Resistive force on a bullet
Research Question: What would be the optimal thickness of the Kevlar to prevent lethal damage and
provide mobility for daily usage of an average police officer?
Subject: Physics
Word Count: 3989
Table of Contents
Introduction ................................................................................................................................... 2
Background Information.............................................................................................................. 2
Resistive force ................................................................................................................................3
Conservation of momentum............................................................................................................4
Conservation of kinetic energy.........................................................................................................4
Pressure .........................................................................................................................................5
Experime nt .................................................................................................................................... 6
Aim of the experiment ....................................................................................................................6
Methodology..................................................................................................................................7
Procedure ......................................................................................................................................8
Safety Precautions ..........................................................................................................................9
Results of the experiment ...............................................................................................................9
Table 5: Average momentum transferred from metal balls to the kitchen sponge. ............................ 16
Evaluation of the data ................................................................................................................... 17
Errors and further improvements .................................................................................................. 19
Predictions ................................................................................................................................... 21
What will be the predictions about? .............................................................................................. 21
Gathering required data for the prediction ..................................................................................... 21
Conclusion.................................................................................................................................... 23
Bibliography ................................................................................................................................ 25
2
Introduction
Since my childhood I always had seen war movies or played video games about wars. So
seeing a bulletproof vest was not really something amazing for me. However, lately I realised that
it is actually a weird piece of equipment because it is just a piece of cloth that can prevent a bullet
from penetrating into a persons body and decreases the chances of taking a lethal damage. A bullet
may not be very heavy so that the force exerted by a high-speed bullet is not much. However, at
first contact bullet exerts a pressure that is way too high for skin to handle due to the bullets pointy
head. While human skin has no chance against such a pressure, a bulletproof vest can prevent most
of the damage. However, in a combat situation, mobility is an important factor. So the weight of
the vest must be taken into account.
Nowadays a material called Kevlar is the most popular when it comes to bulletproof vest
production. Kevlar is a synthetic plastic that is light and is capable of preventing bullets from
penetrating into the body. This essay will focus on investigating the optimal thickness of the Kevlar
suit so that it can provide its user both mobility and lethal damage protection. All the interpretatio ns
will be made with the experiment done which did not involve any humans, bullets or Kevlar vests
in it.
Background Information
When it comes to a bullet colliding with a persons body, there are some useful concepts to
determine the spesifications of the collisions. Those spesifications can also be useful when labeling
a collision as lethal or non-lethal.
3
Resistive force
In this investigation resistive force is the most important concept because it shows how
much force is exerted by a persons body (or by the Kevlar vest) to the bullet to stop it. As the
thickness of the Kevlar vest increases, the resistive force that must be applied by the persons body
decreases. Therefore most of the force becomes absorbed by the Kevlar vest. So damage done on
a persons body is directly related with the thickness of the Kevlar vest and it is inverse ly
proportional. Resistive force is basically formulated like this,
𝐹𝑟 = 𝑚 𝑏 𝑎
𝐹𝑟 : total resistive force.
𝑚 𝑏: mass of the bullet.
𝑎: deceleration of the bullet while it is penetrating.
To find that deceleretaion,
𝑣 = 𝑢 + 𝑎𝑡
𝑣 − 𝑢 = 𝑎𝑡
𝑎=
𝑣 −𝑢
𝑡
Therefore,
𝐹𝑟 = 𝑚 𝑏
(𝑣 − 𝑢)
𝑡
𝑣: final speed of the bullet which is 0 when it reaches the deepest point.
4
𝑢: initial speed (the speed when the bullet comes in contact with the surface) or in this case the
muzzle speed of the bullet.
𝑡: the time it takes for bullet to reach from the surface of the Kevlar vest to the deepest point.
Conservation of momentum
When a bullet hits a human, it passes its momentum to that persons body without bouncing
off. Most of the time bullets do not go through a persons trunk considering that the weapon fired
is not a sniper rifle. In the end, what happens is usually an inelastic collision.
