Average air resistance force formula. Drag (aerodynamics)

As a result of numerous experiments, studies and theoretical generalizations, a formula has been established for calculating the force of air resistance

where S is the cross-sectional area of ​​the bullet,

c is the mass of air under given atmospheric conditions;

bullet speed;

- experimental coefficient, depending on the formula of the bullet and the number, which is taken from pre-compiled tables.

The magnitude of the resistance force depends on the following factors:

The cross-sectional area of ​​the bullet. Therefore, the force of air resistance is directly proportional to the cross-sectional area of ​​the bullet;

- air density. From the formula it can be seen that the force of air resistance is directly proportional to the density of the air. Firing tables are compiled for normal atmospheric conditions. In case of deviation of the actual temperature and pressure from the normal values, it is necessary to make corrections when using the firing tables;

- bullet speed. The dependence of the force of air resistance on the speed of the bullet is expressed by a complex law. The formula includes terms V 2 and, establishing the dependence of the air resistance force on the speed. To study this dependence, consider a graph showing how the bullet speed affects the air resistance force (Fig. 8).

Chart 1 - Dependence of drag force on bullet speed

Graphs similar in appearance are obtained for artillery shells. From the graph it follows that the force of air resistance increases with increasing bullet speed. The increase in the resistance force to a speed of 240 m/sec is relatively slow. At a speed close to the speed of sound, the force of air resistance increases sharply. This is due to the formation of a ballistic wave and, in connection with this, an increase in the difference in air pressure on the head and longitudinal parts of the bullet;

- bullet shape. The shape of the bullet significantly affects the function included in the formula. The question of the most advantageous bullet shape is extremely complex and cannot be decided on the basis of external ballistics alone. A very important factor in choosing the shape of a bullet is: the purpose of the bullet, the way it is guided along the rifling, the caliber and weight of the bullet, the design of the weapon for which it is intended, etc.

To reduce the effect of excess air pressure, it is necessary to sharpen and lengthen the head of the bullet. This causes some rotation of the head wave front, due to which the overpressure air on the head of the bullet. This phenomenon can be explained by the fact that as the head part becomes sharper, the speed with which air particles are repelled away from the surface of the bullet decreases.

Experience shows that the shape of the bullet head plays a minor role in air resistance. The main factor is the height of the head part and the way it is paired with the leading part. Usually, an arc of a circle is taken as the generatrix of the head of the bullet, the center of which is either at the base of the head, or slightly below it (Fig. 9). The tail part is most often made in the form of a truncated cone with an inclination angle of the generatrix (Fig. 10).

Figure 8 - The shape of the ogival part of the bullet

Figure 9 - The shape of the bottom of the bullet

Air flow around the tapered tail is much better. Region low pressure almost absent and vortex formation is much less intense. From the point of view of external ballistics, it is advantageous to make the leading part of the bullet possibly shorter. But with a short leading part, the correct influence of the bullet along the rifling of the barrel is difficult: it is possible to dismantle the bullet shell. It should be noted that the most advantageous form of a bullet can only be discussed for a certain speed, since for each speed there is its own most advantageous form.

On fig. 9 shows the most advantageous projectile shapes for various velocities. The horizontal axis shows the projectile velocities, and the vertical axis shows the projectile heights in calibers.

Figure 9 - Dependence of the relative length of the projectile on the speed

As can be seen, with increasing speed, the length of the head part, and total length projectile increase, and the tail section decreases. This dependence is explained by the fact that at high speeds, the main part of the air resistance force falls on the head part. Therefore, the main attention is paid to reducing the resistance of the head part, which is achieved by its sharpening and elongation. The tail of the projectile in this case is made short so that the projectile is not too long.

At low projectile speeds, the air pressure on the head part is small and the vacuum behind this part, although less than at high speeds, still makes up a significant proportion of the entire air resistance force. Therefore, it is necessary to make a relatively long conical tail of the projectile to reduce the effect of the discharged space. The head part may be shorter, since it is long, in this case it matters less. The sharpening of the tail is especially great for projectiles whose speed is less than the speed of sound. In this case, the drop-shaped form is the most advantageous. This form is given to mines and bombs.

Experiments by definition

Since 1860 different countries experiments were carried out with shells of various calibers and shapes in order to determine.

