International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 12, December 2013
ISSN 2319 - 4847
Ricochet Angle for Armament Shapes
Vijayalakshmi Murali1, S. D. Naik2,
1
Department of Engineering Mathematics, MESCOE (Wadia), University of Pune, India.
Applied Mathematics and Reliability Group, Armament Research & Development Establishment (ARDE),
Defence Research and Development Organisation (DRDO), Pune, India.
2
Abstract
Research has established a critical angle determining water ricochet of projectiles. The first part of the paper deals with the
formulation of an analytical model to estimate the critical ricochet angle, and applying it for real world armament shapes. The
shapes under consideration are Power Series (0.5 and 0.75), HAACK Series and Elliptical Series. The effect of nose shape on
ricochet angle is studied and simulation is done by varying different parameters like mass, diameter and geometry of the
projectile and the critical angle is found to vary as a dimensionless parameter μ.
Keywords: Ricochet, Projectile, Critical angle and nose shape.
1. INTRODUCTION
May 1943 can be considered as the start of a new era in warfare when the British attacked the German dams. The British
had researched and identified that the valley of Ruhr and Ruhr dams as a strategic point to attack. It was the heart line of
that area providing water and power. The most important one was the Mohne Dam protected by two anti torpedo nets and
anti aircraft guns on the shore and on each side of the tower, which made it difficult to attack. Sir Barnes Wallis came up
with the idea of bouncing bombs to breach the dams. The attack by the Royal Air Force was carried out during the period
when the reservoir was completely full and they breached the upper part of the dam. The flooding through the breach
caused enormous damage to the people and the surrounding areas. Wallis’s bouncing bombs were based on the concept of
ricochet and caused more destruction than a direct attacking missile.
Ricochet, is the rebounding of the projectile from any surface, which leads to the round missing its intended target.
Because of the unpredictable nature of ricochet, accidents tend to take place, being dangerous to bystanders. However,
with thorough study of this subject it can be utilised effectively in warfare to cause more damage or to avoid calamities
and casualties. Landing of spacecrafts applies ricochet to reduce its high velocity gradually before touching earth’s
surface, called as skip re-entry.
Forensic science relates to investigating and solving the criminal cases using biological information. Ballistic techniques
are involved in cases where firing has taken place, to identify weapons used, direction of fire, and nature of wounds and
gather other useful information to help solve the criminal cases. In the Warren Commission enquiry of the assignation of
President John.F.Kennedy and wounding of Texas Governor John Connally it was proved by the Wound Ballistics
department of U.S. army that the cartridge and fragments recovered from the accident sight indeed caused the injuries to
both of them.
Ricochet analysis is very useful in crime investigations. T.W.Burke, W.F.Rowe both former police officers have studied
the effect of bullet ricochet in relation to exchange of firing between officers and criminals [1]. Police investigations of
any crime and solving the case is the need of our modern society. Investigators try to reconstruct the shooting scene and
predict the nature of shot and the trajectory of the bullet. During a mission or on duty, ricochet of bullets can be
hazardous to law enforcement officers who are dealing with armed suspects. Safety is also a prime concern for them.
When shooting with a shotgun chances of a ricochet are multiplied by the number of pellets in the cartridge that is used.
It takes only one pellet to cause serious injury or death. Contrary, controlled ricochet of the artillery rounds can prove
beneficial in situations where it can be used to increase their range.
Bullets can ricochet from any surface wall, glass, concrete and from water and even soil. It is unexpected and can be fatal.
A few real life cases of ricocheting of bullets from different surfaces are listed below.
a. A commercial fisherman suspecting a man stealing fish out his nets waited during the night to catch him. When the
person started to take the fish out of the nets, the fisherman shot into the water to scare him. The bullet glanced up and
killed the man. The shooting was ruled accidental after the tests showed it had indeed hit the water first.
b. During a pursuit of a man fleeing from the police, the officer's face was grazed by a ricochet bullet. The bullet
ricocheted after hitting the rim of the tire of the vehicle in which the accused was attempting to flee.
c. Lt. Robert A. Dibb, was killed when a rocket fired at 30 degree and a speed of 280 knots ricocheted 500 feet into the air
and took off one wing of his F6F Hellcat. The rocket body struck him from above carrying away the right wing hinge
fitting causing the right wing to come down.
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 12, December 2013
ISSN 2319 - 4847
d. Lt. Donald Kirkpatrick beginning to climb the aircraft was struck by a ricocheting projectile which pierced the engine
cowling. The engine failed completely, short of the emergency strip and he made a full stall landing in bumpy desert
terrain, which collapsed the landing gear resulting in strike damage to the aircraft [2].
