Bullet flight trajectory, its elements, properties. Types of trajectories and their practical significance. The shape of the trajectory of a bullet and its meaning What is the trajectory of a bullet
trajectory called the curved line described by the center of gravity of the bullet in flight.
A bullet flying through the air is subjected to two forces: gravity and air resistance. The force of gravity causes the bullet to gradually descend, and the force of air resistance continuously slows down the movement of the bullet and tends to topple it. As a result of the action of these forces, the bullet's flight speed gradually decreases, and its trajectory is an unevenly curved curved line in shape. Air resistance to the flight of a bullet is caused by the fact that air is elastic medium and therefore part of the energy of the bullet is expended on movement in this medium.
The force of air resistance is caused by three main causes: air friction, the formation of vortices and the formation of a ballistic wave.
The shape of the trajectory depends on the magnitude of the elevation angle. As the elevation angle increases, the height of the trajectory and the total horizontal range of the bullet increase, but this occurs up to a certain limit. Beyond this limit, the trajectory height continues to increase and the total horizontal range begins to decrease.
The angle of elevation at which the full horizontal range of the bullet is greatest is called the angle longest range. The value of the angle of greatest range for bullets various kinds weapons is about 35 °.
Trajectories obtained at elevation angles, smaller angle longest range are called flat. Trajectories obtained at elevation angles greater than the angle largest angle longest range are called mounted. When firing from the same weapon (with the same initial speeds) you can get two trajectories with the same horizontal range: flat and hinged. Trajectories having the same horizontal range and swarms of different elevation angles are called conjugated.
When firing from small arms only flat trajectories are used. How flatter trajectory, the greater the extent of the terrain, the target can be hit with one sight setting (the less impact on the results of shooting has an error in determining the setting of the sight): this is practical value trajectories.
The flatness of the trajectory is characterized by its greatest excess over the aiming line. At a given range, the trajectory is all the more flat, the less it rises above the aiming line. In addition, the flatness of the trajectory can be judged by the magnitude of the angle of incidence: the trajectory is the more flat, the smaller the angle of incidence. The flatness of the trajectory affects the range direct shot, affected, covered and dead space.
Trajectory elements
Departure point- the center of the muzzle of the barrel. The departure point is the start of the trajectory.
Weapon horizon is the horizontal plane passing through the departure point.
elevation line- a straight line, which is a continuation of the axis of the bore of the aimed weapon.
Shooting plane- a vertical plane passing through the line of elevation.
Elevation angle- the angle enclosed between the line of elevation and the horizon of the weapon. If this angle is negative, then it is called the angle of declination (decrease).
Throw line- a straight line, which is a continuation of the axis of the bore at the time of the bullet's departure.
Throwing angle
Departure angle- the angle enclosed between the line of elevation and the line of throwing.
drop point- the point of intersection of the trajectory with the horizon of the weapon.
Angle of incidence- the angle enclosed between the tangent to the trajectory at the point of impact and the horizon of the weapon.
Total horizontal range- the distance from the point of departure to the point of fall.
final speed- the speed of the bullet (grenade) at the point of impact.
Total flight time- the time of movement of a bullet (grenade) from the point of departure to the point of impact.
Top of the path - highest point trajectories over the horizon of the weapon.
Trajectory height- the shortest distance from the top of the trajectory to horizon arms.
Ascending branch of the trajectory- part of the trajectory from the departure point to the top, and from the top to the drop point - the descending branch of the trajectory.
Aiming point (aiming)- the point on the target (outside it) at which the weapon is aimed.
line of sight- a straight line passing from the shooter's eye through the middle of the sight slot (level with its edges) and the top of the front sight in aiming point.
aiming angle- the angle enclosed between the line of elevation and the line of sight.
Target elevation angle- the angle enclosed between the aiming line and the horizon of the weapon. This angle is considered positive (+) when the target is higher and negative (-) when the target is below the weapon's horizon.
Sighting range- distance from the departure point to the intersection of the trajectory with the line of sight. The excess of the trajectory over the line of sight is the shortest distance from any point of the trajectory to the line of sight.
target line- a straight line connecting the departure point with the target.
Slant Range- distance from the departure point to the target along the target line.
meeting point- point of intersection of the trajectory with the surface of the target (ground, obstacles).
Meeting angle- the angle enclosed between the tangent to the trajectory and the tangent to the target surface (ground, obstacles) at the meeting point. The meeting angle is taken as the smaller of the adjacent angles, measured from 0 to 90 degrees.
Rice. 1. Artillery battleship"Marat"
Ballistics(from the Greek βάλλειν - to throw) - the science of the movement of bodies thrown in space, based on mathematics and physics. It deals mainly with the study of the movement of projectiles fired from firearms, rocket projectiles and ballistic missiles.
Basic concepts
Rice. 2. Elements of firing naval artillery
The main objective of shooting is to hit the target. To do this, the tool must be given a strictly defined position in the vertical and horizontal planes. If we aim the gun so that the axis of the bore is directed at the target, then we will not hit the target, since the trajectory of the projectile will always pass below the direction of the axis of the bore, the projectile will not reach the target. To formalize the terminological apparatus of the subject under consideration, we introduce the main definitions used when considering the theory of artillery firing.
