**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

**SUBJECT ** **FORENSIC SCIENCE **

**Paper No. and Title ** **Paper No. 6: Forensic Ballistics **
**Module No. and Title ** **Module No. 13: External Ballistics - I **

**Module Tag ** **FSC_P6_M13 **

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

**TABLE OF CONTENTS **

**1. Learning Outcomes **

**2. Introduction – External Ballistics **
**3. Trajectory of Bullet in Vacuum **
**4. Calculation of Trajectory Parameters **
** 4.1 Ballistic Coefficient **

**4.2 Vertical Range Determination **
**4.3 Bullet Drop **

**4.4 Calculation of Remaining Velocity **
**5. Summary **

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

**1. Learning Outcomes **

After studying this module, you shall be able to

Know about the external ballistics

Evaluate the velocity of bullet in air

Analyze the path of bullet in air

### 2. Introduction – External Ballistics

It would be more appropriate if it is said that the external ballistics deals with, the motion of projectiles/ bullet exiting from the muzzle of the weapon, to the target or till it drops down under the influence of gravity. The trajectories of projectiles are parabolic in form, but would differ in curvatures and lengths. The study of external ballistics assumes great importance due to its contribution especially in the following studies:

Determination of different kind of ranges namely fatal range, effective range and extreme range.

Reconstruction of sequence of events in different cases involving criminal angles.

Problems involving ricochet of bullets.

Problems relating to safe zones or danger portions or danger spaces.

There are three basic considerations regarding flight of projectiles, which are mentioned, in brief, below:

(i) Trajectory means the path of the bullet from the muzzle to the striking point on the target and it is in the form of a parabola.

(ii) The flight of all projectiles whether through the air or in the vacuum without any air resistance in governed by Newton‟s laws of motion.

(iii) Many factors influence the flight of the bullet, but the two main factors are Gravitational pull of the earth, which brings it to the ground or target, and resistance of the air which reduces its velocity.

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

### 3. Trajectory of a Bullet in Vacuum

Let a bullet be fired from a firearm in the direction OZ at an angle α (Alpha) with the horizon with a velocity V as shown below:

If there are no factors influencing the initial velocity V like air resistance or gravitational pull of the earth as can happen in space, the projectile would travel in a straight line covering equal distances in every second. According to Newton‟s first law of motion, a body at rest or in motion would continue to be in that position unless some force is applied to change that position. Thus the positions of the bullet during four seconds are shown at A, B, C and D after covering distances v, 2v, 3v and 4v. The velocity V of the bullet is along OZ hence its velocity towards OX and OY would be V cos α and V sin α respectively

In actual practice both air resistance and gravitational pull plays a very important role and can greatly influence the motion of projectiles and to illustrate this point some cases of firing in air are cited below:

(i) People celebrate different occasions like marriages, festivals and victories in court cases by firing weapons in air to express their happiness.

(ii) Sometimes crowd needs to be dispersed and Police fires in air to disperse the same.

(iii) Manufacturer of firearms and ballistic experts may fire in air in connection with technical work concerning weapons

(iv) Criminals may use firearms in cases involving homicides

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

The effect of air resistance and gravitational pull would be discussed together to know the change in the trajectory of bullet in air. Air will offer resistance to the moving bullet, resulting in the change (decrease) in its velocity. The gravitation pull will make a further change in the path, which will attain the form of a parabola as shown below.

Positions of bullet shown at A, B, C, D are in case of vacuum firing, but the position will change when firing takes place in air, as air offers resistance to the bullet and decreases its velocity and hence the distance travelled would be lesser, giving rise to new positions A‟, B,‟ C,‟ and D‟.

Gravitational pull will bring the bullet downwards whose trajectory forms a parabola.