W. Goldsmith has given an overview of ricochet in his review of non-ideal impacts on targets [3]. In his review, he has
classified the targets as liquid target, soft target and metallic target. Impact of a body striking the water was studied by
Leon Trilling [4]. Much experimental work and research has been conducted to understand the ricocheting bullets and its
consequences. I.M.Hutchings modified Birkoff’s formulae by Rayleigh’s pressure formulae for finding critical angle of
ricochet for spheres and spinning cylinders [5]. In the late seventies W.Johnson, G.H.Daneshi and Reid did extensive
work on water ricochet of spheres and G.Wijk analyzed the ricochet of conical nose shaped projectiles [6, 7, 8, 9].
Johnson has classified water impacts as bouncing, broaching, penetration and ricochet. He specifies ricochet as the water
entry where the body is never completely immersed in water. This characteristic of ricochet is used in almost all analysis.
In our present work, a model is developed to find the critical angle of ricochet of some important nose shaped projectiles.
2. WATER RICOCHET MODEL
Ricochet of bullets in water is similar to skipping of a stone. A bullet’s interaction with the surface of water, however, is
different as compared to its trajectory in air. The equation of motions of skipping stone was derived and interpreted by L.
Bocquet [10]. Park, et al., studied the impact forces using numerical methods [11]. The nature of impact depends on the
angle of impact, the impact velocity, the geometry of the projectile, mass and the density of the projectile and that of the
target. Projectile Shape is one of the major factors in water ricochet as the force exerted by the target on the body depends
on the immersed area. The critical angle gives the limiting impact angle determining the occurrence of ricochet. This is
obtained by balancing the momentum of the forces acting on the body [12]. As the projectile enters the water surface, the
lift force acting on it is proportional to its wetted area in water. The lift force is calculated by taking into consideration the
actual projected wetted area on the water surface times the square of its velocity.
2.1. Methodology
A projectile with mass ‘m’, and diameter of the cylindrical rear end of the nose ‘d’, impacts the surface with an initial
velocity ‘v’ making an angle ‘α’ with the normal to the water surface. We define θ as the ratio
where L is the
length of the nose and R is the radius of the nose end or d/2.
Figure 1 Projectile entering the water surface.
The projectile depth at any time t is denoted as ξ and its depth ξ* when the front tip of the bullet just touches the water
surface is derived as,
(1)
.
The areas of the projection of the wetted portion of the projectile on the water surface are considered, (Fig 2.), B(ξ) the
projected area of the nose part and C(ξ) the projected area of the back cylindrical part of the projectile. Calculations are
done by taking the individual governing equations of each shape and integrating it.
The total projected wetted area is
(2)
Figure 2 Projected areas
The momentum N in the normal upwards direction to the water surface due to the lift force corresponding to dynamic
pressure is:
(3)
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The maximum depth of cut ξ# for a given projectile is obtained when N, momentum due to the retarding force of the
projected area D(ξ) on the water surface is balanced with I, impending projectile body momentum along the normal to
the water surface. The immersed area changes with the nose shape of the projectile. The most essential condition for a
projectile to ricochet is that the projectile must impact the water laterally or the front tip should not touch the water
surface first. In such cases, the lift force due to the projected area is able to overcome and sustain the weight of the
projectile. The critical angle analysis is based on the fact that the projectile will ricochet only if its depth ξ in water is less
than ξ* the depth when the front tip makes contact with water [9, 15, 18]. The determining condition for ricochet to take
place is ξ# has to be less than ξ*. A threshold for the critical angle for ricochet α* is obtained by equating ξ# and ξ* for a
projectile of given dimensions [12]. We find an expression for critical angle for Power series, elliptical and hack series
nose shape and compare it with our previous results of conical and ogive nose shape projectile.
3. GENERAL DESCRIPTION AND EQUATIONS OF ARMAMENT SHAPES
The geometry of the nose shape can be governed by a mathematical equation. We consider the symmetrical cross section
of the nose cone. The equation is obtained by taking the origin at the tip of the nosecone, x-axis along the axis of the nose
cone, y is the radius at any point x, as x varies from 0 at the tip to L, where we define L as the overall length of the
nosecone, and R as the radius of the base of the nosecone. There are exact equations to define the 2-dimensional profile
of the nose shape. The full body of revolution of the nosecone is formed by rotating the profile around the centreline (C/L)
Figure 3. Cross section of nose shape.
3.1 Power Series
The power series nose shape follows the power law to formulate a polynomial that is characterized by a rounded nose tip.
There is always a discontinuity at the nosecone body joint which gives a distinct non aerodynamic look to the component,
which is modified and smoothened out [13].