Departure point
called the center of the muzzle of the gun.
drop point
called the point of intersection of the trajectory with the horizon of the gun.
horizon guns
called the horizontal plane passing through the departure point.
Elevation line
called the continuation of the axis of the bore of the pointed gun.
Throwing line
OB is the continuation of the axis of the bore at the time of the shot. At the moment of the shot, the gun shudders, as a result of which the projectile is thrown not along the line of elevation of the OA, but along the line of throwing of the OV (see Fig. 2).
Goal line
OC is the line connecting the gun to the target (see Fig. 2).
Line of sight (sight)
called the line running from the gunner's eye through the optical axis of the sight to the aiming point. When firing direct fire, when the line of sight is directed at the target, the line of sight coincides with the line of the target.
Falling line is called the tangent to the trajectory at the point of incidence.
Rice. 3. Shooting at an overlying target
Rice. 4. Shooting at the underlying target
Elevation (greek phi)
called the angle between the line of elevation and the horizon of the gun. If the bore axis is directed below the horizon, then this angle is called the angle of descent (see Fig. 2).
The firing range of the gun depends on the elevation angle and firing conditions. Therefore, in order to throw the projectile to the target, it is necessary to give the gun such an elevation angle at which the firing range will correspond to the distance to the target. The firing tables indicate which aiming angles must be given to the gun in order for the projectile to fly to the desired range.
Throwing angle (Greek theta zero)
the angle between the line of throw and the horizon of the gun is called (see Fig. 2).
Departure angle (Greek gamma)
called the angle between the line of throw and the line of elevation. In naval artillery, the departure angle is small and is sometimes not taken into account, assuming that the projectile is thrown at an elevation angle (see Fig. 2).
Aiming angle (Greek alpha)
the angle between the line of elevation and the line of sight is called (see Fig. 2).
Target elevation angle (greek epsilon)
called the angle between the line of the target and the horizon of the gun. When a ship fires at sea targets, the elevation angle of the target is equal to zero, since the target line is directed along the horizon of the gun (see Fig. 2).
Incident angle (Greek theta s Latin letter from)
the angle between the target line and the fall line is called (see Fig. 2).
Meeting angle (Greek mu)
is the angle between the line of incidence and the tangent to the target surface at the meeting point (see Fig. 2).
The value of the value of this angle greatly affects the resistance of the armor of the ship, which is fired at, to penetration by shells. Obviously, the closer this angle is to 90 degrees, the higher the probability of penetration, and the opposite is also true.
Shooting plane
called the vertical plane passing through the line of elevation. When the ship fires at sea targets, the aiming line is directed along the horizon, in this case the elevation angle equal to the angle aiming. When a ship fires at coastal and air targets, the elevation angle is equal to the sum of the aiming angle and the elevation angle of the target (see Fig. 3). When firing a coastal battery at sea targets, the elevation angle is equal to the difference between the aiming angle and the elevation angle of the target (see Fig. 4). Thus, the magnitude of the elevation angle is equal to the algebraic sum of the aiming angle and the elevation angle of the target. If the target is above the horizon, the target elevation angle is "+", if the target is below the horizon, the target elevation angle is "-".
The influence of air resistance on the trajectory of the projectile
Rice. 5. Changing the trajectory of the projectile from air resistance
The flight path of a projectile in airless space is a symmetrical curved line, called a parabola in mathematics. The ascending branch coincides in shape with the descending branch and, therefore, the angle of incidence is equal to the angle of elevation.
When flying in the air, the projectile spends part of its speed to overcome air resistance. Thus, two forces act on the projectile in flight - the force of gravity and the force of air resistance, which reduces the speed and range of the projectile, as illustrated in Fig. 5. The magnitude of the air resistance force depends on the shape of the projectile, its size, flight speed and air density. The longer and more pointed the head of the projectile, the less air resistance. The shape of the projectile is especially affected at flight speeds exceeding 330 meters per second (that is, at supersonic speeds).
Rice. 6. Short-range and long-range projectiles
On fig. 6, on the left, is a short-range, old-style projectile and a more oblong, pointed, long-range projectile on the right. It can also be seen that a long-range projectile has a conical narrowing at the bottom. The fact is that a rarefied space and turbulence are formed behind the projectile, which significantly increase air resistance. By narrowing the bottom of the projectile, a decrease in the amount of air resistance resulting from rarefaction and turbulence behind the projectile is achieved.
The force of air resistance is proportional to the speed of its flight, but not directly proportional. Dependence is formalized more difficult. Due to the action of air resistance, the ascending branch of the projectile's flight path is longer and delayed than the descending one. The angle of incidence is greater than the angle of elevation.
In addition to reducing the range of the projectile and changing the shape of the trajectory, the force of air resistance tends to overturn the projectile, as can be seen from Fig. 7.
Rice. 7. Forces acting on a projectile in flight
Therefore, a non-rotating elongated projectile will roll over under the action of air resistance. In this case, the projectile can hit the target in any position, including sideways or bottom, as shown in Fig. 8.
Rice. 8. Rotation of a projectile in flight under the influence of air resistance
So that the projectile does not roll over in flight, it is given rotary motion using rifling in the bore.
