External Ballistics


It deals with the flight of projectiles from the muzzle end of the weapon to the target. This is a complicated subject involving parameters such as the shape of the bullet, sectional density, atmospheric conclusions, and even rotation of the earth in larger caliber weapons, etc. It deals with the flight of the bullet from the muzzle of the weapon to the target.

It would be more appropriate if it is said that the external ballistics deals with the motion of projectiles/bullets 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.

There are various studies in which, the study of external ballistics has great importance. Some of them are following:

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

• Reconstruction of a 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 the flight of projectiles, which are mentioned below in brief:

(i) Trajectory means the path of the bullet from the muzzle to the striking point on the target. 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 is governed by Newton’s laws of motion.

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

The exact shape of this trajectory can be predetermined by knowing:

• Gravitational effect
• Muzzle velocity
• The angle of elevation of the barrel
• The sectional density of the bullet
• Bullet shape
• Factors affecting external ballistics
• Trajectory
• Range
• Drop of fall
• Angle of fall
• Remaining velocity
• Maximum range or extreme range
• Spin and drift
• Structure of projectiles
• Sectional density
• Gravitational pull
• Air resistance
• Weather condition
• Muzzle velocity
• Muzzle energy
• Momentum
• Trajectory formation

When a projectile comes out of the muzzle is should go along the axis of the barrel of the firearm and travel equal distances in an equal time interval. If no external force is acting upon it to change the course of the projectile.

Vacuum Trajectories and Air Trajectories

Both vacuum trajectories and air trajectories are important in external ballistics. Vacuum trajectories have 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 the case of extremely low-velocity projectiles.

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 the air to pay respect to the 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 concerning trajectories both in the air as well as in vacuum.

Calculation of Trajectory Parameters

There are many trajectory parameters. Their determination will be given below:

• Vertical range & maximum vertical range
• A drop of the bullet
• Remaining velocity
• Horizontal range
• Striking angle
• Velocity of escape
• Factors Affecting Trajectory

There are internal as well as external factors affecting the trajectory of the projectile which are discussed below:

• The velocity of the projectile
• The spin of the projectile
• The angle of fire
• The most important factor affecting the trajectory shape is the velocity of the bullet.
• Resistance of air
• Wind deflection
• The gravitational pull on the projectile
• Rotation of the earth

Advantages of Flat Trajectories

• Deflection of the projectile due to wind gets reduced to the extent of becoming negligible.
• The time, for which the projectile remains in the barrel gets reduced.
• When the errors in the estimation of distance are reduced, then the effective shooting range is increased.
• Small variations in individual cartridges such as mass, leakage of gases, or temperature become insignificant at higher velocities, which is a very important and significant factor in making trajectories flatter.
• Shooters feel more comfortable concerning personal factors.

Range Determination

Determination of range for vacuum trajectory is based upon the fact that in absence of air resistance, the horizontal component of velocity of projectile does not change and the force of gravity remains constant. The distance measured along the line of aim.

Effective range is a range of a projectile’s trajectory at which it damages the target effectively or the minimum range at which a projectile causes maximum damage to the target.

Extreme range is the maximum range or maximum distance for a projectile at which it reaches the target whether it causes damage to the target or not.

Angle of Fall

It is the angle at which a projectile drops down to the surface before reaching the target or after being hit to the target (if in case it did not enter to the target surface or its velocity is not sufficient to penetrate the target).

Escape Velocity

It is the velocity with which a projectile should be projected in the vertically upward direction so that it does not return to the earth. The gravitational pull of the earth is overcome by the projectiles flying at a very high velocity. Missiles flying over eight kilometers per second would orbit the earth in a trajectory.

If the velocity is more than eleven kilometers per second, the object will fly away into space and will not return to earth. The velocity at which an object escapes the gravitational pull is termed escape velocity.

The following relation gives the value of escape velocity
g = gravitational acceleration
R = Radius of earth

The value of escape velocity is 11 Kilometers per second. The rotation of the earth around its axis tends to hurl away the projectiles, but the effect is too little to affect practically the small ammunition trajectories.

The Velocity of Projectile

Since the velocity of the projectile is the most important factor affecting the shape of the trajectory, it is accordingly adjusted suitably to make the trajectory flatter. Flatter trajectories are very valuable for several reasons and an increase in the velocity of projectiles helps make them flatter, because the greater the velocity, the flatter will the trajectory become.