So assuming that the person is stationary,
𝑚 𝑏 𝑣𝑏 = (𝑚 𝑏 + 𝑚 ℎ + 𝑚 𝑘 )𝑣𝑟
𝑣𝑏: speed of the bullet
𝑚 ℎ: mass of the human
𝑣𝑟 : resultant speed.
Conservation of kinetic energy
Although bullets are not very heavy, they have an immense amount of speed. Therefore
bullets can have large kinetic energies. The kinetic energy of a bullet can be formulated like this,
𝐸𝑏 =
𝐸𝑏 : kinetic energy of the bullet.
1
𝑚 𝑣2
2 𝑏 𝑏
5
Normally, kinetic energy is not conserved in a inellastic collision. However in this
investigation it will be assumed that when bullet collides with the skin and the Kevlar vest, it is
going to transfer all of its energy to the person and the vest without any loss. So we can say that,
𝐸𝑏 = 𝐸ℎ + 𝐸𝑘 + 𝐸𝑏′
1
1
1
1
𝑚 𝑏 𝑣𝑏2 = 𝑚 ℎ 𝑣𝑟2 + 𝑚 𝑘 𝑣𝑟2 + 𝑚 𝑏 𝑣𝑟2
2
2
2
2
𝐸ℎ : kinetic energy of the person
𝐸𝑘 : kinetic energy of the Kevlar vest
𝐸𝑏′ : reduced kinetic energy of the bullet
Although this formula could be shrinked, it is kept like this in order to see individual kinetic
energies of each object.
Pressure
Pressure is the force per unit area. In this investigation pressure is important because a
bullet is able to penetrate a surface due to its high pressure. This will be useful because as the
thickness of the vest increases amount of pressure exerted on the body should be decreased. When
the thickness of the vest increases, the area of the colliding area increases therefore pressure must
decrease. To formulate it pressure of a bullet is simply,
𝑃𝑏 =
𝑃𝑏: pressure that bullet exerts
𝐹𝑏 :force that the bullet exerts
𝐹𝑏
𝐴
6
𝐴: is the collision area (which is the same as the area of the head of the bullet)
This will be useful for labeling the lethal damage.
Since head of a bullet is circular pressure can be written like this,
𝑃𝑏 =
𝐹𝑏
𝜋𝑟𝑏2
𝑟𝑏: radius of the head of the bullet
Note: While resistive force is the main focus of this investigation, kinetic energy, pressure
and momentum will mostly be useful when labeling the spesifications of the collisions.
Experiment
Aim of the experiment
Aim of this experiment is to determine the change in resistive force when the thickness
changes and to see the decrease of distance covered in the body after the bullet collides with the
body. All the further interpretations will be made on the basis of this experiment. Hypothesis of
this experiment is that as the thickness of the vest increases resistive force must increase therefore
damage on the body must decrease.
7
Methodology
Required items for this experiment are the
followings;
-
A scale ( Max capacity 500gr, ±0.1g) (a.)
-
Jenga bricks (Amount depends on the
size of the scale) (b.)
-
A kitchen sponge (c.)
-
A fitness resistance band with the range
of 2.5-9.5 kg (d.)
-
A wooden plate (e.)
Figure 1: How setup should look in the end
-
Cardboard (f.)
-
A video camera or a smartphone (g. is where camera
should be placed)
-
3 Metal balls at different weights (12.0 ±0.1g, 28.4
±0.1g, 55.8 ±0.1g) (h.)
-
A ruler (±0.05 cm)
-
Superglue
-
Scissors
Figure 2: Actual footage of the setup
8
Independent variables of this experiment are the mass of the metal balls and the thickness
of the fitness resistance band as they are not affected by the process. Dependent variable is the
weight measured on the scale when metal balls collide with the fitness resistance band and the
sponge. Control variable is the height which the metal balls are dropped from which is 67 ± 2.0
cm. For simplicity of the evaluation and calculations height will be always taken as 67 cm as it is
the average drop height.
Procedure
1- Put the scale on the wooden plate and glue it tightly.
2- Glue the Jenga bricks on the wooden plate to the left and right side of the scale.