If we consider the effect of air on a rotating projectile, we can see that this leads to a lateral deviation of the trajectory from the plane of fire, as shown in Fig. nine.
Rice. 9. Derivation
derivation called the deviation of the projectile from the plane of fire due to its rotation. If the rifling twists from left to right, then the projectile deflects to the right.
The influence of the angle of elevation and the initial velocity of the projectile on the range of its flight
The range of a projectile depends on the elevation angles at which it is thrown. An increase in the flight range with an increase in the elevation angle occurs only up to a certain limit (40-50 degrees), with a further increase in the elevation angle, the range begins to decrease.
Range limit angle
called the elevation angle at which the greatest firing range is obtained for a given initial velocity and projectile. When firing in an airless space, the greatest range of the projectile is obtained at an elevation angle of 45 degrees. When firing in the air, the maximum range angle differs from this value and is not the same for different guns (usually less than 45 degrees). For ultra-long-range artillery, when the projectile flies for a significant part of the path high altitude in highly rarefied air, the maximum range angle is more than 45 degrees.
For a gun of this type and when firing a certain type of ammunition, each elevation angle corresponds to a strictly defined range of the projectile. Therefore, in order to throw the projectile at the distance we need, it is necessary to give the gun an elevation angle corresponding to this distance.
The trajectories of projectiles fired at elevation angles smaller than the maximum range angle are called flat trajectories
.
The trajectories of projectiles fired at elevation angles greater than the maximum range angle are called " hinged trajectories" .
Projectile dispersion
Rice. 10. Dispersion of projectiles
If several shots are fired from the same gun, with the same ammunition, with the same direction of the gun barrel, under the same, at first glance, conditions, then the shells will not hit the same point, but will fly along different trajectories, forming a bundle of trajectories, as illustrated in fig. 10. This phenomenon is called projectile dispersion .
The reason for the dispersion of projectiles is the impossibility of achieving exactly the same conditions for each shot. The table shows the main factors that cause projectile dispersion and possible ways reduce this dispersion.
| The main groups of causes of dispersion | Conditions that give rise to the causes of dispersion | Control measures to reduce dispersion |
|---|---|---|
| 1. Variety of starting speeds |
(dimensions and location of the leading belt, sending shells).
|
|
| 2. Variety of throwing angles |
|
|
| 3. A variety of conditions in the flight of a projectile |
Variety of influence of the air environment (density, wind). |
The area on which projectiles fired from a gun with the same direction of the barrel bore fall is called scattering area
.
The middle of the scattering area is called midpoint of fall
.
An imaginary trajectory passing through the departure point and middle point fall is called average trajectory
.
The scattering area has the shape of an ellipse, so the scattering area is called scattering ellipse
.
The intensity with which the projectiles hit different points of the dispersion ellipse is described by a two-dimensional Gaussian (normal) distribution law. From here, if we follow exactly the laws of probability theory, we can conclude that the scattering ellipse is an idealization. The percentage of shells hitting inside the ellipse is described by the three-sigma rule, namely, the probability of shells hitting the ellipse, the axis of which is equal to three times square root from the variances of the corresponding one-dimensional Gaussian distribution laws is 0.9973.
Due to the fact that the number of shots from one gun, especially large caliber, as already mentioned above, due to wear often does not exceed one thousand, this inaccuracy can be neglected and it can be assumed that all shells fall into the dispersion ellipse. Any section of a beam of projectile flight paths is also an ellipse. The dispersion of projectiles in range is always greater than in the lateral direction and in height. The value of the median deviations can be found in the main shooting table and the size of the ellipse can be determined from it.
Rice. 11. Shooting at a target with no depth
Affected space is the space over which the trajectory passes through the target.
According to fig. 11, the affected space is equal to the distance along the horizon AC from the base of the target to the end of the trajectory passing through the top of the target. Each projectile that fell outside the affected space either passed above the target or fell before it. The affected space is limited by two trajectories - the OA trajectory passing through the base of the target, and the OS trajectory passing through the top point of the target.
Rice. 12. Shooting at a target with depth
In case the target to be hit has depth, the amount of space to hit is increased by the value of the target's depth, as illustrated in Fig. 12. The depth of the target will depend on the size of the target and its position relative to the plane of fire. Consider the most likely target for naval artillery - an enemy ship. In such a case, if the target is coming from us or towards us, the depth of the target is equal to its length, when the target is perpendicular to the plane of fire, the depth is equal to the width of the target, as illustrated in the figure.
Given the fact that the scattering ellipse has great length and a small width, it can be concluded that at a shallow target depth, fewer projectiles hit the target than at a large depth. That is, than more depth target, the easier it is to hit. With an increase in the firing range, the affected target space decreases, as the angle of incidence increases.
Straight shot a shot is called, in which the entire distance from the point of departure to the point of impact is the affected space (see Fig. 13).
Rice. 13. Direct shot
This is obtained if the height of the trajectory does not exceed the height of the target. The range of a direct shot depends on the steepness of the trajectory and the height of the target.
Range of a direct shot (or range of flattening) called the distance at which the height of the trajectory does not exceed the height of the target.