Chart 2 - Curves for various forms shells: 1, 2, 3 - similar in shape; 4 - light bullet

Considering the curves for projectiles of similar shape, you can see that they also have a similar shape. This makes it possible to approximately express for some projectile in terms of another projectile, taken as if for a standard, using a constant factor i:

This multiplier, or the ratio of a given projectile to another projectile taken as a standard, is called the projectile shape factor. To determine the shape factor of any projectile, it is necessary to experimentally find the force of air resistance for it for any speed. Then the formula can be found

Dividing the resulting expression by we get the form coefficient

Different scientists have given different mathematical expressions for the calculation. For example, Shiachi (Graph 3) expressed the law of resistance with the following formula

where F(V) - resistance function.

Chart 3 - Law of Resistance

Resistance function N.V. Maievsky and N.A. Zabudsky is smaller than the Siacci resistance function. Conversion factor from Siacci's law of resistance to N.V.'s law of resistance Mayevsky and N.A. Zabudsky is on average 0.896.

At the Military Engineering Artillery Academy. F.E. Dzerzhinsky derived the law of air resistance for long-range projectiles. This law was obtained on the basis of processing the results of special firing with long-range projectiles and bullets. The resistance functions in this law are chosen so that in ballistic calculations for long-range projectiles, as well as for bullets and feathered projectiles (min), the form factor is as close to unity as possible. The function for velocities less than 256 m/s or greater than 1410 m/s can be expressed as a monomial Define the coefficient

For V< 256 м/ сек

For V > 1410 m/s

When specifying the form factor, you should always indicate in relation to which law of resistance it is given. In the formula for determining the force of air resistance, replacing we get with, we get

The average value of the shape factor for the Siacci drag law is given in Table. 3.

Table 3 - i values ​​for various projectiles and bullets

When any object moves on the surface or in the air, forces arise that prevent it. They are called forces of resistance or friction. In this article, we will explain how to find the resistance force and consider the factors that affect it.

To determine the resistance force, it is necessary to use Newton's third law. This value is numerically equal to the force that must be applied to make an object move uniformly on a flat horizontal surface. This can be done with a dynamometer. The resistance force is calculated by the formula F=μ*m*g. According to this formula, the desired value is directly proportional to body weight. It is worth considering that for the correct calculation it is necessary to choose μ - a coefficient depending on the material from which the support is made. The material of the object is also taken into account. This coefficient is selected according to the table. For the calculation, the constant g is used, which is equal to 9.8 m/s2. How to calculate resistance if the body does not move in a straight line, but along an inclined plane? To do this, you need to enter the cos of the angle in the original formula. It is from the angle of inclination that the friction and resistance of the surface of bodies to movement depend. The formula for determining friction on an inclined plane will look like this: F=μ*m*g*cos(α). If the body moves at a height, then the air friction force acts on it, which depends on the speed of the object. The desired value can be calculated by the formula F=v*α. Where v is the speed of the object, and α is the drag coefficient of the medium. This formula is only suitable for bodies that move at low speed. To determine the drag force of jet aircraft and other high-speed units, another one is used - F = v2 * β. To calculate the friction force of high-speed bodies, the square of the speed and the coefficient β are used, which is calculated for each object separately. When an object moves in a gas or liquid, when calculating the friction force, it is necessary to take into account the density of the medium, as well as the mass and volume of the body. Drag significantly reduces the speed of trains and cars. Moreover, two types of forces act on moving objects - permanent and temporary. The total friction force is represented by the sum of two quantities. To reduce resistance and increase the speed of the machine, designers and engineers invent a variety of materials with a sliding surface from which air is repelled. That is why the front of high-speed trains has a streamlined shape. The fish move very quickly in the water thanks to the streamlined body, covered with mucus, which reduces friction. The resistance force does not always have a negative effect on the movement of cars. To pull the car out of the mud, it is necessary to pour sand or gravel under the wheels. Thanks to the increase in friction, the car copes well with swampy soil and dirt.

Air resistance is used during skydiving. As a result of the resulting friction between the dome and the air, the speed of the skydiver is reduced, which allows parachuting without damage to life.

One of the manifestations of the force of mutual gravity is gravity, i.e. force of attraction of bodies to the Earth. If only the force of gravity acts on the body, then it makes a free fall. Therefore, free fall is the fall of bodies in airless space under the influence of attraction to the Earth, starting from a state of rest.

This phenomenon was first studied by Galileo, but due to the lack of air pumps he could not conduct an experiment in a vacuum, so Galileo made experiments in the air. Discarding all minor phenomena encountered during the movement of bodies in the air, Galileo discovered the laws of free fall of bodies. (1590)

• 1st law. Free fall is a rectilinear uniformly accelerated motion.
• 2nd law. The free fall acceleration at a given place on the Earth is the same for all bodies; its average value is 9.8 m/s.