This shape is obtained by rotating about the axis the equation
(4)
The base of the nose cone is parallel to the latus rectum of the parabola and the bluntness of the shape is controlled by the
factor ‘n’,
. The value of n less than 0.7 results in blunt nose and above which it becomes sharp and as given
below.
n = 1 for a cone
n = 0.75 for a ¾ power
n = 0.5 for a ½ power (parabola)
n=0
for a cylinder
The equation of 0.5 power series is
0.5
x
y  R 
 L .
(5)
As the projectile impacts the water surface the complete area of the projection of the wetted portion of the projectile is
given by
(6)
The momentum N in the normal upwards direction to the water surface due to the lift force corresponding to dynamic
pressure for the above is got by substituting is D(ξ) in equation (3).
I is the downward momentum due to penetration and is given as
(7)
A balance between the momentum N and I is achieved and the corresponding maximum depth of cut ξ#, is obtained by
solving N = I,
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(8)
Ricochet takes place for all values of ξ# < ξ*. The critical angle is found as the intersection of the two curves ξ# and ξ*.
The graph gives the critical angle as the intersection of the straight line ξ* and the curve of ξ#.
Figure 4 Graph showing intersecion of ξ# and ξ*.
Similarly, we derive the projected area for 0.75 power series by taking the corresponding equation and the projected area
in water is,
(9)
Comparison between N and I gives the maximum depth of cut for 0.75 power series parabola as
(10)
The maximum depth of cut for a conical nose shaped projectile when n=1 is given as
(11)
The areas of the projection for ogive nose shaped projectile are calculated by taking into consideration the ogive equation
(12)
where γ is the radius of the circle that forms the ogive and is called the Ogive Radius and it is related to the length and
base diameter of the nose cone by the formula
(13)
Simulation is carried out for finding the critical ricochet angle for different mass and diameter and using the condition
#
has to be less than ξ*.
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3.2 Elliptical Series
The elliptical series nose is formed by incorporating one-half of the ellipse, with the major axis being the centreline and
the minor axis being the base of the nosecone. The rotation of the half ellipse about the major axis is called as prolate
hemispheroid. Note also, that if R equals L, this shape is a hemisphere.
The curve is governed by the equation:
(14)
The complete projected area is given as,
(15)
Comparison between N and I reduces the equation to
(16)
where z =ξ/d.
The equation is solved using the MATLAB solver and simulations are done to get the critical angle for varying mass,
diameter and tip angle θ.
3.2 HAACK Series
The HAACK series shapes are not characterized as the previous shapes based on geometry, instead are derived by
mathematical formulations with the purpose to minimize drag. The formulation contains a variable C that determines the
outcome of the HAACK shapes. However two particular values of C are important; that is when C=0, also known as ‘LD’
which signifies minimum drag for a given length and diameter, and C=1/3 also known as ‘LV’ indicating minimum drag
for a given length and volume. It is governed by the equation:
(17)
Where
(18)
The results for C=0, and C=1/3 are similar for small values δ.
Correspondingly, we get the projected areas for HAACK series as
(19)
N = I give the roots as:
(20)
4. SIMULATION AND RESULTS
Taking a sample of projectiles of standard dimension, we conducted parametric simulation analysis on the above said
shapes and the following results were obtained. It was observed that there exists a dimensionless parameter for a target of
given density
which affects the critical angle of the projectile directly. In our analysis of water ricochet since
density is constant we found that for same values of µ for different mass and diameter the critical angle is same to be true
for all the above said nose shapes. Simulations were carried out for all nose shapes and the results are given in Table 1.
4.1 Table1: Critical angle for different nose shapes for same value of µ.
Mass
In Kg.
Diamete
r
Critical angle α*
Cone
ogive
82.3797
α*-ogive
83.1818
In
0.9084
6
metres.
0.0604
4.12282588
4
Volume 2, Issue 12, December 2013
Power
Series0.5
Power
Series0.75
Elliptic
Shape
Haack
Series
81.9786
81.4056
80.8327
81.0046
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Volume 2, Issue 12, December 2013
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1.3602
8
4.12281806
8
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.0691
1.8133
9
4.13095476
7
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.076
2.7195
7
4.11571957
8
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.0871
4.0809
1
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.1
4.08091
5.4426
9
4.12281375
3
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.1097
8.1649
9
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.1256
4.12084936
83.1818
81.9786
81.4056
80.8327
81.0046
0.1382
4.12460627
5
82.3797
10.887
14.515
7
4.12525633
7
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.1521
29.030
6
4.12088146
1
82.3797
83.1818
81.9786
81.4056
80.8327
81.0046
0.1917
5. RESULT:
The critical angle was found to be same for projectiles of different mass and diameter having the same value µ for each
nose shaped projectiles. For projectiles of varying mass and diameter but of constant ratio µ the critical angle is same for
all nose shapes.
