How does the force of air resistance depend on the shape of an object and its mass.

As a result of numerous experiments, studies and theoretical generalizations, a formula was 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;

- an experimental coefficient depending on the bullet formula and a number that is taken from pre-compiled tables.

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

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

- air density. The formula shows that the force of air resistance is directly proportional to the density of the air. The shooting tables are compiled for normal atmospheric conditions. In case of deviation actual temperature and pressure from normal values it is necessary to make corrections when using shooting 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 air resistance force on speed. To study this dependence, consider a graph showing how bullet speed affects the force of air resistance (Fig. 8).

Schedule 1 - Dependence of drag force on bullet speed

Similar-looking graphs are obtained for artillery shells. From the graph it follows that the force of air resistance increases with increasing bullet speed. The increase in drag force up to a speed of 240 m/sec is relatively slow. At speeds close to the speed of sound, the force of air resistance increases sharply. This is explained by the formation of a ballistic wave and, in connection with this, an increase in the difference in air pressure on the head and bottom parts of the bullet;

- bullet shapes. 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 resolved on the basis of external ballistics. Very important factor When choosing the shape of a bullet, it is important to consider: the purpose of the bullet, the method of guiding it along the rifling, the caliber and weight of the bullet, the device 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 front of the head wave, due to which the overpressure air onto the head of the bullet. This phenomenon can be explained by the fact that as the head part becomes sharper, the speed at which air particles are repelled to the sides 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 connected to the leading part. Usually, the generatrix of the head part of the bullet is taken to be an arc of a circle, the center of which is either at the base of the head part or slightly below it (Fig. 9). The tail part is most often made in the form of a truncated cone with an angle of inclination of the generatrix (Fig. 10).

Figure 8 - Shape of the ogive part of the bullet

Figure 9 - Shape of the bottom of the bullet

The air flow around the conical tail section 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 on the rifling of the barrel is difficult: dismantling of the bullet casing is possible. It should be noted that we can talk about the most advantageous shape of a bullet only for a certain speed, since for each speed there is its own most advantageous shape.

In Fig. 9 shows the most advantageous shapes of projectiles for various speeds. The horizontal axis shows projectile velocities, and the vertical axis shows projectile heights in calibers.

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

As you can see, with increasing speed, the length of the head part and the total length of the projectile increase, and the tail part decreases. This dependence is explained by the fact that at high speeds the main share 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 sharpening and elongating it. The tail part 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, makes up a significant proportion of the total air resistance force. Therefore, it is necessary to make a relatively long conical tail part of the projectile to reduce the effect of the discharged space. The head part can be shorter, since its length is of less importance in this case. The sharpening of the tail is especially high for projectiles whose speed less speed sound. In this case, the teardrop shape is most advantageous. This shape is given to mines and air bombs.

Experiments by definition

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

Schedule 2 - Curves for different shapes of projectiles: 1, 2, 3 - similar in shape; 4 - light bullet

By examining the curves for projectiles of similar shape, one can be convinced that they also have similar appearance. This makes it possible to approximately express for a certain projectile in terms of another projectile, taken as 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 coefficient. To determine the shape coefficient of any projectile, it is necessary to experimentally find the air resistance force for it for any speed. Then using the formula you can find

Dividing the resulting expression by we obtain the form factor

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

where F(V) - resistance function.

Graph 3 - Law of Resistance

N.V. resistance function Maievsky and N.A. Zabudsky is less 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's average is 0.896.

At the Military Engineering Artillery Academy named after. F.E. Dzerzhinsky derived the law of air resistance for long-range projectiles. This law was obtained based on processing the results of special shooting with long-range shells and bullets. The resistance functions in this law are chosen such that in ballistic calculations for long-range projectiles, as well as for bullets and feathered projectiles (mines), the shape coefficient is as close to unity as possible. The function for speeds less than 256 m/sec or greater than 1410 m/sec can be expressed as a monomial. Let us determine the coefficient

For V< 256 м/ сек

For V > 1410 m/s

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

The average value of the shape coefficient for Siacci's law of resistance is given in table. 3.