The dependences between the kinematic characteristics of free fall are obtained from the formulas for uniformly accelerated motion, if we put a = g in these formulas. For v0 = 0 V = gt, H = gt2 \2, v = √2gH .

In practice, air always resists the movement of a falling body, and for a given body, the greater the air resistance, the greater the falling speed. Therefore, as the speed of fall increases, the air resistance increases, the acceleration of the body decreases, and when the air resistance becomes equal to strength gravity, the acceleration of a freely falling body becomes zero. In the future, the movement of the body will be uniform movement.

The real movement of bodies in earth's atmosphere occurs along a ballistic trajectory, which differs significantly from a parabolic one due to air resistance. For example, if a bullet is fired from a rifle at a speed of 830 m/s at an angle α = 45o to the horizon and the actual trajectory of the tracer bullet and the place of its fall are recorded using a movie camera, then the flight range will be approximately 3.5 km. And if you calculate by the formula, then it will be 68.9 km. The difference is huge!

Air resistance depends on four factors: 1) SIZE of the moving object. A large object will obviously receive more resistance than a small one. 2) SHAPE of a moving body. A flat plate of a certain area will provide much more resistance to the wind than a streamlined body (drop shape) having the same cross-sectional area for the same wind, actually 25 times more! The round object is somewhere in the middle. (This is the reason why the hulls of all cars, airplanes and paragliders are as rounded or teardrop-shaped as possible: it reduces air resistance and allows you to move faster with less effort on the engine, and therefore, with less fuel). 3) DENSITY OF AIR. We already know that one cubic meter weighs about 1.3 kg at sea level, and the higher you go, the less dense the air becomes. This difference may play some practical role when taking off only from very high altitudes. 4) SPEED. Each of the three factors considered so far makes a proportional contribution to air resistance: if you double one of them, the resistance also doubles; if you halve any of them, the resistance drops by half.

AIR RESISTANCE is HALF THE DENSITY OF AIR times RESISTANCE COEFFICIENT times SECTION AREA times SQUARE OF SPEED.

We introduce the following symbols: D - air resistance; p - air density; A - sectional area; cd is the drag coefficient; υ - air speed.

Now we have: D \u003d 1/2 x p x cd x A x υ 2

When a body falls in real conditions, the acceleration of the body will not be equal to the acceleration of free fall. In this case, Newton's 2nd law will take the form ma = mg - Fresist -Farch

Farx. =ρqV , since the air density is low, can be neglected, then ma = mg - ηυ

Let's analyze this expression. It is known that a resistance force acts on a body moving in air. It is almost obvious that this force depends on the speed of movement and the dimensions of the body, for example, the cross-sectional area S, and this dependence is of the type "the more υ and S, the greater F". You can still refine the form of this dependence, based on considerations of dimensions (units of measurement). Indeed, force is measured in newtons ([F] = N), and N = kg m/s2. It can be seen that the second squared is included in the denominator. From here it is immediately clear that the force must be proportional to the square of the body's velocity ([υ2] = m2/s2) and density ([ρ] = kg/m3) - of course, of the medium in which the body moves. So,

And to emphasize that this force is directed against the velocity vector.

We have already learned a lot, but that's not all. Surely the resistance force (aerodynamic force) also depends on the shape of the body - it is no coincidence that aircraft are made “well streamlined”. To take into account this supposed dependence, it is possible to introduce a dimensionless factor into the ratio (proportionality) obtained above, which will not violate the equality of dimensions in both parts of this ratio, but will turn it into an equality:

Let's imagine a ball moving in the air, for example, a shotgun shot horizontally with an initial speed - If there were no air resistance, then at a distance x in time the shot would move vertically down by. But due to the action of the resistance force (directed against the velocity vector), the time of flight of the pellet to the vertical plane x will be greater than t0. Consequently, the force of gravity will act on the pellet for a longer time, so that it will fall below y0.

And in general, the pellet will move along another curve, which is no longer a parabola (it is called a ballistic trajectory).

In the presence of an atmosphere, falling bodies, in addition to the force of gravity, experience the forces of viscous friction against the air. In a rough approximation, at low speeds, the force of viscous friction can be considered proportional to the speed of motion. In this case, the equation of motion of the body (Newton's second law) has the form ma = mg - η υ

The viscous friction force acting on spherical bodies moving at low speeds is approximately proportional to their cross-sectional area, i.e. the square of the radius of the bodies: F = -η υ= - const R2 υ

The mass of a spherical body of constant density is proportional to its volume, i.e. cube of radius m = ρ V = ρ 4/3π R3

The equation is written taking into account the downward direction of the OY axis, where η is the air resistance coefficient. This value depends on the state of the environment and body parameters (body weight, size and shape). For a spherical body, according to the Stokes formula η =6(m(r where m is the mass of the body, r is the radius of the body, ( is the coefficient of air viscosity.