The most important works on ballistics
17th century
- - Tartaglia theory,
- 1638- labor Galileo Galilei about the parabolic motion of a body thrown at an angle.
- 1641- a student of Galileo - Toricelli, developing the parabolic theory, derives the expression horizontal range, which later formed the basis of artillery firing tables.
- 1687- Isaac Newton proves the influence of air resistance on a thrown body, introducing the concept of the shape factor of the body, and also drawing a direct dependence of the movement resistance on the cross section (caliber) of the body (projectile).
- 1690— Ivan Bernoulli mathematically describes main task ballistics, solving the problem of determining the motion of a ball in a resisting medium.
18th century
- 1737- Bigot de Morogues (1706-1781) published a theoretical study of the issues internal ballistics, which laid the foundation for the rational design of tools.
- 1740- the Englishman Robins learned to determine the initial speeds of the projectile and proved that the projectile flight parabola has a double curvature - its descending branch is shorter than the ascending one, in addition, he empirically concluded that the air resistance to the flight of projectiles at initial speeds above 330 m / s increases abruptly and should calculated using a different formula.
- Second half of the 18th century
- Daniel Bernoulli deals with the issue of air resistance to the movement of projectiles;
- mathematician Leonhard Euler develops the work of Robins, Euler's work on internal and external ballistics forms the basis for the creation of artillery firing tables.
- Mordashev Yu. N., Abramovich I. E., Mekkel M. A. Textbook of deck artillery commander. M.: Military publishing house of the Ministry armed forces Union of the SSR. 1947. 176 p.
Flight of a bullet in the air
Having flown out of the bore, the bullet moves by inertia and is subjected to the action of two forces of gravity and air resistance
The force of gravity causes the bullet to gradually descend, and the force of air resistance continuously slows down the movement of the bullet and tends to topple it. To overcome the force of air resistance, part of the energy of the bullet is expended
The force of air resistance is caused by three main reasons: air friction, the formation of eddies, and the formation of a ballistic wave (Fig. 4)
The bullet collides with air particles during flight and causes them to oscillate. As a result, the air density increases in front of the bullet and sound waves are formed, a ballistic wave is formed. The force of air resistance depends on the shape of the bullet, flight speed, caliber, air density
Rice. 4. Formation of air resistance force
In order to prevent the bullet from tipping over under the action of air resistance, it is given a rapid rotational movement with the help of rifling in the bore. Thus, as a result of the action of gravity and air resistance on the bullet, it will not move uniformly and rectilinearly, but will describe a curved line - a trajectory.
trajectory called the curved line described by the center of gravity of the bullet in flight.
To study the trajectory, the following definitions are adopted (Fig. 5):
· departure point - the center of the muzzle of the barrel, in which the center of gravity of the bullet is located at the time of departure. The moment of departure is the passage of the bottom of the bullet through the muzzle of the barrel;
· weapon horizon - a horizontal plane passing through the departure point;
· elevation line - a straight line, which is a continuation of the axis of the bore at the moment of departure;
· shooting plane - a vertical plane passing through the line of elevation;
· throw line - a straight line, which is a continuation of the axis of the bore at the time of the bullet's departure;
· throw angle - the angle enclosed between the line of throw and the horizon of the weapon;
· departure angle - the angle enclosed between the line of elevation and the line of throwing;
· drop point - the point of intersection of the trajectory with the horizon of the weapon,
· injection fall – the angle at the point of impact between the tangent to the trajectory and the horizon of the weapon,
· full horizontal range - distance from point of departure to point of fall,
· top of the trajectory the highest point of the trajectory;
· trajectory height - the shortest distance from the top of the trajectory to the horizon of the weapon,
· ascending branch of the trajectory - part of the trajectory from the departure point to its top;
· descending branch of the trajectory - part of the trajectory from the top to the point of fall,
· meeting point - intersection of the trajectory with the surface of the target (ground, obstacles),
· meeting angle - the angle enclosed between the tangent to the trajectory and the tangent to the target surface at the meeting point;
· aiming point - the point on or off the target at which the weapon is aimed,
· line of sight - a straight line from the shooter's eye through the middle of the sight slot and the top of the front sight to the aiming point,
· aiming angle - the angle enclosed between the aiming line and the elevation line;
· target elevation angle the angle enclosed between the aiming line and the horizon of the weapon;
· effective range – distance from the point of departure to the intersection of the trajectory with the line of sight;
· excess of the trajectory over the aiming line - the shortest distance from any point of the trajectory to the line of sight;
· elevation angle - the angle enclosed between the line of elevation and the horizon of the weapon. The shape of the trajectory depends on the elevation angle
Rice. five. Bullet trajectory elements
The trajectory of a bullet in the air is following properties:
The descending branch is steeper than the ascending one;
the angle of incidence is greater than the angle of throw;
The final velocity of the bullet is less than the initial one;
The lowest speed of a bullet when shooting at high angles of throw
on the descending branch of the trajectory, and when firing at small throwing angles - at the point of impact;
the time of movement of the bullet along the ascending branch of the trajectory is less than
descending;
· the trajectory of a rotating bullet due to the decrease under the action of gravity and derivation is a line of double curvature.
