Study of critical ricochet angle for conical nose shape projectiles

Vijayalakshmi Murali, Manish G. Law, and Smita D. Naik

Citation: AIP Conference Proceedings 1482, 58 (2012); doi: 10.1063/1.4757438

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 Study Of Critical Ricochet Angle For Conical Nose Shape
Projectiles
Vijayalakshmi Murali1, Manish G. Law2, Smita D. Naik3
1
Department of Engineering Mathematics, MESCOE (Wadia), University Of Pune, India.
2
Department of Mechanical Engineering, MESCOE (Wadia), University Of Pune, India.
3
Department of Applied Mathematics ,Defence Institute of Advanced Technology, Pune, India.

Abstract. The purpose of this research is to formulate a generic analytical model to assess the phenomena of water ricochet
for a conical nose shaped projectile. A theoretical model is analyzed to study the critical angle of conical nose shaped
projectile entering in water and is extended for different mediums as normal sand and mercury. Numerical Simulation has
been carried out to find the effect of tip angle of the conical nose shaped projectile on the critical angle.
Critical angle is defined as that angle of impact of the projectile above which ricochet will occur. This angle is obtained by
balancing the momentums acting on the projectile at the time of impact on the basis of Newtonian theory. Major factors
affecting critical angle are impact velocity, impact angle, density of the projectile and the target. An attempt has been made
to study the effect of longitudinal spin of the projectile on the critical angle.

Keywords: Conical, Density, Nose shape, Ogive, Projectile, Ricochet..

PACS: 89.20.Dd

INTRODUCTION greater will ricochet, critical angle being measured

from the normal to the water surface.
Ricochet, is the rebounding of the projectile from Empirically it has been found that non-spinning
any suitable surface such as wall water or armor spherical projectiles, the critical angle (measured in
plating and leads to the round missing its intended degrees from the surface of the water) for ricochet is
target. The ricochet’s result is indiscriminate and approximately given by [1]:
unpredictable, placing all in close proximity in danger
and maybe striking an innocent bystander. Study of θ c = 18/() (1)
ricochet helps to predict the trajectory and analyze
safety range considerations and hence reduce its Where  is the specific gravity (the density
hazards. relative to water) of the projectile.
Theory of ricochet plays a major role in Police Projectile Shape is one of the major factors in
Investigations to trace the path of the shot by ricochet. There has been much work done in ricochet
analyzing the nature of wounds in shooting victims, of spherical and cylindrical projectiles [2, 3, 4] both
the deformation of the projectile and the targets. In analytically and experimentally, but not much in
military applications it was used for siege operations conical and ogive shaped projectiles. In our work the
to bounce cannon balls through an enemy’s tightly study of ricochet of a conical nose shaped projectile
packed ranks or along a defended rampart. This impacting a water surface is attempted.
technique was used to increase the range of cannons, The analysis has been done for simplified models
as well as increase the damage inflicted on the target with the following assumptions:
(hitting near the water line is more damaging than i) Water is calm and non wavy.
being hit by a descending shot). ii) Yaw variation is ignored.
Models of ricochet process can be developed iii) During the ricochet phase, the velocity of the
analytically or experimentally and the process can be projectile is constant.
simulated for optimizing the parameters. Water
ricochet is motivated by experimental study of stone
skipping on the water surface. Non-spinning WATER RICOCHET MODEL
projectiles encountering the water surface at angles
lower than the critical angle will enter the water; Study of ricochet of conical projectiles is based on
projectiles encountering the water surface at angles the work done by Gunnar Wijk [5] in which a partial
International Conference on Fundamental and Applied Sciences 2012
AIP Conf. Proc. 1482, 58-63 (2012); doi: 10.1063/1.4757438
© 2012 American Institute of Physics 978-0-7354-1094-7/$30.00

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 portion of the projectile wetted area had been The areas of the projection (Fig 2.) of the wetted
considered. The model has been modified by these portion of the projectile on the water surface are
authors by considering both the nose and cylindrical considered, B(ξ) the projected area of the conical part
wetted part of the projectile. The drag and lift forces of the projectile and C(ξ) the projected area of the
are obtained by considering the actual projected wetted cylindrical part of the projectile.
area of the nose and cylindrical body on the surface of
water (shaded region in Fig.1). A condition for critical
angle of ricochet is obtained by comparing the c b
momentums. The same model is also extended for
sand and mercury as medium.
A projectile having a mass ‘m’, a cylindrical rear 2a
end with diameter ‘d’ and a conical front end with top
angle ‘2 θ’ is considered to impact the water surface. It
impacts at time t=0 with the velocity ‘v’ at an angle ‘α’
to the normal direction of the water surface. In practice
α is always very close to π/2. The analysis is based on FIGURE 2. Projected areas.
the fact that the projectile does not ricochet if its front
tip is the first part to make contact with water[1,5,7]. B(ξ) = ab (4)
If α + θ < π/2 then the front tip of the projectile is the
first part to make contact with the water and under
such circumstances it is assumed that the projectile ξ
Where, a =  2 − [ − ]2 (5)
sin α
will not be ricochet, but it will be buried in the water.
−ξ ξ
And b = + π
tan α tan [θ−( +α)]
2
−ξ
≈ (6)
cot (θ+α)