Consider, for example, balls falling from different material. Take two balls of the same diameter, plastic and iron. Let us assume for clarity that the density of iron is 10 times greater than the density of plastic, so the iron ball will have a mass 10 times greater, respectively, its inertia will be 10 times higher, i.e. under the same force, it will accelerate 10 times slower.

In a vacuum, only gravity acts on the balls, on iron balls 10 times more than on plastic ones, respectively, they will accelerate with the same acceleration (10 times great strength gravity compensates for 10 times the inertia of the iron ball). With the same acceleration, both balls will cover the same distance in the same time, i.e. in other words, they will fall at the same time.

In air: aerodynamic drag and Archimedean force are added to the effect of gravity. Both of these forces are directed upwards, against the action of gravity, and both depend only on the size and speed of the balls (do not depend on their mass) and, at equal speeds of movement, are equal for both balls.

T.o. the resultant of the three forces acting on the iron ball will no longer be 10 times greater than the similar resultant of the wooden one, but more than 10, while the inertia of the iron ball remains greater than the inertia of the wooden one by the same 10 times .. Accordingly, the acceleration of the iron ball will be greater than that of the plastic one, and he will fall earlier.

The magnitude of the air resistance force depends on the shape of the projectile, the state of the surface of its body, its largest cross-sectional area, air density, projectile velocity relative to air, sound propagation velocity and the position of the projectile longitudinal axis relative to the projectile velocity vector.

Let us briefly consider how the factors listed above affect the magnitude of the air resistance force.

The shape and condition of the surface of the projectile. The influence of the shape of the projectile and the state of its surface on the magnitude of the air resistance force was indicated when considering the factors that determine the occurrence of the air resistance force.

Rice. 12. The influence of the shape of the projectile on the formation of the head and tail

waves and swirls behind the projectile:

a- cylindrical projectile; b - ball projectile (core); in - an oblong projectile with a cylindrical rim (an old high-explosive grenade);

G- oblong projectile with a conical zapoyaskovy part

The dependence of the magnitude of the wave and vortex resistances on the shape of the projectile is clearly seen in Fig. 12, which shows snapshots of projectiles fired at approximately the same muzzle velocity.

The smallest waves and turbulences are obtained from a projectile having the most pointed head and a beveled bottom, the largest waves and eddies are obtained from a cylindrical projectile.

But it should be borne in mind that when choosing the optimal shape of the projectile, it is necessary, along with a decrease in air resistance, to ensure the stability of the flight of the projectile, rational use metal, equipment and the effective action of the projectile at the target; so shells various types have a different shape.

The dependence of the air resistance force on the shape of the projectile is expressed by the shape factor i.

For projectile of this type, the shape of which is taken as a standard, the shape factor is taken equal to one. When changing the shape of the projectile relative to the reference shape factor is determined empirically.

The area of ​​the largest cross section. If the nutation angle δ = 0, then the number elementary particles air that the projectile will meet on its way, ceteris paribus, will depend on the area of ​​​​its largest cross-section. How more area the cross section of the projectile, the more elementary particles of air will act on the projectile, the greater will be the air resistance force. Experimental data show that the force of air resistance changes in proportion to the change in the cross-sectional area of ​​the projectile.

Air density. Air density is the mass of air per unit volume. A change in the mass of air per unit volume can occur due to a change in the number of elementary particles (molecules) per unit volume, or due to a change in the mass of each particle. If, for example, the density of air has increased, then this means that either the number of elementary particles in each unit of air volume has increased, or the mass of particles has increased (or both), and if so, then the force of air impact on each unit of surface area projectile will increase, therefore, the total air resistance will also increase.

It is established that the force of air resistance changes in proportion to the change in air density.

Projectile speed. Research shows that the force of air resistance is directly proportional to the square of the speed of the projectile relative to the air. If, for example, the speed of a projectile relative to the air is doubled, then the force of air resistance will increase four times.

This is explained by the fact that, firstly, with an increase in the speed of the projectile, in each unit of time it will meet more elementary air particles on its way and, secondly, the inertia of air particles at a higher speed "must be overcome by the projectile in a shorter time , which will cause more resistance from air particles.

The speed of sound propagation in air. The formation of wave resistance, as shown above, occurs at the moment when the projectile speed becomes equal to the speed of sound, i.e. at the moment when,

where v- projectile speed and a is the speed of sound in air.