The shape of the trajectory depends on the magnitude of the elevation angle (Fig. 6). As the elevation angle increases, the height of the trajectory and the total horizontal range of the bullet increase, but this occurs up to a certain limit. Beyond this limit, the trajectory height continues to increase and the total horizontal range begins to decrease.
Rice. 6. Angle of greatest reach, flat,
hinged and conjugate trajectories
The angle of elevation at which the full horizontal range of the bullet is at its greatest is called the angle of greatest range. The value of the angle of greatest range for small arms is 30-35 degrees, and for the range artillery systems 45-56 degrees.
Trajectories obtained at elevation angles smaller than the angle of greatest range are called flat.
Trajectories obtained at elevation angles greater than the angle of greatest range are called mounted. When firing from the same weapon, you can get two trajectories with the same horizontal range - flat and mounted. Trajectories having the same horizontal range at different elevation angles are called conjugated.
Flat trajectories allow:
1. It is good to hit openly located and fast moving targets.
2. Successfully fire from guns at a long-term firing structure (DOS), a long-term firing point (DOT), from stone buildings at tanks.
3. The flatter the trajectory, the greater the extent of the terrain, the target can be hit with one sight setting (the less impact on the results of shooting is caused by errors in determining the sight setting).
Mounted trajectories allow:
1. Hit targets behind cover and in deep terrain.
2. Destroy the ceilings of structures.
These different tactical properties of flat and overhead trajectories can be taken into account when organizing a fire system. The flatness of the trajectory affects the range of a direct shot, the affected and covered space.
Aiming (aiming) weapons at the target.
The task of any shooting is to hit the target in the most a short time and with the least amount of ammunition. This problem can be solved only in close proximity to the target and if the target is motionless. In most cases, hitting a target is associated with certain difficulties arising from the properties of the trajectory, meteorological and ballistic conditions shooting and the nature of the target.
Let the target be at point A - at some distance from the firing position. In order for the bullet to reach this point, the barrel of the weapon must be given a certain angle in the vertical plane (Fig. 7).
But from the wind, lateral deflections of the bullet can occur. Therefore, when aiming, it is necessary to take a lateral correction for the wind. Thus, in order for the bullet to reach the target and hit it or the desired point on it, it is necessary to give the axis of the bore a certain position in space (in the horizontal and vertical plane) before firing.
Giving the axis of the bore of a weapon the position in space necessary for firing is called aiming or pointing. Giving the axis of the bore of the weapon the required position in the horizontal plane is called horizontal pickup, and in the vertical plane - vertical pickup.
Rice. 7. Aiming (aiming) with open sight:
O - front sight, a - rear sight, aO - aiming line; сС - the axis of the bore, оО - a line parallel to the axis of the bore: H - the height of the sight, M - the amount of displacement of the rear sight;
a - aiming angle; Ub - angle of lateral correction
Accurate solution of aiming problems of any type sights depends on the correct alignment of them on the weapon. Alignment of sights of small arms for shooting at ground targets carried out in the process of checking the combat of the weapon and bringing it to normal combat.
external ballistics. Trajectory and its elements. Exceeding the trajectory of the bullet above the point of aim. Trajectory shape
External ballistics
External ballistics is a science that studies the movement of a bullet (grenade) after the cessation of the action of powder gases on it.
Having flown out of the bore under the action of powder gases, the bullet (grenade) moves by inertia. A grenade with a jet engine moves by inertia after the expiration of gases from the jet engine.
Bullet trajectory (side view)
Formation of air resistance force
Trajectory and its elements
A trajectory is a curved line described by the center of gravity of a bullet (grenade) in flight.
A bullet (grenade) when flying in the air is subjected to two forces: gravity and air resistance. The force of gravity causes the bullet (grenade) to gradually lower, and the force of air resistance continuously slows down the movement of the bullet (grenade) and tends to overturn it. As a result of the action of these forces, the speed of the bullet (grenade) gradually decreases, and its trajectory is an unevenly curved curved line in shape.
Air resistance to the flight of a bullet (grenade) is caused by the fact that air is an elastic medium and therefore part of the energy of the bullet (grenade) is expended on movement in this medium.
The force of air resistance is caused by three main causes: air friction, the formation of vortices and the formation of a ballistic wave.
Air particles in contact with a moving bullet (grenade), due to internal adhesion (viscosity) and adhesion to its surface, create friction and reduce the speed of the bullet (grenade).
The layer of air adjacent to the surface of the bullet (grenade), in which the movement of particles changes from the speed of the bullet (grenade) to zero, is called the boundary layer. This layer of air, flowing around the bullet, breaks away from its surface and does not have time to immediately close behind the bottom.
A rarefied space is formed behind the bottom of the bullet, as a result of which a pressure difference appears on the head and bottom parts. This difference creates a force directed in the direction opposite to the movement of the bullet, and reduces the speed of its flight. Air particles, trying to fill the rarefaction formed behind the bullet, create a vortex.
A bullet (grenade) in flight collides with air particles and causes them to oscillate. As a result, air density increases in front of the bullet (grenade) and sound waves are formed. Therefore, the flight of a bullet (grenade) is accompanied by a characteristic sound. At a bullet (grenade) flight speed that is less than the speed of sound, the formation of these waves has little effect on its flight, since the waves propagate faster speed flight of a bullet (grenade). When the speed of the bullet is higher than the speed of sound, a wave of highly compacted air is created from the incursion of sound waves against each other - a ballistic wave that slows down the speed of the bullet, since the bullet spends part of its energy to create this wave.
