Ammunition Dynamics

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 ABSTRACT Ballistic Flight concerns the science of observing differences between theoretical flight paths and

those observed in reality. A number of different measurements were taken so as to evaluate the

effect of elevation angle on flight characteristics. Thus if different pellets; with different muzzle

velocities; different angles; and different dynamics; affect the drop data. In this report it is

established by close analysis of the data collected, what actions took place after the pellets exited

the barrel and before they hit the target. Furthermore work has been placed towards the

aerodynamic drag and gravity of ammunition illustrated during this experiment and the way

different pellets act towards these forces. The analysis linked to this report associates the factor of

a higher pellet muzzle velocity, with a more accurate shoot. It also illustrates that the heavier the

ammunition type; the better it’s ballistics performance. Furthermore the lower the pellet drag

observed; the lower the drop allowing for greater ballistic coefficient.

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 METHODOLOGY

The effect of elevation angle in ballistic flight

One framed mounted air pistol was used to fire two different ammunition types; Bisley Magnum .22

Pellets (5.5mm) and Umarex Hobby- Sport Pellets (0.77g 11.9gr; Kal 5.5mm Cal .22). Firstly 10 of

each type of lead pellets were weighted by using the supplied balance, so as to calculate a mean

mass value. Then a check was undertone to ascertain that the pistol barrel was horizontal and the

distance between the pistol muzzle and the paper clipped on the box was taken. After the laser

attached to the barrel gave a marked position on the graph paper and the pistol was fired, so as to

take the velocity measurements from the chronographs and note the position of any holes in the

target papers. The same procedure was repeated five time a -3°, 0° and +3° angles.

Aerodynamics

On the second part of the experiment the ammunition dynamics were measured in a chamber

model that housed the different ammunition used during the flight and held a rail system connected

to a fan. By an anemometer adjacent to the cone in the test chamber a measurement of the

airflow velocity that the pellet model experienced during different fan speeds was recorded. Then

the drag force was linked to a computer using Cassy software, that calculated the average drag

force for a fixed duration of time. Five reading were recorded for each pellet as well as the

diameters of these models used, so as to be accessed during further data analysis.

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 RESULTS Important numerical values collected during the practical class. First Part : The effect of elevation angle in ballistic flight

• Range distance 5.2 m

• Weight of lead pellets (weight of 10 units/pellets)

10 Hobby - Sport Line Pellets +7.81 g (0.781g mean)

10 Bisley Magnum .22 Pellets + 13. 65 g (1.365 mean)

• Measured Muzzle Velocity and Average Muzzle Velocity at 0° elevation

Hobby-Sport Line Pellets measure muzzle velocity (m/s)

Bisley Magnum .22 Pellets measured muzzle velocity (m/s)

1 99.18 74.78

2 100.4 73.60

3 102.8 74.99

4 101.3 76.46

5 101.4 76.46

Average muzzle velocity (m/s) 101.016 75.258

• Drop Distance (measuring the distance between the laser mark and the actual mark produced by

the pellet on the graph paper), measurements that will help ascertain the accuracy and damage caused by different type of ammunition. Elevation Hobby-Sport Line

Pellets (cm) Bisley Magnum .22 Pellets(cm)

0° 1 0.3 1.4 2 0.5 1.2 3 0.5 1.8 4 0.9 1.4 5 0.4 1.2 +3° 1 31.6 32.1 2 31.8 32.0 3 31.9 31.9 4 31.5 31.4 5 31.1 31.6 -3° 1 27.8 27.3 2 27.5 27.5 3 28.0 27.0 4 28.2 27.5

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 5 28.0 27.3 Mean drop distance 0° 0.52 1.4 Mean drop distance +3°

31.58 31.8

Mean drop distance -3°

27.9 27.32

Second Part : Aerodynamics

• Diameter of hobby “flat head” projectile model : 55.25mm

`Cross sectional area: 3.06916x10-3m

• Diameter of magnum “round head” projectile model : 56.48mm

Cross sectional area: 3.1787044x10-3m

• Diameter of actual hobby/magnum pellets : 5.5mm

Drag Force (N) Flow Velocity (m/s)

Hobby projectile model

1 0.05325±0.00010 8.3

2 0.1178 ±0.0017 10.8

3 0.2024±0.0010 16.9

4 0.3113±0.0018 19.4

5 0.3835±0.011 23.9

Magnum projectile model

1 0.0630±0.0003 10.9

2 0.0722±0.0003 14.4

3 0.1154±0.0010 16.4

4 0.1343±0.0007 20.9

5 0.1537±0.0008 22.6

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 ANALYSIS

First Part :The effect of elevation angle on flight characteristics

• Using the mean mass, measured range, and measured drop data calculate the theoretical muzzle velocity of the pellets using Newtonian mechanics (SUVATs). Used to give us more data about the ballistic flight mechanisms observed in the experiment as well as to bring perspective towards the realistic expectations that one can have when using different ammunition types.