New positions of the bullet as a result of air resistance and bullet drop will be E, F, G and H to give the trajectory a form of a parabola. The exact shape of the trajectory can be pre- determined by including additional factors like shape of the bullet and its sectional density which are defined as follows:

Sectional density of a bullet Weight of the bullet_{2}

= (Diameter)

The difference in air resistance is referred to in ballistics as “Form Factor”.

The efficiency of the flight of the projectile depends upon its sectional density- the greater, the sectional density, the better is the efficiency. Lead is therefore a good material for the manufacturing of projectiles.

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

**Vacuum Trajectories and Air Trajectories **

Both the vacuum trajectories and air trajectories are important in external ballistics.

Vacuum trajectories have assumed great importance in space travel but derivations and formulations applicable to vacuum trajectories cannot be applied to the real trajectories in air and for small arms ammunition. At best it could be a rough estimate in case of extremely low velocity projectiles.

The firing of guns in air is a frequent phenomenon in several cases already mentioned, namely police firing in air for dispersing of crowd or use by experts & research works in the field of ballistics.

In Forensic science, frequent examination of improvised firearms is required, as they are used in many criminal cases involving homicides & murders and this may involve firing in the air.

Many people use firearms in air to pay respect to National flag.

Both non-standard and standard weapons are used for committing crimes, where air does play an important part, hence the study of external ballistics is an important field with respect to trajectories both in the air as well as in vacuum.

**4. Calculation of Trajectory Parameters **

There are many trajectory parameters. Their determination will be followed, later on.

They are given below:

Vertical range & maximum vertical range Drop of the bullet

Remaining velocity Horizontal range Striking angle Velocity of escape

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

Before discussing about evaluation of these parameters, it is necessary that various terms used during evaluation are properly explained.

Let a bullet be projected from a point O in the direction OC as shown below:

*OX* Represents maximum horizontal range & horizontal axis

*OY* Represent the vertical axis
Angle *COX* Angle of projection

Curve *ODX* Represent the trajectory i.e. path traced by the bullet

*OAC* Direction of projection of bullet

*CX* Represents the bullet drop
*DZ*Represents the midrange height
Angle *OXB*Represents angle of impact
**4.1 Ballistic Coefficient **

(a) Ballistic coefficient is usually represented by C and indicates the ability of the projectile to overcome air resistance, and its efficiency in flight.

(b) The sectional density is not the only factor affecting the retardation (the degree of velocity loss due to the air) of a bullet, as shape of the bullet also plays a very important part.

(c) If form factor is taken in to consideration and sectional density is divided by it, the term so obtained is called Ballistic coefficient.

*C*= *w*
*id*^{2}

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

Where,

C = Ballistic coefficient W = Weight of the bullet i = Form factor

d = Diameter of the bullet

(d) Form factor is basically a measure of how streamlined a bullet is.

(e) Thus, the larger the ballistic coefficient, the better the bullet will retain its velocity and lower the bullet drop for any given distance.

**Example **

Sectional density of 0.38‟ special bullet having a diameter of 0.357” and weight 150 gr
will be 12.39 gr/*inch*^{2}

A much lighter bullet, 12.5grams having the same caliber will give sectional density of 125gr 2

= 980 gr/ inch 0.357× 0.357

Sectional density of the lighter bullet is lower indicating lower carrying capacity though caliber of both bullets is the same, their carrying capacities are different.

**4.2 Vertical Range Determination **

It has been clear, earlier, that a projectile when projected at an angle Alpha

###

^{}

^{to the }

horizontal axis with velocity V will have the following components:

Horizontal Component = V cos ^{}
Vertical Component = V sin^{}

If the firearm is fired in vacuum where there is absence of air and hence no air resistance, it would be correct to say that by the time the projectile loses all its velocity due to gravitational pull of the earth and attains zero velocity it can no more go upwards and that would be the vertical range attained by the bullet knowing that final velocity of the bullet will be equal to initial velocity = -gt (where, the time of flight is t and g is acceleration due to gravity).