The speed of sound in air is not constant (it depends on the temperature and humidity of the air). Consequently, at the same projectile speed, due to a change in the speed of sound in air, the magnitude of the wave resistance and the air resistance force as a whole can be different. The dependence of the air resistance force on the speed of sound propagation is taken into account by a special coefficient. Value , depends on the size and shape of the projectile. The graph of this dependence is shown in Fig. 13.

Rice. 13. Function Graph:

a.- a projectile with a cylindrical zapoyaskovy part (an old high-explosive grenade);

b - an oblong projectile with a conical zapoyaskovy part

The position of the longitudinal axis of the projectile relative to the tangent to the trajectory (velocity vector). The flight of a projectile in air is accompanied by complex oscillatory movements around the center of gravity, as a result of which the longitudinal axis of the projectile is not aligned with the direction of flight (with the velocity vector), i.e., nutation angles appear.

When a nutation angle occurs, the projectile no longer flies with its head part forward, but exposes a part of the side surface to the oncoming air flow. The conditions for the air flow around the projectile also deteriorate sharply because of this.

All this dramatically increases the force of air resistance. To reduce the influence of this factor, measures are taken to stabilize the flight of the projectile, i.e., to reduce the nutation angles.

So, the influence of various factors on the magnitude of the air resistance force is complex and multifaceted. Therefore, the air resistance force is usually determined empirically for the conditions that the air resistance force is applied to its center of gravity throughout the movement and is directed tangentially to the trajectory, i.e., there are no nutation angles.

The magnitude of the air resistance force is expressed by various empirical formulas. One of the most common has the form

(1.7)

where R- the magnitude of the air resistance force, kg;

i- form factor;

S- projectile cross-sectional area, m 2;

ρ - air density (mass 1 m 3 given air, it is equal to,

where P- weight 1 m 3 air, or weight density of air);

v is the speed of the projectile relative to the air, m/s;

Empirical coefficient taking into account the influence of the quantity

the ratio of projectile speed to the speed of sound, depending on the shape of the projectile.

In formula 1.7, the value has an independent meaning, because it is nothing but kinetic energy, or living force 1 m 3 air. This value is called velocity head.

Lecture 10

Topic 4. Activity 2. Projectile movement in the field

1. The quickened strength of the support of the resurrection. Transverse tension and ballistic coefficient.

2. The need to take the world to ensure the stability of the projectile in the field.

3. The movement of a shvidko-wrapped projectile in the field. Derivation.

To determine the strength resistance air create conditions under which the body will begin to move uniformly and rectilinearly under the influence of gravity. Calculate the value of gravity, it will be equal to the force of air resistance. If a body moves in the air, picking up speed, its resistance force is found using Newton's laws, and the air resistance force can also be found from the conservation law mechanical energy and special aerodynamic formulas.

You will need

• rangefinder, scales, speedometer or radar, ruler, stopwatch.

Instruction

• Determination of air resistance to a uniformly falling body Measure the mass of the body using a balance. After dropping it from a certain height, make sure that it moves evenly. Multiply the mass of the body in kilograms by the acceleration of gravity, (9.81 m/s²), the result is the force of gravity acting on the body. And since it moves uniformly and in a straight line, the force of gravity will be equal to the force of air resistance.
• Determining the air resistance of a body picking up speed Determine the mass of the body using a balance. After the body has begun to move, use a speedometer or radar to measure its instantaneous initial speed. At the end of the section, measure its instantaneous final speed. Speeds are measured in meters per second. If the instruments measure it in kilometers per hour, divide the value by 3.6. In parallel, using a stopwatch, determine the time during which this change occurred. Subtracting the initial speed from the final speed and dividing the result by the time, find the acceleration with which the body moves. Then find the force that causes the body to change speed. If the body falls, then this is the force of gravity, if the body moves horizontally, it is the traction force of the engine. Subtract the product of the body's mass and its acceleration from this force (Fc=F+m a). This will be the force of air resistance. It is important that when moving the body does not touch the ground, for example, moving on an air cushion or falling down.
• Determination of the air resistance of a body falling from a height Measure the mass of a body and drop it from a height that is known in advance. On contact with the ground, record the speed of the body using a speedometer or radar. After that, find the product of the free fall acceleration of 9.81 m / s² and the height from which the body fell, subtract the squared speed from this value. Multiply the result obtained by the mass of the body and divide by the height from which it fell (Fc \u003d m (9.81 H-v²) / H). This will be the force of air resistance.