Topic 4 - external ballistics

Question 1

what is the path of a projectile dictated by?

Answer 1

• Firearm projectiles do not follow true ‘ballistic arcs’.

The path of the projectile is dictated by:
* Gravity.
* Angle of launch (elevation).
* Velocity.
* Air density, temperature and humidity.
* Projectile shape (and drag coefficient) plus overall projectile stability.

Question 2

what is a realistic flight path like?

Answer 2

• A real arc, when air resistance
  is taken into consideration.
• The second half is more
  ‘truncated’. = shortened in duration or extent.
• Trajectory is uneven either
  side of the maximum height.

Question 3

when can SUVAT equations be used?

Answer 3

• Remember that in the absence of air resistance, calculations can be
  conducted using the SUVATS

S = u x t + 1/2 a x t^2

Vx = Sx / Ttotal

Question 4

Aerodynamic forces and gravity

Answer 4

• Aerodynamic forces and gravity are the two most important factors in calculating the true dynamics of a projectile.
• Aerodynamic drag is typically proportional to the square of the velocity, so drag builds significantly with velocity.
• There are other properties that also contribute to drag, including:
• The profile or shape of the projectile.
• The cross-sectional area of the projectile in the direction of travel.
• Air density.

Question 5

centre of mass/centre of gravity

Answer 5

• The centre of mass (AKA centre of gravity or CoM/CoG) is the point where the bullet balances its weight (W = mass × gravity). Think of it like the pivot point on a see-saw.

Question 6

centre of pressure/centre of mass

Answer 6

• Ideally, for the best flight stability, the CoP should be rearward of the CoM and very close to it. Fin stabilisation (more generally known as drag stabilisation) facilitates this by creating additional aerodynamic forces at the back of the projectile.
• Normal ‘spitzer’ bullet shapes actually have the CoP significantly forward of the CoM (as shown in the below figure). This naturally makes the bullet want to
  tumble as this creates a major ‘turning moment’.
• This is why bullets need gyroscopic stabilisation to overcome the want to tumble and keep the bullet point directed towards the impact point.

Question 7

Drag Stabilisation

Answer 7

• Drag stabilisation is typically achieved by
  adding fins to the projectile, which creates
  additional drag forces at the rear of the
  projectile (like the rocket in the image, where
  the fins make it less classically ‘aerodynamic’
  than a bullet shape).
• This brings the CoP rearward of the CoM,
  leading to a smaller and less influential turning
  moment.
• This results in a stable flight path without the
  need for any gyroscopic stabilisation.
• Gyroscopic stability would not be possible with this
  design anyway as the fins would not fit in the barrel.

Question 8

what equation would you use to calculate drag coefficient (Cd)

Answer 8

The total drag force that is experienced by the projectile can be calculated using the following equation:

Fd = 1/2 Cd AV^2 p

Where:
Fd = drag force in N
Cd = drag coefficient (no units)
V = flow velocity (for the air or projectile) in m.s-1
A = cross sectional area in m2
ρ = air density @ sea level, which is about 1.2 kg.m-3

Question 9

why do we use wind tunnels?

Answer 9

• Wind tunnels can be used to measure the drag
  force experienced by a projectile design as air
  is blown over a stationary object.
• In general, they provide us with the data to
  calculate the drag coefficients for a projectile
  (or other moving object).
• Force gauges can be attached to scale models
  of the object (or the real item can be used,
  depending on real size).
• Using high velocity wind tunnels is an
  extremely expensive endeavour – a cheaper
  alternative is to use computational fluid
  dynamics (CFD).

Question 10

what equation do we use to calculate drag coefficeint from wind tunnel data?

Answer 10

Cd = 2Fd / AV^2 p

Question 11

how do we calculate sectional density?

Answer 11

S = m / d^2

Question 12

what is sectional density? non ballistics definition

Answer 12

Mass of projectile divided by its cross-sectional area.

Question 13

what is sectional density? ballistics definition

Answer 13

mass of a projectile divided by its maximum diameter squared.

Question 14

what is a ballistic coefficient (Cb) ?

Answer 14

• This is a measure of the aerodynamic
  forces exerted on a particular bullet in flight
  and is known as Cb.
• The coefficient is specific to an individual
  bullet design and size and can be used to
  calculate real-time trajectory values during
  flight.
• It relates the bullet’s sectional density to its
  drag coefficient.
• Cb has units of kg.m-2

Question 15

what is modern Cb?

Answer 15

• The most up to date method is derived from the bullet’s cross-sectional
  area, drag coefficient, and mass.
• In an out-dated system, Cb was quoted as being between zero and one.
• The calculation method we will use yields a value with no theoretical
  upper limit.
• In reality, most commercially available ammunition falls into the 50 to
  500 range (when using SI units in the calculation).
• The exception is DU (depleted uranium) ammunition which is much higher due to its very high density (and mass), but this is not commonly encountered and is highly illegal!

Question 16

how do you calculate Cb?

Answer 16

Cb = Cg / Cd . m / d^2 = Cg / Cd . S

Once you have found Cd, you can then calculate Cb.

• It’s important to note that each Cd value leads to a separate Cb value.
• Never average your Cd and Cb values.