Table 3 - i values for various projectiles and bullets

One of the manifestations of mutual gravitational force is gravity, i.e. the force of attraction of bodies towards the Earth. If only the force of gravity acts on a body, then it undergoes free fall. Consequently, free fall is the fall of bodies in airless space under the influence of gravity towards 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 experiments in airless space, so Galileo carried out experiments in air. Discarding all secondary phenomena encountered when bodies move 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 acceleration of gravity in a given place on the Earth is the same for all bodies; its average value is 9.8 m/s.

The dependencies between the kinematic characteristics of free fall are obtained from formulas for uniformly accelerated motion, if in these formulas we put a = g. At 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 speed of fall, the greater the air resistance. Consequently, as the speed of falling increases, air resistance increases, the acceleration of the body decreases, and when the air resistance becomes equal strength gravity, the acceleration of a freely falling body will become zero. In the future, the movement of the body will be a uniform movement.

Real movement of bodies in earth's atmosphere occurs along a ballistic trajectory, significantly different from a parabolic one due to air resistance. For example, if you fire a bullet from a rifle at a speed of 830 m/s at an angle α = 45° to the horizon and use a movie camera to record the actual trajectory of the tracer bullet and the location of its impact, then the flight range will be approximately 3.5 km. And if you calculate it using the formula, 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. Flat plate certain area will provide much greater wind resistance than a streamlined body (droplet shape) having the same cross-sectional area for the same wind, actually 25 times greater! The round object is somewhere in the middle. (This is the reason why the bodies of all cars, airplanes and paragliders are rounded or teardrop-shaped whenever possible: it reduces air resistance and allows you to move faster with less effort on the engine, and therefore less fuel). 3) AIR DENSITY. 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 can play some practical role when taking off only from a very high altitude. 4) SPEED. Each of the three factors considered so far makes a proportional contribution to air drag: if you double one of them, the drag also doubles; if you reduce either one by half, the resistance drops by half.

AIR RESISTANCE is equal to HALF THE DENSITY OF THE AIR multiplied by the DRAG COEFFICIENT multiplied by the SECTIONAL AREA and multiplied by the SQUARE OF VELOCITY.

Let's introduce the following symbols: D - air resistance; p - air density; A - cross-sectional area; cd - resistance coefficient; υ - air speed.

Now we have: D = 1/2 x р x cd x A x υ 2

When a body falls into real conditions the acceleration of the body will not be equal to the acceleration of gravity. In this case, Newton’s 2nd law will take the form ma = mg – Fresist –Farch

Farkh. =ρqV , since the air density is low, it can be neglected, then ma = mg – ηυ

Let's analyze this expression. It is known that a drag force acts on a body moving in the air. It is almost obvious that this force depends on the speed of movement and the size of the body, for example, the cross-sectional area S, and this dependence is of the type “the larger υ and S, the larger F.” You can also clarify the type 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 speed ([υ2] = m2/s2) and density ([ρ] = kg/m3) - of course, 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 drag 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 expected dependence, it is possible to introduce a dimensionless factor into the relation (proportionality) obtained above, which will not violate the equality of dimensions in both parts of this relation, but will turn it into equality:

Let's imagine a ball moving in the air, for example, a pellet flying horizontally from initial speed- If there were no air resistance, then at a distance x in time the pellet would move vertically downwards by. But due to the action of the drag 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 longer, so that it will fall below y0.

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

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

The force of viscous friction acting on spherical bodies moving at low speeds is approximately proportional to their cross-sectional area, i.e. squared body radius: 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 air viscosity coefficient.

Consider, for example, the fall of balls from different materials. Let's 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, and accordingly 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 it is 10 times more than on plastic ones; accordingly, they will accelerate with the same acceleration (10 times great strength gravity compensates for 10 times greater inertia iron ball). With the same acceleration, both balls will travel the same distance in the same time, i.e. in other words, they will fall simultaneously.

In the air: the force of aerodynamic drag and the Archimedean force are added to the action of gravity. Both of these forces are directed upward, 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 the movements 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, and 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.