s = 𝒖𝒕 +𝟏

𝟐a𝒕𝟐 𝐫𝐞𝐚𝐫𝐫𝐚𝐧𝐠𝐞𝐝 𝐟𝐨𝐫 𝐭 = √

𝟐𝐬

𝐚

s = distance (M)

u= initial velocity (m/s)

a=acceleration (m/s²)

t=time (s)

Hobby – Sport Line Bisley Magnum .22

t = √2s

a t = √

2s

a

𝑡 = √2 ×(5.2 ×10−3)

9.81 𝑡 = √

2 ×( 0.0140)

9.81

𝑡 = √0.0104

9.81 𝑡 = √

0.028

9.81

𝑡 = 0.033𝑠𝑒𝑐 𝑡 = 0.053𝑠𝑒𝑐

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 • Calculation for speed, used to evaluate the impact of different pellet muzzle velocity to the efficiency of a weapon (due to the fact that audience background knowledge in ballistics would effectively observe the importance of the difference in speed). Thus the higher the speed; the more accurate the ammunition (the more damage observed).

𝒔𝒑𝒆𝒆𝒅 =𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞

𝒕𝒊𝒎𝒆

s= speed (m/s)

d=distance (M)

t= time (s)

Hobby – Sport Line Bisley Magnum .22

𝑠𝑝𝑒𝑒𝑑 =distance

𝑡𝑖𝑚𝑒 𝑠𝑝𝑒𝑒𝑑 =

distance

𝑡𝑖𝑚𝑒

𝑠𝑝𝑒𝑒𝑑 = 5.2𝑚

0.033𝑠 𝑠𝑝𝑒𝑒𝑑 =

5.2𝑚

0.053𝑠

𝑠𝑝𝑒𝑒𝑑 = 157.57𝑚/𝑠2 𝑠𝑝𝑒𝑒𝑑 = 98.11𝑚/𝑠2

• Acceleration due to gravity (hobby, 0°)

Using the measured range and drop data gathered, a theoretical value was calculated for acceleration due to gravity using Newtonian mechanisms. The reason for that would be that the gravity is of major importance to our report, since gravity is a concept that differs between different geographical locations ( i.e. earth vs. moon, the University of Kent vs. the highest point of Everest mountain).

𝒂 =𝟐𝐬

𝐭²

a= acceleration (m/s²)

s= distance (M)

t= time (s)

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 Hobby – Sport Line Bisley Magnum .22

𝑎 =2s

t² 𝑎 =

2s

t²

𝑎 = 2 ×(5.2×10−3)

0.033 𝑎 =

2 ×0.0140

0.053

𝑎 = 0.0104

0.0332 𝑎 = 0.028

0.0532

𝑎 = 9.56m/𝑠2 𝑎 = 9.97m/𝑠2

• Acceleration due to gravity (hobby, +3°)

𝒕 =𝒔𝒙

𝒗(𝒄𝒉𝒓𝒐𝒏𝒐𝒈𝒓𝒂𝒑𝒉) 𝐜𝐨𝐬𝟑

t= time (s)

s= distance (M) in the x direction

v= final velocity (m/s)

Hobby- Sport Line Bisley Magnum . 22

𝑡 =𝑠𝑥

𝑣(𝑐ℎ𝑟𝑜𝑛𝑜𝑔𝑟𝑎𝑝ℎ) cos3 𝑡 =

𝑠𝑥

𝑣(𝑐ℎ𝑟𝑜𝑛𝑜𝑔𝑟𝑎𝑝ℎ) cos3

𝑡 = 5.2𝑚

101.016𝑐𝑜𝑠(+3) 𝑡 = 5.2𝑚

75.258𝑐𝑜𝑠(+3)

𝑡 = 0.05154763799𝑠𝑒𝑐 𝑡 = 0.06919046745𝑠𝑒𝑐

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015

𝒖 = 𝒗(𝐜𝐡𝐫𝐨𝐧𝐨𝐠𝐫𝐚𝐩𝐡) 𝐬𝐢𝐧 𝟑


v= final velocity (m/s)

Hobby- Sport Line Bisley Magnum .22

𝑢 = 101.061sin (+3) 𝑢 = 75.258 sin(+3)

𝑢 = 5.289124074𝑚/𝑠 𝑢 = 3.938699395𝑚/𝑠

𝒂 =𝟐(𝐬 − 𝐮𝐭)