The elliptic nose shaped projectile has least critical angle or more range of ricochet and the ogive nose shaped projectile
has the least range of ricochet. From Table 2 it can be seen that for projectiles of same mass as the diameter increases the
critical angle decreases or the range of ricochet is increases.
We validate this relation between critical angle α* and µ, by finding the correlation coefficient between them which is
very close to1 as shown in Table 3, which implies α* is proportional to µ a dimensionless parameter. A graph depicting
the relation between µ and α* is also shown.
Table 2: Critical angle for projectiles of same mass but varying diameter
Critical angle α*
Ma
ss
m
in
Kg
.
Diam
eter
d
In
metr
es
Cone
ogive
α*_cone
α*-ogive
Power
Series0.5
Power
Series0.75
Elliptic
Shape
Haack
α*_ps.5
α*_ps.75
α*_ellipse
α*_haack
Series
0.0
02
0.005
16
84.51106
432
86.03513
206
84.21885
585
84.006861
46
83.812055
81
83.76048
961
0.0
02
0.005
45
12.3549
4
84.13291
218
85.57103
624
83.82351
497
83.571413
54
83.296393
8
83.29066
422
0.0
02
0.005
7
10.7995
4
83.93237
695
85.31320
524
83.61152
058
83.336500
84
83.015644
48
83.03856
279
0.0
02
0.006
4
83.38806
704
84.60273
757
83.04429
237
82.700517
69
82.259340
19
82.34528
386
0.0
03
0.005
63
16.8111
84.57981
926
86.12107
573
84.29334
036
84.087075
55
83.909458
64
83.84643
328
0.0
03
0.006
4
11.4440
9
84.01832
062
85.42779
679
83.70319
383
83.439633
25
83.135965
61
83.14742
477
7.62939
5
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Table 3: Correlation coefficient between µ and critical angle α*.
Nose
shapes
Correlatio
n coefficient
µ and
cone α*
µ and
ogive α*
0.956091
537
0.942402
961
µ and
elliptical
α*
µ and
power
series
0.5 α*
0.954389
665
0.954086
964
µ and
power
series
0.75 α*
0.950014
294
µ and
haack
series
α*
0.949935
262
.
Figure 5. Relation between dimensionless parametr µ and critical angle α*
5. Conclusion
Critical angle of ricochet is found to vary with µ a dimensionless parameter. µ is an important parameter for ricochet
studies at the design level. It can predict the dimension of bigger projectiles for the given ricochet angle with the help of
available previous data.
References
[1] Burke, T.W. and Rowe, W.F. “Bullet Ricochet: A comprehensive Review,” Journal of Forensic Sciences, JFSCA,
Vol. 37, No. 5,September 1992, pp. 1254-1260.
[2] http://www.chinalakealumni.org/accidents.htm
[3] Goldsmith W. Non-ideal projectile impact on targets. Int J Impact Engg. 1999; 22: 95-395
[4] Leon Trilling, “The Impact of a Body on a Water Surface at an Arbitrary Angle,” J. Appl. Phys. 21, 61, (1950);
http://dx.doi.org/10.1063/1.1699617
[5] L.M.Hutchings, “The Ricochet of Spheres and Cylinders from the Surface of Water” in Int. Journal of Mech. Sci.,
Vol. 18, 1976, pp. 243-247.
[6] Johnson, W. and Reid, S. "Ricochet of Spheres off Water," Journal of Mechanical Engineering Science, (17): 71-81,
1975
[7] Johnson: "Ricochet of Non-Spinning Projectiles, Mainly from Water. Part I: Some Historical Contributions" in Int.
Journal of Impact Engg., Vol. 21, Nos. 1-2, 1998, pp.15-24.
[8] Johnson: "The Ricochet of Spinning and Non-Spinning Spherical Projectiles, Mainly from Water. Part II: An Outline
of
[9] Theory and Warlike Applications" in Int. Journal of Impact Engg., Vol.21, Nos. 1-2, 1998, pp.25-34.
[10] G.Wijk: "A Water Ricochet Model", Defence Research Establishment, Weapons and Protection Division, 98-03-06.
[11] L. Rosellini, F. Hersen, C. Clanet, and L. Bocquet, "Skipping stones," in Journal of Fluid Mechanics, vol. 543, 2005,
pp. 137-146.
[12] M.S. Park, et al., " Numerical study of impact force and ricochet behavior of high speed water-entry bodies", in
Computers and Fluids, vol.32, 2003,pp.939-951.
[13] Vijayalakshmi Murali, Law.M., S.D. Naik, “Study of critical ricochet angle for conical nose shape projectiles,” AIP
Conf. Proc. 1482, pp. 58-63.
[14] The Descriptive Geometry of Nose Cones, © 1996 Gary A. Crowell Sr.
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