The resultant (total) of all forces resulting from the influence of air on the flight of a bullet (grenade) is the force of air resistance. The point of application of the resistance force is called the center of resistance.
The effect of the force of air resistance on the flight of a bullet (grenade) is very large; it causes a decrease in the speed and range of the bullet (grenade). For example, a bullet mod. 1930 at a throw angle of 15 ° and an initial speed of 800 m / s in airless space would have flown at a distance of 32,620 m; the flight range of this bullet under the same conditions, but in the presence of air resistance, is only 3900 m.
The magnitude of the air resistance force depends on the flight speed, the shape and caliber of the bullet (grenade), as well as on its surface and air density.
The force of air resistance increases with the increase in the speed of the bullet, its caliber and air density.
At supersonic bullet speeds, when the main cause of air resistance is the formation of an air seal in front of the head (ballistic wave), bullets with an elongated pointed head are advantageous. At subsonic grenade flight speeds, when the main cause of air resistance is the formation of rarefied space and turbulence, grenades with an elongated and narrowed tail section are beneficial.
The effect of the force of air resistance on the flight of a bullet: CG - center of gravity; CA - center of air resistance
The smoother the surface of the bullet, the lower the friction force and. force of air resistance.
The variety of shapes of modern bullets (grenades) is largely determined by the need to reduce the force of air resistance.
Under the influence of initial perturbations (shocks) at the moment the bullet leaves the bore, an angle (b) is formed between the bullet axis and the tangent to the trajectory, and the air resistance force acts not along the bullet axis, but at an angle to it, trying not only to slow down the movement of the bullet, but and knock her over.
In order to prevent the bullet from tipping over under the action of air resistance, it is given a rapid rotational movement with the help of rifling in the bore.
For example, when fired from a Kalashnikov assault rifle, the speed of rotation of the bullet at the moment of departure from the bore is about 3000 revolutions per second.
During the flight of a rapidly rotating bullet in the air, the following phenomena occur. The force of air resistance tends to turn the bullet head up and back. But the head of the bullet, as a result of rapid rotation, according to the property of the gyroscope, tends to maintain the given position and deviates not upwards, but very slightly in the direction of its rotation at right angles to the direction of the air resistance force, i.e., to the right. As soon as the head of the bullet deviates to the right, the direction of the air resistance force will change - it tends to turn the head of the bullet to the right and back, but the head of the bullet will not turn to the right, but down, etc. Since the action of the air resistance force is continuous, but its direction relative to the bullet changes with each deviation of the bullet axis, then the head of the bullet describes a circle, and its axis is a cone with a vertex at the center of gravity. The so-called slow conical, or precessional, movement occurs, and the bullet flies with its head part forward, i.e., as it were, follows the change in the curvature of the trajectory.
Slow conical movement of the bullet
Derivation (Trajectory top view)
The effect of air resistance on the flight of a grenade
The axis of slow conical motion lags somewhat behind the tangent to the trajectory (located above the latter). Consequently, the bullet collides with the air flow more with its lower part and the axis of the slow conical movement deviates in the direction of rotation (to the right when the barrel is right-handed). The deviation of the bullet from the plane of fire in the direction of its rotation is called derivation.
Thus, the causes of derivation are: the rotational movement of the bullet, air resistance and the decrease under the action of gravity of the tangent to the trajectory. In the absence of at least one of these reasons, there will be no derivation.
In shooting charts, derivation is given as heading correction in thousandths. However, when shooting from small arms, the magnitude of the derivation is insignificant (for example, at a distance of 500 m it does not exceed 0.1 thousandth) and its effect on the results of shooting is practically not taken into account.
The stability of the grenade in flight is ensured by the presence of a stabilizer, which allows you to move the center of air resistance back, behind the center of gravity of the grenade.
As a result, the force of air resistance turns the axis of the grenade to a tangent to the trajectory, forcing the grenade to move forward.
To improve accuracy, some grenades are given slow rotation due to the outflow of gases. Due to the rotation of the grenade, the moments of forces that deviate the axis of the grenade act sequentially in different directions, so the shooting improves.
To study the trajectory of a bullet (grenade), the following definitions are adopted.
The center of the muzzle of the barrel is called the departure point. The departure point is the start of the trajectory.
Trajectory elements
The horizontal plane passing through the departure point is called the weapon's horizon. In the drawings depicting the weapon and the trajectory from the side, the horizon of the weapon appears as a horizontal line. The trajectory crosses the horizon of the weapon twice: at the point of departure and at the point of impact.
A straight line, which is a continuation of the axis of the bore of the aimed weapon, is called the line of elevation.
The vertical plane passing through the line of elevation is called the shooting plane.
The angle enclosed between the line of elevation and the horizon of the weapon is called the angle of elevation. If this angle is negative, then it is called the angle of declination (decrease).
The straight line, which is a continuation of the axis of the bore at the moment the bullet takes off, is called the line of throw.