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

Final Velocity (O) = *V*sinµ-gt
Or *t*=*V*sinµ

*g*

Now using the formula = (Final velocity)^{2} – (Initial velocity)^{2} = 295
Where,

S is the distance travelled by the bullet *O V* ^{2}sin^{2}2*gs*

S is equal to height reached by the bullet and is therefore equal to H (S=H)

2 2

sin 2

*V* *gh* where, H is the height reached by the bullet
Or *H* =*V*^{2}sin^{2}µ

2*g* =Vertical range

**Calculation of Maximum Vertical Range **

By having the maximum value of *Sin*^{2} , we shall have maximum value of vertical
range.

Maximum value of sin is one when µ=90°, so Maximum vertical range

2

*V*2
*g* , so
when the bullet is fired vertically upward, range will be maximum and equal

2

162
*V*
*g*

**4.3 Bullet Drop**

The rate of fall of a bullet is determined by the use of well-known formula which is stated below:

1 2

2

*h* *gt*

Where, *h*drop of the bullet

*g* acceleration due to gravitational pull of the earth. Its value is 32.17 feet /sec.

It is generally taken as 32 feet per second per second.

*t*time in seconds

Thus a bullet will fall through a distance of 16 feet as per calculations shown below:

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

Bullet Drop ^{1} ^{2}

2

*h* *gt*

When *t*1

1 1

32 16

2 2

*h* *g* feet

It may be clearly understood that the drop of bullet does not depend either on the mass of the bullet or its initial velocity. It is totally independent of both the mass and velocity of the bullet whether the velocity was initial or attained during the flight. Thus all bullets whether moving will small velocity 200 feet/ second or large velocities of about 4000 feet/sec will have a fall of 16 feet in one second or 4 feet in a quarter of a second. The difference in bullet velocities will have effect on the distance travelled as bullets with higher velocity will take shorter time for a given distance.

The bullet does not continue with the same velocity during its flight. As the pressure on its nose causes resistance, it will result in gradual reduction in its velocity.

** 4.4 Calculation of Remaining Velocity **

The remaining velocity at any time “t” after its projection can be ascertained.

The resultant velocity of a projectile projected at an angle with the horizontal axis can be calculated (say after time t) from the square root of the sum of squares of its vertical and horizontal components.

Let a bullet be projected with a velocity V at an angle to the horizon. The vertical and horizontal components of velocity V after time „t‟ would be, V sin gtin the vertical direction V cos in the horizontal direction. Horizontal component has not changed because there is no gravitational pull in that direction but the vertical component which had its value V sin at the start gets reduced to V sin gt after a time t due to gravitational pull of the earth

Resultant Velocity = ^{Vr}^{}

###

^{V}^{sin}

^{ }

^{gt}###

^{2}

^{}

^{V}^{cos}

^{}

###

^{2}

2 2 2 2 2 2

sin 1 2 sin cos

*V* *g t* *gt V* *V*

###

2 sin2 cos2 2 sin 2 2

*V* *V gt* *g t*

###

2 2 2 2 2

2 sin sin cos 1

*Vr* *V* *g V* *t* *g t*

**FORENSIC SCIENCE ** **PAPER No. 6: Forensic Ballistics ** **MODULE No. 13: External Ballistics - I ** ** **

**5. Summary **

External ballistics deals with, the motion of projectiles/ bullets from the muzzle end of the weapon, to the target or till it drops down under the influence of gravity.

Two factors which influence the flight of the bullet are:

o Gravitational pull of the earth which brings it to the ground or target and o Resistance of the air which reduces its velocity.

A projectile when projected at an angle Alpha

###

^{}to the horizontal axis with velocity V will have the following components:

Horizontal Component = V cos ^{}
Vertical Component = V sin^{}

Vertical range is calculated by *H* =*V*^{2}sin^{2}µ
2g

The rate of fall of a bullet (Bullet Drop) is determined by ^{1} ^{2}

2

*h* *gt*