3- Cut the kitchen sponge so that it should be just below the Jenga bricks in height.
4- Wrap the cardboard so that all the metal balls can go through it easily and glue the wrapped
cardboard to a wall over the scale.
5- Measure the height between the top of a Jenga brick and top of the cardboard.
6- Place the camera in front of the scale. Make sure the camera can see the readings from the scale
clearly.
7- Drop each metal ball 7 times from the top of the cardboard. Make sure the camera is recording
the readings from the scale.
8- Measure the distance between the left Jenga bricks and the right ones.. Then cut 4 pieces from
the resistance band according to that length. Then glue one piece on top of the Jenga bricks.
9
9- Repeat step 7. Glue one more piece on top of the previous one every time all balls are dropped
7 times. Do it until all 4 pieces are glued. Remember that glue should only be at the left and right
sides of the pieces. Record all the readings.
10- After all recordings are done, write down the highest readings from the scale when the metal
balls collide with the resistance band.
Note: Because the balls were dropped, acceleration of the balls will be taken as 9.8 ms -2 for further
calculations.
Safety Precautions
This experiment does not involve any danger in it. However, there is a chance that metal
ball could bounce to you. So using a pair of gloves and plastic goggles could prevent any possible
damage.
Results of the experiment
First of all, there are some measurements that should be gathered from the videos recorded
and through calculations because they are highly important for the results and evaluation. First
measurement is the speed of the balls when they are at the surface of the resistance band. The
height they were dropped from was 67 cm and their initial speeds was 0. As mentioned before they
have an acceleration of 9.8 ms-2 due to gravity which gives the speed as 3.6 ms-1 (see appendix).
Second important measurement is the time it takes for the balls to reach to the deepest point
of the resistance band. For this measurement videos are used. The videos I captured was shoot
with a 30 fps camera which means a second was divided into 30 pieces. By using the frame by
frame feature of the software called VLC Video Player I was able to see that longest time it took
10
for a ball to reach the deepest point was about 3 frames which is 0.1 seconds and the shortest time
was about 2 frames which is approximately 0.0667 seconds. As 30 frames is low for this
experiment, I had to divide the difference between longest time and shortest time into 4 pieces so
that there would be 5 timelines to calculate the deceleration of the balls.
These calculations are highly unlikely to be not accurate as air resistance is neglected so
the time it takes for all three balls to reach to the ground is same. The reason for that is because
mass is not a factor that affects the speed of an object due to gravity.
Results are written by recording the highest mass read on the scale after the balls bounced
off the scale. Due to the scale’s measuring intervals highest reading came after the balls bounced.
Greatest reading is the point where most of the resistive force is applied by the sponge and the
resistance band.s
Mass of the ball
Radius of the ball
Circular area
Kinetic Energy
(g)
(mm)
(mm2 )
(J)
Error: ±0.1g
Error:±0.5mm
Error:±0.25mm2
-
Ball #1
12.00
7.20
162.90
0.07776
Ball #2
28.40
9.50
283.50
0.18403
Ball #3
55.80
12.00
452.40
0.36158
Number of balls
Table 1: Spesifications of the metal balls.
11
60
50
48.7
45.2
MASS (G)
40
35.4
30
30.1
29.9
23.8
23.3
20
15.1
10
18.8
13.5
15.4
13.5
8.6
11
8.1
0
0
0 . 0 25
0.05
0 . 0 75
0.1
THICKNESS (MM)
Ball #1
Ball #2
Ball #3
Graph 1: The average masses read on the scale for all the balls as the thickness of the resistance
band was increasing.