C(ξ) = ac (7)

Where,  = () (8)

Thus we get the corresponding areas as,

ξ3/2 d−ξ/ sin α

B(ξ) = −
(cot (θ+α)) √sin α)
(9)

ξ 3/2 tan αd−ξ/ sin α

FIGURE 1. Schematic of a Conical Projectile. ( ) = (10)
√sin α

The total area is D(ξ) = B(ξ) + C(ξ) (11)

The projectile penetrates into the water with
specific angle α and follows the initial trajectory with
the velocity v and continues for some time, ‘t’. At time The momentum N in the normal upwards direction to
t the projectile has cut to the depth, ξ, where, the water surface due to the lift force corresponding to
dynamic pressure is:
ξ =vtcos(α) (2)

ξ ρv 2 (sin (α))2 D(ξ’)d ξ’

N = ∫0
If this continues until the front tip of the projectile 2∗v∗cos (α)
5 5
2 ξ ξ
reaches the water, the depth of the cut can be derived 0.8ρvd 3 (tan (α) − tan (θ+α))( ( )2 (sin ( α))2 − 0.154 ( )5 )
5 d d
as, =
sin (2α)
− cos (α +θ)
ξ∗ = (3) (12)
2 sin (θ)

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 A balance between the momentum N and I is
achieved and the corresponding depth, ξ#, is found,
where N is the momentum due to the lift force
corresponding to the dynamic pressure and, I is the
downward momentum due to penetration and is given
as

I = mv cos(α) (13)

ξ # the maximum depth of cut is obtained by solving

N = I,
2
5 FIGURE 3. Critical angles for water, sand and
25 125
 sin (α)5 − μ cos (α) sin (2α)
ξ# 5 5
16 16 mercury for θ=10o
=  sin(α) − 
2 
d 4 tan (α)−tan (α +θ)

(14)

Where μ = ( 3 ).
ρd

When ξ#, the maximum depth of cut, is compared

with ξ*, the depth of cut when the tip just touches the
water, the criteria for water ricochet conditions are
observed. For ricochet to take place the front end tip of
the projectile must not reach the water surface first.
Hence we get the condition for ricochet as:

ξ#<ξ (15) FIGURE 4. Critical angles for water, sand and

mercury for θ=15o
or
2
5
25 125
5 5  (sin (α))5 − μ cos (α) sin (2α)
16 16
 sin(α)2 −   =
4 tan (α)−tan (α +θ)

cos (α+θ)
(16)
2sin (θ)

which gives the Critical angle for ricochet as α*.

The same model is applied to sand and mercury as
medium. The density of normal sand is 1.8 kg/dm3
and for mercury is13.6 kg/dm3.
FIGURE 5. Critical angles for water, sand and
mercury for θ=20o
Numerical Example And Simulation
TABLE 1. Critical Ricochet Angle.
Simulation has been performed by taking a conical Medium Theta(θ in deg.) Ricochet Angle
nose shaped projectile with m=17 kg and diameter, (α* in deg.)
d=10.5 cm. With the density of water as ρ=1kg/dm3 Water 10 85.93
and the tip angle θ=10°, 15°, 20° the model gives the Sand 10 85.13
corresponding α*. The critical angle α* is obtained by Mercury 10 82.61
Water 15 86.10
changing the parameters θ, ρ and represented
Sand 15 85.19
graphically as alpha versus ξ* and alpha versus ξ#. Mercury 15 81.41
Fig.3, 4 and 5 gives the critical angle for water, sand Water 20 86.45
and mercury by changing theta to 10o, 15o, and 20o. Sand 20 85.42
The same model can be analyzed by changing the Mercury 20 80.83
mass and the diameter of the cone.

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 RESULT AND DISCUSSION
ξ#
(i) When the tip angle is varied for the cone with d
water as medium for given geometry and mass the
critical angle α* increases gradually as θ increases and ⎛5
= ⎜ sin(α)5/2
when θ is 45o, α* is 87.5o and thus reducing the range 4
of ricochet angle at θ = 45o by about 2.5o . It was
⎝
observed from the analysis that as θ approaches π/2, 5/2
blunt nose projectile, α* is almost equal to 90o and the 25 sin(α)5 − 125 μ cos(α)(1 − ωd/2v) sin(2α) ⎞
projectile will not ricochet but get submerged. 16 16
−⎛ ⎞⎟
(ii) The variations of θ does not have much effect tan(α) − tan(α + θ)
on α* in sand and mercury as compared to water as ⎝ ⎠⎠
medium. α* varies from 85.13 o to 86.73 o as θ varies
from 10 o to 45 o for sand and for mercury the critical (19)
decreases gradually. .
(iii) Effect of density: As the density of the The analysis of spin was performed varying ωd/2v
medium increases, the critical angle α* decreases, values between 0.3-0.5 which limits the angular
implying that, more dense the medium, more the velocity of the projectile between
chance of ricochet. It is observed from the graph that (3v/5d) to (v/d).
critical angle for mercury is almost 5 o more than that
of water.