𝐭²

a= acceleration (m/s²)

s= distance (M)


t= time (s)

Hobby- Sport Line Bisley Magnum .22

𝑎 =2(s−ut)

t² 𝑎 =

2(s−ut)

t²

𝑎 =2(0.3158−5.289124074 ×0.05154763799)

0.05154763799² 𝑎 =

2(0.3180−3.938699395 ×0.06919046745)

0.06919046745²

𝑎 = 0.0863162939

2.657158982 × 10−3 𝑎 =

0.09095909543

0.06919046745²

𝑎 = 32.4844296 𝑚/𝑠2 𝑎 = 19.0000001𝑚/𝑠2

• Acceleration due to gravity (hobby, - 3°)


𝒕 =𝒔𝒙

𝒗(𝒄𝒉𝒓𝒐𝒏𝒐𝒈𝒓𝒂𝒑𝒉) 𝐜𝐨𝐬𝟑

t= time (s) s= distance (M) in the x direction v= final velocity (m/s) Hobby- Sport Line Bisley Magnum . 22

𝑡 =𝑠𝑥

𝑣(𝑐ℎ𝑟𝑜𝑛𝑜𝑔𝑟𝑎𝑝ℎ) cos3 𝑡 =

𝑠𝑥

𝑣(𝑐ℎ𝑟𝑜𝑛𝑜𝑔𝑟𝑎𝑝ℎ) cos3

𝑡 = 5.2𝑚

101.016𝑐𝑜𝑠(−3) 𝑡 = 5.2𝑚

75.258𝑐𝑜𝑠(−3)

𝑡 = 0.05154763799𝑠𝑒𝑐 𝑡 = 0.06919046745𝑠𝑒𝑐

𝒖 = 𝒗(𝐜𝐡𝐫𝐨𝐧𝐨𝐠𝐫𝐚𝐩𝐡) 𝐬𝐢𝐧 𝟑

u= initial velocity (m/s) v= final velocity (m/s) Hobby- Sport Line Bisley Magnum .22

𝑢 = 101.061sin (−3) 𝑢 = 75.258 sin(−3)

𝑢 = −5.289124074𝑚/𝑠 𝑢 = −3.938699395𝑚/𝑠


𝒂 =𝟐(𝐬 − 𝐮𝐭)

𝐭²

a= acceleration (m/s²) s= distance (M) u= initial velocity (m/s) t= time (s) Hobby- Sport Line Bisley Magnum .22

𝑎 =2(s−ut)

t² 𝑎 =

2(s−ut)

t²

𝑎 =2(0.3158−(−5.289124074 ×0.05154763799)

0.05154763799² 𝑎 =

2(0.3180−(−3.938699395 ×0.06919046745)

0.06919046745²

𝑎 = 1.176883706

2.657158982 × 10−3 𝑎 =

1.181040905

0.06919046745²

𝑎 = 442.9105348 𝑚/𝑠2 𝑎 = 246.7018521𝑚/𝑠2

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 Second Part : Aerodynamics

𝐅𝐝 =𝐂𝐝𝐀𝐕𝟐𝛒

𝟐 𝐫𝐞𝐚𝐫𝐫𝐚𝐧𝐠𝐞𝐝 𝐟𝐨𝐫 𝐂𝐝 =

𝟐𝐅𝐝

𝐀𝐕²𝛒

Fd = drag force (N)

Cd = drag coefficient

𝝆 = density of medium (Kgm-3) , which is 1.2 Kgm-3

V = flow velocity (m/s)

A = cross sectional area (m2)

• Drag coefficients for hobby pellet

1. Cd =2Fd

AV²ρ =

2 ×0.053225

3.06916𝑥10−3𝑚 ×8.32×1.2 =2.517328866

2. Cd =2Fd

AV²ρ =

2 ×0.1178

3.06916𝑥10−3𝑚 ×10.82×1.2 =0.5484373196

3. Cd =2Fd

AV²ρ =

2 ×0.2024

3.06916𝑥10−3𝑚 ×16.92×1.2 =0.3848276947

4. Cd =2Fd

AV²ρ =

2 ×0.3113

3.06916𝑥10−3𝑚 ×19.42×1.2 =0.4491639382

5. Cd =2Fd

AV²ρ =

2 ×0.3835

3.06916𝑥10−3𝑚 ×23.92×1.2 =0.3645849915

PS556 Ammunition Dynamics Dimitra Riga (dr280) 27 Mar 2015 • Drag coefficients for magnum pellet