Black powder does not generate a very high velocity in small arms ammunition without an inconvenient increase in the recoil or weight of the weapon or both. Smokeless powders are now being used and success to a significant extent has been achieved in obtaining flat trajectories in small arms ammunition especially in military rifles.

Bullet Spin

The spin of the bullet is its revolving around its longitudinal axis or gyration. The rifling in the barrel of a firearm called a rifled weapon gives a spin to the bullet resulting in a reduction of air resistance because it disperses the air. In addition, it keeps the bullet in the line of fire and reduces the loss of velocity. If the spin of the bullet is not available, several problems which have been solved by it would appear including inaccurate aims.

The projectile passing through the barrel of a non rifled weapon would jostle with the sides of the barrel and acquire a directional motion, which would carry the projectile away from the aim. Accumulation of an increasing mass of air in the case of a non-rotation projectile will not only retard the velocity of the projectile but also change its flight path, making it inefficient. The rifling in the barrel overcomes all defects pointed out above and gives a spin to the bullet around its axis resulting in the decrease of air resistance also reducing the loss of velocity and keeping the projectile in the line of fire.

The Angle of Fire or Angle of Elevation

Angle with which the barrel axis line makes with baseline.

Wind of Deflection

If the wind is against the projectile then the velocity is decreased. If the wind is along the direction of the projectile then the velocity is increased.

Zero Range

It is the range where a projectile intersects the line of sight. It occurs twice, once on the way up and the other on the way down.


Drift in the flight takes place, significantly, due to the following reasons:

• So far the velocity of the wind is concerned it will make drift to the left when blowing from the right of the bullet to its left.
• If the wind direction is from the left of the bullet to its right, the bullet will get a drift to its right.
• If the motion of the wind is in the same direction as that of the bullet, it adds to the velocity of the projectile.
• Similarly, the wind blowing against the direction of the bullet will decrease the velocity of the bullet.


In cold temperature, more density causes decreases in velocity, and in cold temperature low chamber pressure develops which also reduces the initial velocity.

In hot temperatures, less density causes less air resistance that is very less reduction in velocity.

Ricochet of Bullet

It is a phenomenon that refers to the deflection of the projectile from its trajectory on the line of motion after hitting hard objects even from bones. A ricocheted bullet is deviated from its trajectory by striking an intermediary object.

Critical Angle

It is the minimum angle at which the bullet may produce a ricochet phenomenon is known as the critical angle.
Ricochet angle is the actual degree to which a bullet-producing ricochet from the surface is known as a ricochet angle.

Ricochet of a bullet may very with:
• Bullet shape
• Bullet material
• Bullet spin
• Velocity of bullet
• Intermediary surface
• Angle of incidence
• Nose (tip) of bullet

The effect caused due to ricochet:

A bullet that strikes the body after ricochet may abrade or bruise the body surface.

The bullet may enter the body, the entrance wound assumes a bizarre shape due to deformity. It may enter on its broadside and produce a keyhole entrance wound. Extensive focal lacerations of tissues may be found in the bullet ricochet after entering the body and hitting a bone. It may fracture the bone.

The path of a bullet that has a ricochet inside the body is less predictable and unusual deviations may be found.

Different Behaviors of Bullet

Sometimes when a bullet leaves the barrel, the bullet is in the slightly unstable condition which is due to three main factors:
• Yaw
• Precession
• Nutation

Yaw and Stability

Yaw is something, which only has real relevance to rifled ammunition. This is due to the slight destabilization of the bullet as it leaves the barrel and is probably the result of excessive spin on the bullet. This causes the bullet to describe an air spiral while at the same line having a spin around its tail axis.

At close ranges, this results in a large target group than would be expected. As the range becomes greater the effect disappears and the target groups return to their normal expected dimensions.

The effect is very similar to the spinning top, which wobbles slightly before settling down into a stable, spinning condition. In other words, yaw is over stabilization of the bullet as it leaves the muzzle resulting in the bullet describing an air spiral that at the same time spins around its tail axis.

The deviation between the longitudinal axis of the bullet and the axis of the path of the bullet is known as Yaw and the between it is termed as Angle of Yaw.


In this, the rotation of the bullet away from the line of flight.
Rotational effect of bullet around its mid axis.
It also decreases as the distance of bullets from the barrel increases.


It refers to the small circular movement at the tip of the bullet. It is a progressive cock-crew motion of the bullet.

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