EFFECT OF SPIN

Effect of spin comes into play when a velocity

vector acts perpendicular to the longitudinal axis of the
projectile due to cross wind in air and cross currents in
water. We can see that there is almost no significant
change in the critical angle but only a slight increase in
α* due to positive spin and a slight decrease in α*
due to negative spin. By adding a spin component in
equation (13), we get the resulting equations below.

ωd
I = m(v + ) cos(α) (17)
2

And the maximum depth of cut ξ# for positive spin as, FIGURE 6. Critical angles for θ=10o with and without
spin.
ξ#
d

⎛5
= ⎜ sin(α)5/2
4
⎝
5/2

25 sin(α)5 − 125 μ cos(α)(1 + ωd/2v) sin(2α) ⎞

⎛ 16 16 ⎞⎟
−
tan(α) − tan(α + θ)
⎝ ⎠⎠

(18)

And for negative spin as, FIGURE 7. Critical angles for θ=15o with and without spin.

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 CONCLUSION
(i) The critical angle increases with tip angle
reducing the effective range of ricochet angle.
(ii) A blunt nose shape projectile will not ricochet.
It will either broach the water surface or sink.
(iii) For more dense medium critical angle
decreases.
(iv) Slight increase/decrease in critical angle due to
positive/negative spin.

FUTURE SCOPE
FIGURE 8. Critical angles for θ=20o with and without spin.
The model is limited only for the case when the
projectile axis follows the flight trajectory, ignoring
the yaw angle. Waves on water surface complicate the
DISCUSSION OF SPIN analysis and can be studied by considering the
amplitude and wavelength. The simple model can be
For given mass and given geometry of the conical projectile,
considered as apart of larger model to analyze the
positive spin gives a slight increase in critical angle α* and different targets. The study can be extended for
negative spin decreases the critical angle, but the effect is projectiles of different basic shapes as well as hybrid
minimal. shapes.

Note: There are certain observations when simulations are

done. As the value of α is varied from π/2- θ to π/2, the value NOTATIONS
of ξ# can become imaginary. As theta varies from 10 to 45
degrees the range of alpha for which the values inside the
square root sign in equation (6) becomes negative is shown m - Projectile mass
in Fig 8 and gives imaginary roots of ξ#. d - Projectile diameter
r - Projectile radius
l - Ogive nose length
2θ - Tip angle of conical nose of projectile
t - Time
v - Projectile velocity
α - Angle of impact
ξ - Depth of cut
ρ - Density of target material
μ - Dimensionless parameter
N - Momentum due to retarding force on area B(ξ)
I - Momentum along the normal to fluid surface
a - Base of the cone projected on the fluid surface
FIGURE 9. Alpha vs. Discriminant of the Equation. b - Perpendicular distance from the conical portion
to ‘a’
TABLE 2. Critical Ricochet Angle With Spin For Water. c - Perpendicular distance from the cylindrical
Spin Theta(θ in deg.) Ricochet Angle portion to ‘a’
(α* in deg.) D(ξ) -Total projected area on the fluid surface of the
Positive 10 86.50 p projectile
No Spin 10 85.93 B(ξ) - Projected area on the fluid surface for
Negative 10 85.02 conical/ogive part
Positive 15 86.73 C(ξ) - Projected area on the fluid surface for
No Spin 15 86.10
cylindrical part.
Negative 15 85.02
Positive 20 87.02
No Spin 20 86.45
Negative 20 85.24

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 REFERENCES
1. 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.
2. A.S.Soliman, et al, “The Effect of Spherical Projectile
Speed in Ricochet off Water and Sand” in Int. Journal of
Mech. Sci., Vol. 18, 1976, pp. 279-284.
3. 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.
4. Johnson: "The Ricochet of Spinning and Non-Spinning
Spherical Projectiles, Mainly from Water. Part II: An
Outline of Theory and Warlike Applications" in Int.
Journal of Impact Engg., Vol.21, Nos. 1-2, 1998, pp.25-
34.
5. G.Wijk: "A Water Ricochet Model", Defence Research
Establishment, Weapons and Protection Division, 98-03-
06.
6. Goldsmith W., Review Non-ideal projectile impact on
targets, International Journal of Impact Engg. vol.22,
1999, pp. 95-395..
7. L. Rosellini, F. Hersen, C. Clanet, and L. Bocquet,
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