1. Cd =2Fd

AV²ρ =

2 ×0.0630

3.06916𝑥10−3𝑚 ×10.92×1.2 =0.287949795

𝟐. Cd =2Fd

AV²ρ =

2 ×0.0722

3.06916𝑥10−3𝑚 ×14.42×1.2 =0.1890781888

𝟑. Cd =2Fd

AV²ρ =

2 ×0.1154

3.06916𝑥10−3𝑚 ×16.42×1.2 =0.2329953925

4. Cd =2Fd

AV²ρ =

2 ×0.1343

3.06916𝑥10−3𝑚 ×20.92×1.2 =0.1669600863

6. Cd =2Fd

AV²ρ =

2 ×0.1537

3.06916𝑥10−3𝑚 ×22.62×1.2 =0.1634128519


𝑪𝒃 =𝐦

𝑪𝒅𝑪𝑮𝟏

× 𝒅𝟐

Cb = ballistic coefficient (kgm-2)

m = mass (Kg)

Cd = drag coefficient

CG1= drag coefficient of G1 bullet, 0.5191

• Ballistic Coefficient for hobby pellet

1. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0007812.517328866

0.5191×0.00552

= 5.323983832

2. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0007810.5484373196

0.5191×0.00552

= 24.43710102

3. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0007810.3848276947

0.5191×0.00552

= 34.8265428

4. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0007810.4491639382

0.5191×0.00552

= 29.83814381

5. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0007810.3645849915

0.5191×0.00552

= 36.76021039


• Ballistic Coefficient for magnum pellet

1. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0013650.287949795

0.5191×0.00552

= 81.34699745

2. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0013650.1890781888

0.5191×0.00552

= 123.8844702

3. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0013650.2329953925

0.5191×0.00552

= 100.5514341

4. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0013650.1669600863

0.5191×0.00552

= 138.1116276

5. 𝐶𝑏 =m

𝐶𝑑𝐶𝐺1

×𝑑2 =

0.0013650.1634128519

0.5191×0.00552

= 143.3415485

One can observe certain discrepancies between the theoretical and observed muzzle velocitiy as

well as the acceleration (take into account gravity). The muzzle velocity for the magnum was

90.144m/s, but the calculated velocity was 157.55m/s. Also the observed velocity for hobby was

114.1m/s, but the calculated one was 98.11m/s. This can be accounted for by looking at the air

resistance as well as the pressure and humidity.

Furthermore the elevation observed affected more the hobby pallet, rather than the magnum pallet

that was heavier in weight, even though it should be considered that we had a range of 5.2m.


The change in flow velocity’s effect on drag coefficient

From the graphs one can see that the relationship between drag force generated and flow velocity differ a lot from the hobby pellet that is a lot more unsteady than the magnum pellet. With the magnum pellet demonstrating a steadier trend of results.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15 20 25 30

Flo

w v

elo

city

(m

/s)

Drag Force (N)

Relationship between drag force generated and flow velocity for the hobby pellet

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 5 10 15 20 25

Flo

w v

elo

city

(m

/s)

Drag Force (N)

Relationship between drag force generated and flow velocity for the magnum pellet


Relationship between drag coefficient and flow velocity

One can clearly see that the projectile design can affect the ballistic coefficient. Thus the projectile design is directly linked with its aerodynamic ability, the heavier a projectile; the larger the surface area that drags the projectile and makes it slower.

Furthermore it is obvious that the hobby pellets have significantly worst values, with the highest of

them being 36.7 Kgm-2 whilst the magnum’s was 143.3 Kgm-2 . The magnum’s smoother head

makes it easier to go against the drag force and that makes this pellet’s design more aerodynamic.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20 25

Dra

g co

eff

icie

nt

Flow Velocity (m/s)

Relationship between drag coefficient and flow velocity for the hobby pellet

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30

Dra

g C

oe

ffic

ien

t

Flow Velocity (m/s_

Relationship between drag coefficient and flow velocity for the hobby pellet


CONCLUSION

One can observe that on the first part of the experiment concerning the effect of the elevation angle on ballistic flight, it is obvious that the heavier pellet (magnum) does travel slower on an +3 angle, but far more accurately than the lighter pellet (hobby) that experience a faster travel but is affected more by the gravity and loses the ability to maintain its velocity. Furthermore gravity can be accounted for by looking at how long the pellets stay on air, since the longer they are between our range distance; the more the values will be effected. The magnum thus is a heavier projectile, that travels slowly and accurately. At the same time the hobby is lighter, faster and achieves better data.

Furthermore as far as the second part of the experiment is concerned the magnum has a better coefficient, since it can be affected less from the drag since it is a lot heavier than the hobby. Also the different between the height observed while air drop was not really drastic keeping both pellets.

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Ammunition Dynamics