12
Thickness of the
resistance band
(mm)
Error: ±0.003cm
Weight measured on the scale (g)
Trial #
Ball #1
Ball #2
Error: ±0.1g
1
16.2
30.7
2
13.7
32.0
3
14.1
23.9
0
4
15.3
30.1
5
15.2
30.1
6
14.8
33.0
7
16.3
29.5
1
11.6
21.1
2
13.2
23.8
3
14.4
24.2
0.025
4
14.1
22.2
5
14.9
25.6
6
14.2
22.8
7
12.1
23.6
1
11.9
19.1
2
9.8
20.0
3
10.7
18.7
0.05
4
10.3
17.1
5
10.9
18.0
6
11.6
18.5
7
11.9
20.5
1
9.2
15.2
2
7.0
16.8
3
8.2
17.8
0.075
4
9.9
4.6
5
9.0
7.8
6
8.7
15.7
7
7.9
16.6
1
8.7
16.0
2
8.5
14.4
3
7.6
15.2
0.1
4
7.8
14.9
5
8.3
15.9
6
7.9
15.6
7
8.0
15.5
Table 2: Masses read on the scale when the metal balls were dropped. Raw
Ball #3
47.6
43.7
55.6
44.9
53.7
48.8
46.3
44.0
46.7
47.2
47.1
43.5
43.6
44.4
31.2
36.7
36.6
37.4
34.0
35.2
36.6
24.5
31.2
27.2
14.7
23.4
20.7
24.8
28.2
24.3
32.4
31.9
31.7
30.5
31.5
data.
13
Thickness of the resistance band
Number of the ball
Resistive Force(N)
-
-
Ball #1
-0.5436
Ball #2
-1.0764
Ball #3
-1.7532
Ball #1
-0.5301
Ball #2
-0.9150
Ball #3
-1.7750
Ball #1
-0.4751
Ball #2
-0.8120
Ball #3
-1.5289
Ball #1
-0.4126
Ball #2
-0.6477
Ball #3
-1.1419
Ball #1
-0.4372
Ball #2
-0.8311
Ball #3
-1.6245
(cm)
Error: ±0.003cm
0.000
0.025
0.050
0.075
0.100
Table 3: The average resistive forces applied by the kitchen sponge and the resistance band to
the bullet.
The reason why resistive force has a negative (-) sign is because force is a vectoral
quantity and if the force applied by the bullet has a positive (+) sign, resistive force must be
negative as it is an opposite force.
14
Decrease of pressure
Thickness of the resistance band
Pressure
Number of the ball
(cm)
as percentage
(Pa)
(%)
Error: ±0.003 cm
0.000
0.025
0.05
0.075
0.100
-
-
-
Ball #1
3337.00
-
Ball #2
3796.80
-
Ball #3
3875.30
-
Ball #1
3254.10
2.48
Ball #2
3227.50
14.99
Ball #3
3923.50
-
Ball #1
2916.50
12.60
Ball #2
2864.20
24.45
Ball #3
3379.50
12.79
Ball #1
2532.80
24.10
Ball #2
2284.70
39.83
Ball #3
2524.10
34.87
Ball #1
2683.90
19.57
Ball #2
2931.60
22.79
Ball #3
3590.80
7.34
Table 4: Average pressure applied by the metal balls to the resistance band and the kitchen
sponge.
15
Thickness of the resistance
band (cm)
Error: ±0.003 cm
0.000
0.025
0.05
0.075
0.100
Number of the
balls
Decrease in kinetic
Average decrease
energy as percentage
as percentage
(%)
(%)
-
Kinetic energy
(J)
-
-
-
Ball #1
0.03184
-
Ball #2
0.0417
-
Ball #3
0.04685
-
Ball #1
0.02599
18.37
Ball #2
0.03785
9.23
Ball #3
0.04438
5.27
Ball #1
0.02331
26.79
Ball #2
0.03466
16.88
Ball #3
0.04197
10.41
Ball #1
0.02064
35.18
Ball #2
0.03184
23.65
Ball #3
0.03988
14.88
Ball #1
0.0183
42.53
Ball #2
0.02958
29.06
Ball #3
0.0381
18.68
Table 5: Average kinetic energies transferred from metal balls to the kitchen sponge.
