball to exit the crater at which point the
trail length was reduced to zero.
From graph (4) it is seen that as the angle of the projectile is increased vertically there
is an increase in the diameter of the minor axis that is due to the fact that as the angle
becomes more vertical there becomes a more even circular spread of the force of
impact. The impact angle has a large error margin of ±5.0 degrees despite the
protractor having a precision of ±1.0 degrees because of a human error where the
angle had a tendency to move slightly as the clamp that was holding it in place was
tightened. From graph and the experiment it was clear that the angle of impact holds
some proportionality to the diameter of the formed crater so if the angle is factored
into equation (1) and since the angle holds no units, you get:
𝐷 ∝ 𝐸!
∙ 𝑆𝑖𝑛 𝜃 (11)
‘n’ is the value of the power law curve, which was extrapolated from the results to be
0.2933 ≈ 0.29 to 2s.f; this power value is +0.05 from the theorized value in equation
(1) and is mostly probable due to the human error from measuring the angle.
y=k.x0.2933
40
45
50
55
60
65
70
75
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
Average
Minor
Axis
±0.2mm
Impact
Angle
±5.0
degrees
Relationship Between Impact Angle vs Minor Axis Diameter When
Impactor Mass Is Constant with A Changing Impact Angle

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Conclusion:
When determining equation (1) during the experimental process it was found that
through using the functions of variables method, the equation determined that the
diameter was found to be proportional to a projectile’s energy increased to the power
of 0.25. In Section one multiple ball bearings were dropped vertically where both
mass and drop height were variables to be altered, the subsequent power law curve
for the energy of the ball vs. the crater diameter produced a power of 0.2462 only
0.004 away from the true value thus suggesting a high validity in the equation
derived. Further research into section ones shows that the experiment could utilize a
much wider variety of masses for the ball bearings to supplement the result whilst
additionally testing the equation for reliability at increased ranges in kinetic energy.
A significant weakness in this experiment was that the depth of the crater could not at
all be measured accurately; had the plaster of Paris been in stock when it was
requested then molds of the excavated crater could have been taken then once the
plaster had dried it could’ve been removed and measured with a vernier caliper to
achieve a precise reading without disturbing the crater. When the experiment that
made use of both sand and flour to act as a replica moon surface was carried out,
what was found was that the crater rim became much more visible after impact
because the powdered flour acted as a good contrast against the sand. Taking this into
consideration any future experiments could include a thin top layer of flour to help
identify the edges of a crater since the added flour was found not to significantly alter
the crater size itself.
For the last part of this research the angle was altered so that the ball bearing hit the
sand at intervals between 30.0-80.0 degrees and it was found that the crater became
elliptical in shape followed by a trail in the sand as the ball slows to a stop, from the
plotted graph of angle vs. minor axis diameter a power law value for n was given as
0.2933 only 0.0433 above the theoretical value suggesting that the result obtained is
accurate however there are clear errors present. To improve this experiment a better
method of securing the plastic tubing is needed to remove the human error of
maintaining the tube at a constant angle. Furthermore during the experiment the balls
that weighed less than roughly 20.0g failed to cause a cratering effecting when hitting
the sand at an angle and simply rolled across the surface so in further tests heavier
balls should be made use of to create the desired effect of cratering.

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References:
[1] http://www.sciencekids.co.nz/sciencefacts/space/cometasteroidmeteoroiddifferences.html [21 Nov 2014]
[2] Title: A Comet Model. II. Physical Relations for Comets and Meteors.
Authors: Whipple, F. L.
Journal: Astrophysical Journal, vol. 113
[3] NASA’s office of Space Science and Public Library
[4] http://en.wikipedia.org/wiki/Chicxulub_crater [21 Nov 2014]
[5] https://www.aiaa.org/uploadedFiles/Education_and_Careers/STEM_K-
12_Outreach/Kids_Place/Solar_System_and_Planets_Activities/Making%20Deep%20Impact%20Craters.pdf [23
Nov 2014]
[6] Title: Functions of a Real Variable: Elementary Theory
Authors: N. Bourbaki
Published (2004)
[7] http://www.Wikipedia.com [29 Nov 2014]
[8] http://www.Wikipedia.com [29 Nov 2014]
[9] http://www.lpi.usra.edu/meetings/lpsc2013/pdf/1455.pdf [1 Dec 2014]
Image 1. http://www.esri.com/~/media/Images/Content/news/arcnews/spring14/p20p2-lg.jpg [2 Dec 2014]
Image 2/3/4. Extracts from Laboratory Book