The angle enclosed between the line of throw and the horizon of the weapon is called the angle of throw.
The angle enclosed between the line of elevation and the line of throw is called the departure angle.
The point of intersection of the trajectory with the horizon of the weapon is called the point of impact.
The angle enclosed between the tangent to the trajectory at the point of impact and the horizon of the weapon is called the angle of incidence.
The distance from the point of departure to the point of impact is called the full horizontal range.
The speed of a bullet (grenade) at the point of impact is called the final speed.
The time of movement of a bullet (grenade) from the point of departure to the point of impact is called full time flight.
The highest point of the trajectory is called the vertex of the trajectory.
The shortest distance from the top of the trajectory to the horizon of the weapon is called the height of the trajectory.
The part of the trajectory from the departure point to the top is called the ascending branch; the part of the trajectory from the top to the point of fall is called the descending branch of the trajectory.
The point on or off the target at which the weapon is aimed is called the point of aim.
The straight line that runs from the shooter's eye through the middle of the sight slot (level with its edges) and the top of the front sight to the aiming point is called the aiming line.
The angle enclosed between the line of elevation and the line of sight is called the angle of aim.
The angle enclosed between the line of sight and the horizon of the weapon is called the elevation angle of the target. The target's elevation angle is considered positive (+) when the target is above the weapon's horizon, and negative (-) when the target is below the weapon's horizon. The elevation angle of the target can be determined using instruments or using the thousandth formula.
The distance from the departure point to the intersection of the trajectory with the aiming line is called the aiming range.
The shortest distance from any point of the trajectory to the line of sight is called the excess of the trajectory over the line of sight.
The straight line connecting the departure point with the target is called the target line. The distance from the departure point to the target along the target line is called the slant range. When firing direct fire, the target line practically coincides with the aiming line, and the slant range with the aiming range.
The point of intersection of the trajectory with the surface of the target (ground, obstacles) is called the meeting point.
The angle enclosed between the tangent to the trajectory and the tangent to the target surface (ground, obstacles) at the meeting point is called the meeting angle. The smaller of the adjacent angles, measured from 0 to 90°, is taken as the meeting angle.
The trajectory of a bullet in the air has the following properties:
The descending branch is shorter and steeper than the ascending one;
The angle of incidence is greater than the angle of throw;
The final speed of the bullet is less than the initial one;
The lowest speed of the bullet when firing at high angles of throw - on the descending branch of the trajectory, and when firing at small angles of throw - at the point of impact;
The time of movement of a bullet along the ascending branch of the trajectory is less than along the descending one;
The trajectory of a rotating bullet due to the drop of the bullet under the action of gravity and derivation is a line of double curvature.
Grenade trajectory (side view)
The trajectory of a grenade in the air can be divided into two sections: active - the flight of a grenade under the action of a reactive force (from the point of departure to the point where the action of the reactive force stops) and passive - the flight of a grenade by inertia. The shape of the trajectory of a grenade is about the same as that of a bullet.
Trajectory shape
The shape of the trajectory depends on the magnitude of the elevation angle. With an increase in the elevation angle, the height of the trajectory and the full horizontal range of the bullet (grenade) increase, but this occurs up to a known limit. Beyond this limit, the trajectory height continues to increase and the total horizontal range begins to decrease.
Angle of greatest range, flat, overhead and conjugate trajectories
The angle of elevation at which the full horizontal range of the bullet (grenade) becomes the greatest is called the angle of greatest range. The value of the angle of greatest range for bullets of various types of weapons is about 35 °.
Trajectories obtained at elevation angles smaller than the angle of greatest range are called flat. Trajectories obtained at elevation angles greater than the angle of greatest range are called hinged.
When firing from the same weapon (at the same initial speeds), you can get two trajectories with the same horizontal range: flat and mounted. Trajectories that have the same horizontal range at different elevation angles are called conjugate.
When firing from small arms and grenade launchers, only flat trajectories are used. The flatter the trajectory, the greater the extent of the terrain, the target can be hit with one sight setting (the less impact on the results of shooting is caused by errors in determining the sight setting); this is the practical significance of the flat trajectory.
Exceeding the trajectory of a bullet above the aiming point
The flatness of the trajectory is characterized by its greatest exceeding the line of sight. At a given range, the trajectory is all the more flat, the less it rises above the aiming line. In addition, the flatness of the trajectory can be judged by the magnitude of the angle of incidence: the trajectory is the more flat, the smaller the angle of incidence.
Bullet flight trajectory, its elements, properties. Types of trajectories and their practical significance
A trajectory is a curved line, described by the center of gravity of a bullet in flight.
A bullet flying through the air is subjected to two forces: gravity and air resistance. The force of gravity causes the bullet to gradually descend, and the force of air resistance continuously slows down the movement of the bullet and tends to topple it.