-
10.96
18.03
24.57
30.09
16
Thickness of the
Average decrease as
Number of the
Momentum
Decrease in momentum
balls
(kg m-1 s-1 )
as percentage (%)
resistance band
percentage
(cm)
(%)
Error: ±0.003 cm
0.000
0.025
0.050
0.075
0.100
-
-
-
Ball #1
0.02299
-
Ball #2
0.02631
-
Ball #3
0.02789
-
Ball #1
0.02117
7.92
Ball #2
0.02507
4.71
Ball #3
0.02714
2.70
Ball #1
0.01967
14.44
Ball #2
0.02399
8.82
Ball #3
0.02639
5.38
Ball #1
0.01851
19.49
Ball #2
0.02299
12.62
Ball #3
0.02573
7.74
Ball #1
0.01743
24.18
Ball #2
0.02216
15.77
Ball #3
0.02515
9.82
-
-
5.11
9.55
13.28
16.59
Table 6: Average momentum transferred from metal balls to the kitchen sponge.
17
Evaluation of the data
Looking at Graph 1 and Table 3, it can be seen that as the thickness of the resistance band
increases, the mass read on the scale decreases. It means that when the thickness was increased
more force was absorbed by the resistance band. It can also be seen that the decrease in readings
are in a linear trend until the thickness of 0.1cm. While the force applied by the ball #1 continued
to decrease at the 0.1 cm thickness, the readings from ball #2 and ball #3 showed an increase. This
could be due to many reasons which will be mentioned in the next section. The results from this
experiment proves that as the thickness increases the damage done to the body decreases. So when
calculating the optimal thickness of a Kevlar vest in the next section, this linearity and the
proportion between 0 cm of thickness and other thicknesses will be taken into account.
At thickness of 0 cm, there was no resistance band. Therefore, resistive force seen at first
trials are completely applied by the kitchen sponge. Although there are some unexpected results,
generally resistive force applied by the kitchen sponge decreases.
18
Thickness of the
Force absorbed by
Number of
resistance band
the resistance band
the balls
(cm)
0.025
0.050
0.075
0.100
Average percentage
Percentage
of force absorbed
(%)
(N)
(%)
Ball #1
0.0135
2.48
Ball #2
0.1614
14.99
Ball #3
-0.0218
-
Ball #1
0.0685
12.60
Ball #2
0.2664
24.75
Ball #3
0.2243
12.79
Ball #1
0.1310
24.10
Ball #2
0.4287
39.83
Ball #3
0.6113
34.87
Ball #1
0.1064
19.57
Ball #2
0.2453
22.79
Ball #3
0.1287
7.34
8.74
16.71
32.93
16.57
Table 7: how much resistive force was applied by the resistance band, their percentages and
average force absorbed by percentage.
Note: At 0.025 cm, ball #3 has an unexpected result due to the unknown errors.
Looking at Table 7 there seems to be an interesting pattern. Every time thickness of the
resistance band was increased by 0.025 cm, average percentage of force absorbed by the resistance
band almost doubled. However, at thickness of 0.100 cm, average force absorbed seems to be
decreased by approximately 16% which is an unexpected result and possible reasons are mentioned
in the next section. If 0.025 cm was one unit thickness, it is safe to say that as the thickness
19
increases by 1 unit, percentage of the absorbed force becomes doubled. However, this may not be
correct at all times, as the percentage starts to approach 100%, thickness of the bulletproof material
may be needed to be increased at a faster pace. Also, it is not wrong to say that if the mass or the
speed of the colliding object becomes too great, force absorbed by the vest may become
insignificant. This insignificancy can be proved by looking at the Table 5 and Table 6. As it can
be seen from those table, as the mass of the metal balls increase, kinetic energy and momentum
transferred from the metal balls to the kitchen sponge greatly increases (While at thickness of
0.100 cm 24.18% of the momentum was absorbed by the resistance band for the smallest ball, this
value went down to 9.82% biggest ball). Therefore it can be said that every new layer of Kevlar
would greatly increase the protection. Looking at their average protections, each new layer gives
about 3.50% more protection against momentum and about 7% protection against kinetic energy.
When looked at Table 4, an unusual result can be seen because unlike the other calculatio ns,
protection against pressure is low for the ball #1, then for ball #2 protection greatly increases and
for the ball #3 protection becomes low again. So it can be said that pressure wise, protection is
more likely to be related with mass of the bullet. Because results from pressure is unexpected,
those numbers will not be used in the predictions as it needs an experiment of its own.