Where:
Cb = Ballistic coefficient
m = Mass of the test bullet
Cd = Drag coefficient
Cg = Drag coefficient of the G1 standard projectile = 0.5191
d = Diameter of the test bullet.
S = Sectional density of the test bullet

As the drag coefficient is determined in a wind tunnel we are usually reliant upon manufacturers for this data.

Question 17

what is gyroscopic stability?

Answer 17

• One of the major contributors to modern projectile accuracy is gyroscopic (or spin) stabilisation, which we have already discussed in this module.
• With the exception of a brief experiment in the 1983, all modern small arms (aside from shotguns) use the barrel to impart spin stabilisation upon the in-flight projectile.
• The exception here is the “Gyro-jet”, a case-less rocket bullet/hybrid which was spin stabilised after leaving the barrel - a good idea which just couldn’t be made to work with the available propellants.

Question 18

how is spin rate calculated? (gyroscopic stability)

Answer 18

spin rate = Vm / Tr , in S^-1

• Where: Vm = Muzzle velocity
  Tr = Rifling twist rate (length for one 360o rotation, which we called ‘L’ previously)
• Note that ‘spin’ relates to the bullet in flight and ‘twist’ relates to the barrel rifling.
• The spin rate is of limited use to a forensic examiner but for the sake of completeness, it’s useful to show how to calculate it here.
• This is the rate of rotation of the bullet about its longitudinal axis after leaving the firearm’s barrel but depends on the twist rate of the barrel, which we have looked at previously as a useful class characteristic of a
  firearm.

Question 19

what is the Greenhill formula?

Answer 19

relationship between the dimensions of a bullet and its optimum twist rate.

T = Cd^2 / L

Where:
T = Optimum twist rate in metres (remember, this is distance for one full rotation).
C = A constant (Use the values of ‘150’ for muzzle velocities under 860 m.s-1 and ‘180’ for over 860 m.s-1
).
d = Bullet diameter in metres.
L = Bullet length in metres.

Question 20

what is aerodynamic lift?

Answer 20

• This is where a ‘boundary layer effect’ occurs whenever air passes over a curved surface.
• It is how planes fly.
• For example, air moving over the curved
  upper wing surface travels further then the
  air passing under the flat lower surface.
• This creates a pressure drop over the wing which generates lift.
• The amount of lift is proportional to the air speed over the surface and the rate of curvature.
• This can be related to a spinning bullet in a crosswind.
• The bullet will tend to lift or drop at a greater than expected rate in a strong cross wind as the wind may blow over variable shape profiles of the bullet depending on wind angle of interaction.

Question 21

what is wind deflection?

Answer 21

Two main types of wind deflection are experienced by a projectile:
1. Aerodynamic: Caused by wind flow over the projectile in flight, generating more lift on one side of the bullet due to spin induced pressure difference.

• This is much the same as that experienced by an aircraft wing.
• This effect is very small and only shows up at very long range.

2. Windage: Deflection caused by constant wind pressure during projectile flight.
   * This has a much more pronounced effect on the overall trajectory.

• A strong breeze can cause several metres of deflection at longer ranges.

Question 22

how do you calculate wind deflection?

Answer 22

• Bullet wind deflection due to windage can be calculated using the vector addition formula.

R = square root [ (Ax + Bx)^2 + (Ay + By)^2 ]

where:
Ax = Acos 0 with line through
Ay = Asin 0 with line through

Bx = Bcos a that looks like a fish
By = Bsin a that looks like a fish

Question 23

what are the causes of abnormal flight characteristics?

Answer 23

Rifling-induced instabilities are caused by a low spin rate being
experienced by the projectile and has three main causes:

1. Low muzzle velocity: in this case, the bullet engages with the rifling but
   never reaches a spin rate that will ensure stability. This could be
   caused by a low or defective charge, excessive bore friction, a bullet of
   incorrect caliber being used in firearm to name a few reasons.
2. High muzzle velocity: this is particularly seen with unjacketed bullets.
   The bullet velocity may be so high that the soft bullet skids over the
   rifling. This is typically caused by a hand-loaded round or an incorrect
   bullet choice. This can lead to distinctive rifling marks.
3. Defective rifling: The bullet fails to engage with the worn or defective
   rifling, leading to a low spin rate (compare to Greenhill optimum),
   aerodynamic imbalance and potential tumbling.

Question 24

what is projectile instability?

Answer 24

• Yaw refers to the lateral
  movement of the nose of
  the bullet away from
  the line of flight.
• Precession refers to
  rotation of the bullet
  around the centre of
  mass.
• Nutation refers to small
  circular movement at
  the bullet tip due to the tip
  not being perfectly round.
• Yaw and precession typically decrease as the distance of the bullet from the
  barrel increases, since gyroscopic stability will keep them under control.

Question 25

what causes yaw?

Answer 25

Main causes of yaw include: * A poorly cast bullet or bad loading, causing the bullet to be off centre in the cartridge case neck. * Irregular rifling or non-optimal spin rate. * This is also the main reason why ‘sabot’ rounds (saboted projectiles shown to the right), though a very good idea, cannot always be made to work in small caliber applications.