As a result of the action of these forces, the bullet's flight speed gradually decreases, and its trajectory is an unevenly curved curved line in shape.
|
Parameter |
Parameter characteristic |
Note |
|
Departure point |
Center of muzzle |
The departure point is the start of the trajectory |
|
Weapon horizon |
Horizontal plane passing through the departure point |
The horizon of the weapon looks like a horizontal line. The trajectory crosses the horizon of the weapon twice: at the point of departure and at the point of impact |
|
elevation line |
A straight line that is a continuation of the axis of the bore of the aimed weapon |
|
|
Shooting plane |
The vertical plane passing through the line of elevation |
|
|
Elevation angle |
The angle enclosed between the line of elevation and the horizon of the weapon |
If this angle is negative, then it is called the angle of declination (decrease) |
|
Throw line |
Straight line, a line that is a continuation of the axis of the bore at the time of the bullet's departure |
|
|
Throwing angle |
The angle enclosed between the line of throw and the horizon of the weapon |
|
|
Departure angle |
The angle enclosed between the line of elevation and the line of throw |
|
|
drop point |
Point of intersection of the trajectory with the horizon of the weapon |
|
|
Angle of incidence |
The angle enclosed between the tangent to the trajectory at the point of impact and the horizon of the weapon |
|
|
Total horizontal range |
Distance from departure point to drop point |
|
|
Ultimate speed |
Bullet speed at point of impact |
|
|
Total flight time |
The time it takes for a bullet to travel from point of departure to point of impact |
|
|
Top of the path |
The highest point of the trajectory |
|
|
Trajectory height |
The shortest distance from the top of the trajectory to the horizon of the weapon |
|
|
Ascending branch |
Part of the trajectory from the departure point to the summit |
|
|
descending branch |
Part of the trajectory from the top to the point of impact |
|
|
Aiming point (aiming) |
The point on or off the target at which the weapon is aimed |
|
|
line of sight |
A straight line passing from the shooter's eye through the middle of the sight slot (level with its edges) and the top of the front sight to the aiming point |
|
|
aiming angle |
The angle enclosed between the line of elevation and the line of sight |
|
|
Target elevation angle |
The angle enclosed between the line of sight and the horizon of the weapon |
The target's elevation angle is considered positive (+) when the target is above the weapon's horizon, and negative (-) when the target is below the weapon's horizon. |
|
Sighting range |
Distance from the point of departure to the intersection of the trajectory with the line of sight |
|
|
Exceeding the trajectory above the line of sight |
The shortest distance from any point of the trajectory to the line of sight |
|
|
target line |
A straight line connecting the departure point with the target |
When firing direct fire, the target line practically coincides with the aiming line |
|
Slant Range |
Distance from point of origin to target along target line |
When firing direct fire, the slant range practically coincides with the aiming range. |
|
meeting point |
Intersection point of the trajectory with the target surface (ground, obstacles) |
|
|
Meeting angle |
The angle enclosed between the tangent to the trajectory and the tangent to the target surface (ground, obstacles) at the meeting point |
The smaller of the adjacent angles, measured from 0 to 90°, is taken as the meeting angle. |
|
Sighting line |
A straight line connecting the middle of the sight slot to the top of the front sight |
|
|
Aiming (pointing) |
Giving the axis of the bore of the weapon the position in space necessary for firing |
In order for the bullet to reach the target and hit it or the desired point on it |
|
Horizontal aiming |
Giving the axis of the bore the desired position in the horizontal plane |
|
|
vertical guidance |
Giving the axis of the bore the desired position in the vertical plane |
The trajectory of a bullet in the air has the following properties:
- the descending branch is shorter and steeper than the ascending one;
- the angle of incidence is greater than the angle of throw;
- the final speed of the bullet is less than the initial one;
- the lowest speed of the bullet when firing at high angles of throw - on the descending branch of the trajectory, and when firing at small angles of throw - at the point of impact;
- the time of movement of the bullet along the ascending branch of the trajectory is less than along the descending one;
- the trajectory of a rotating bullet due to the lowering of the bullet under the action of gravity and derivation is a line of double curvature.
Types of trajectories and their practical significance
When firing from any type of weapon with an increase in the elevation angle from 0° to 90°, the horizontal range first increases to a certain limit, and then decreases to zero (Fig. 5).
The angle of elevation at which the greatest range is obtained is called the angle of greatest range. The value of the angle of greatest range for bullets of various types of weapons is about 35 °.
The angle of greatest range divides all the trajectories into two types: into the trajectories flat and hinged (Fig. 6). 
Flat trajectories are called trajectories obtained at elevation angles smaller than the angle of greatest range (see Fig. trajectories 1 and 2).
Overhead trajectories are called trajectories obtained at elevation angles greater than the angle of greatest range (see Fig. trajectories 3 and 4).
Conjugate trajectories are called trajectories obtained at the same horizontal range by two trajectories, one of which is flat, the other is hinged (see Fig. trajectories 2 and 3).
When firing from small arms and grenade launchers, only flat trajectories are used. The flatter the trajectory, the greater the extent of the terrain, the target can be hit with one sight setting (the less impact on the shooting results is the error in determining the sight setting): this is the practical significance of the trajectory.
The flatness of the trajectory is characterized by its greatest excess over the aiming line. At a given range, the trajectory is all the more flat, the less it rises above the aiming line. In addition, the flatness of the trajectory can be judged by the magnitude of the angle of incidence: the trajectory is the more flat, the smaller the angle of incidence. The flatness of the trajectory affects the value of the range of a direct shot, struck, covered and dead space.