Errors and further improvements
In the experiment done biggest cause of error was a human error. As I was not able to
develop a system that would automatically dropped the ball, I dropped it by my hand. This caused
an uncertainty in the height balls were dropped from and there is a possibility that an external
acceleration might be caused by my hand. In order to minimise those errors, I dropped the balls
from a range which was between 65 cm and 69 cm. The average height is 67 cm and it is what I
used for my calculations. In addition to that I tried to drop the balls without moving my hand too
20
much. Another error was caused by the low framerate of the camera. Due to the low fps I was not
able to see the collision moment exactly so the time taken might be faulty.
Another factor that can be a source of error is that as I approached the end of the
experiment, the scale took too much damage. This was not because the resistance band was unable
to protect it, but because bouncing balls were falling on the scale and therefore damaging it. This
might have affected the latest reading to become faulty.
In order to improve this experiment, there must be an automatic system that would drop
the balls from a certain height. By doing so, all the errors caused by me can be eliminated. Also,
using a faster camera or even a slow motion camera, precision could be greatly increased. In
addition to that by covering the whole area that Jenga bricks and the scale was located in with a
plastic box, damage on the scale can be minimised, decreasing the chances of faulty readings.
Lastly, the scale I used in this experiment does its measurements 4 times a second. This might be
the reason why ball #3 always showed readings lower than its original mass. Eventhough results
from ball #3 may not be accurate, they are precise.
21
Predictions
What will be the predictions about?
The predictions that in the next section will be about the optimal thickness for a Kevlar
vest, and they will be based on the results from the previous experiment . Optimal thickness will
be determined according to the most likely scenario that a pollice officer would be in.
Spesifications such as how much resistive force will be applied by the Kevlar vest, how much
kinetic energy can the vest absorb, how much of the momentum can it absorb and how much did
the pressure applied by the bullet to the body was decreased will all be based on the experime nt
conducted. By using the average volume of a persons trunk and density and thickness of the Kevlar,
total mass of the vest will be calculated. In the end, according to the peak kinetic energy that a
human can reach, optimal vest will be created. Also, how much mobility was lost will be seen
from the increase in mass and decrease in speed in the kinetic energy formula.
Gathering required data for the prediction
In the previous sections, 0.025 cm was accepted as 1 unit thickness and as the thickness
was increased by 1 unit, percentage and ratio wise resistive force applied by the resistance band
was doubled. In the experimental setup, there was a gap between the kitchen sponge and the
resistance band. So it is possible to prevent almost 100% of the force applied by the bullet if there
is a gap between the body and the vest. There was a 0.4 ± 0.05 cm gap between the kitchen sponge
and the resistance band so that it can be expected at 5 unit thickness, metal balls would not even
touch the sponge. A simple prediction is that as the gap between the protection and the body
decreases, required thickness increases. However, this is another experiment and for the end
22
results, gap between the vest and the body will be accepted as 0.4 cm. In the results it was seen
that at 0.025 cm thickness 8% of the resistive force was applied by the resistance band and at 0.075
cm that percentage became approximately 32%. By following that trend it can be assumed that at
5 unit thickness 100% protection may be approached (Readings from 0.1 cm thickness are accepted
as faulty due to the mentioned errors).
According to a report by State of California Department of Justice in 2016, 86.5% of the
firearms used in crimes were handguns. That percentage goes up to 92.6% when it comes to crimes
of violence. It is also common sense as the handguns are the most easily accessible guns. So it can
be said that a police officer will most likely encounter a handgun in a fight. According to the same
report most fired ammunation is 9 mm Luger cartridge. According to a muzzle speed chart of 9
mm Luger published by BallisticsByTheInch.com, if Cor Bon 90 gr. JHP +P type of this
ammunation is fired from a Kimber Target II 1911 handgun, it will have the highest muzzle speed
of 477.01 meters per second. This bullet weighs 90 grains which is equal to 5.832 grams.
Finding an average value for a human’s physical properties is almost impossible as there
are different humans in every where in the world. There are many different ethnicities and races.
By limiting those differences, according to Halls.md, average height of a white male in America
is 178 cm. According to City of Los Angeles Personnel Department, weight of a 178 cm police
officer should be 82.2 kilograms. According to IamLivingIt.com, average speed of a male between
the ages of 18-34 (which are acceptable ages for a police officer) is 13.62 kilometers per hour or
3.78 meters per second. Deducing from those numbers, an average police officer has an
approximate kinetic energy of 587.25 joules while running. By using a body proportion calculator,
it is calculated that at an height of 178 cm and weight of 82 kg, predicted chest circumfere nce
comes out as 102 cm and waist circumference comes out as 89 cm. Also the length between the
23
shoulders and waist is about 51 cm for an average person mentioned above. For the sake of
simplicity, shape of the trunk will be accepted as a cylinder and whole circumference of the
cylinder will be accepted as 102 cm. Radius of such a cylinder is 16.2 cm. Therefore, volume of it
is 42048 cm3 or 0.042 m3 .
Conclusion
At purchase the resistance band had a thickness of 0.025 cm and it was accepted as 1 unit
thickness. Looking at the commercially available Kevlar fabrics, one sheet of Kevlar fabric has a
thickness of approximately 0.0023 cm which is about 0.092 unit thickness. As mentioned in the
previous section, a thickness of 5 units is needed in order to prevent 100% of the damage. It should
be noted that as the Kevlar vest is only in contact with the persons body and nothing else, 100% is
not possible but 5 unit thickness will give the optimal protection.
By doing a simple calculation;
5 × 0.0023 = 0.092 × 𝑡
𝑡=
5 × 0.0023
0.092
𝑡 = 0.125
𝑡: optimal thickness of the Kevlar vest in cm.
This means that in order to prevent maximum damage caused by a handgun, thickness of
the Kevlar vest should be 0.125 cm. This equals to about 54 sheets of Kevlar fabric. According to
information given by the seller, this is well above NIJ level IIIA type protection which is at 24-28
sheets. Deducing from that, the calculated optimal thickness falls between NIJ level IIIA and NIJ
level III type protections. This type of protection is capable of preventing damage from almost any
24
handgun. However, this is type of protection is also too much for daily usage of a police officer.
Normally in a bulletproof vest, there are additional layers which are not Kevlar. So, 54 sheets of
Kevlar fabric will provide maximum protection even if there are no additional layers. Based on
the experiment done it can be said that 54 sheets of Kevlar fabric could absorb up to 37% of the
kinetic energy and 19% of the momentum.
Last part of this prediction is how much mobility will be lost when such a Kevlar vest is
equipped. According to DuPont.com, Kevlar has a density of 1.44 g cm-3 or 1440 kg m-3 . As it was
calculated previously optimal thickness was 0.125 cm. Therefore, Kevlar vest would increases its
users radius by 0.125 cm. So the new radius would be 16.325 cm and the new volume would be;
𝜋 × 16.3252 × 51 = 42699.86
42699.86 cm3 or 0.043 m3 . By substracting this new volume from the original one, it is
found that Kevlar vest occupies a volume of 0.001 m3 . Therefore its mass would 1.44 kg. Earlier
it was found that at 82.2 kg and 3.78 m s-1 a human would have a kinetic energy of 587.25 joules.
Keeping that value as constant;
𝟓𝟖𝟕. 𝟐𝟓 =
𝟏
× (𝟖𝟐.𝟐 + 𝟏. 𝟒𝟒) × 𝒗𝟐𝒓
𝟐
𝒗𝒓 = 𝟑. 𝟕𝟓
𝒗𝒓: resultant speed after equipping the Kevlar vest.
As it can be seen from the calculation above, there is only a 0.03 ms-1 decrease in speed.
So it can bi concluded that at 0.125 cm thickness of Kevlar fabric optimal damage protection can
be achieved with the highest mobility possible.
25
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