It is a component of the total aerodynamic force.

Strength drag usually represented as the sum of two components: drag at zero lift and inductive drag. Each component is characterized by its own dimensionless drag coefficient and a certain dependence on the speed of movement.

Drag can contribute to both icing aircraft(at low temperatures air), and cause heating of the frontal surfaces of the aircraft at supersonic speeds by impact ionization.

Drag at zero lift

This drag component does not depend on the magnitude of the lift force created and consists of the profile drag of the wing, the resistance of aircraft structural elements that do not contribute to the lift, and wave drag. The latter is significant when moving at near- and supersonic speeds, and is caused by the formation shock wave, carrying away a significant portion of the energy of motion. Wave drag occurs when the aircraft reaches a speed corresponding to the critical Mach number, when part of the flow flowing around the aircraft wing acquires supersonic speed. The greater the critical number M is, the greater the wing sweep angle, the more pointed the leading edge of the wing, and the thinner it is.

The resistance force is directed against the speed of movement, its magnitude is proportional to the characteristic area S, the density of the medium ρ and the square of the speed V:

C x 0 is the dimensionless aerodynamic drag coefficient, obtained from similarity criteria, for example, Reynolds and Froude numbers in aerodynamics.

Determination of the characteristic area depends on the shape of the body:

in the simplest case (ball) - cross-sectional area;
for wings and empennage - the area of the wing/empennage in plan view;
for propellers and rotors of helicopters - either the area of the blades or the swept area of the rotor;
for oblong bodies of rotation oriented along flow (fuselage, airship shell) - reduced volumetric area equal to V 2/3, where V is the volume of the body.

The power required to overcome a given component of the drag force is proportional to Cuba speed.

Inductive reactance

Inductive reactance(English) lift-induced drag) is a consequence of the formation of lift on a wing of finite span. Asymmetrical flow around the wing leads to the fact that the air flow escapes from the wing at an angle to the flow incident on the wing (the so-called flow bevel). Thus, during the movement of the wing, there is a constant acceleration of the mass of incoming air in a direction perpendicular to the direction of flight and directed downward. This acceleration, firstly, is accompanied by the formation of a lifting force, and secondly, it leads to the need to impart kinetic energy to the accelerating flow. Quantity kinetic energy, necessary to impart a speed to the flow perpendicular to the direction of flight, and will determine the amount of inductive reactance.

The magnitude of induced drag is influenced not only by the magnitude of the lift force, but also by its distribution along the wing span. The minimum value of inductive drag is achieved with an elliptical distribution of the lifting force along the span. When designing a wing, this is achieved using the following methods:

choosing a rational wing planform;
the use of geometric and aerodynamic twist;
installation of auxiliary surfaces - vertical wing tips.

Inductive reactance is proportional square lift force Y, and inversely proportional wing area S, its elongation λ, medium density ρ and square speed V:

Thus, induced drag makes a significant contribution when flying at low speeds (and, as a consequence, at high angles of attack). It also increases as the weight of the aircraft increases.

Total resistance

Is the sum of all types of resistance forces:

X = X 0 + X i

Since drag at zero lift X 0 is proportional to the square of the speed, and the inductive X i- is inversely proportional to the square of the speed, then they make different contributions at different speeds. With increasing speed, X 0 is growing, and X i- falls, and the graph of the total resistance X on speed (“required thrust curve”) has a minimum at the point of intersection of the curves X 0 and X i, at which both resistance forces are equal in magnitude. At this speed, the aircraft has the least drag for a given lift ( equal to weight), and therefore the highest aerodynamic quality.

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We are so accustomed to being surrounded by air that we often don’t pay attention to it. We are talking here, first of all, about applied technical problems, in the solution of which at first it is forgotten that there is a force of air resistance.

She reminds herself of herself in almost any action. Even if we drive a car, even if we fly on an airplane, even if we just throw stones. So let’s try to understand what the force of air resistance is using simple cases as examples.

Have you ever wondered why cars have such a streamlined shape and smooth surface? But everything is actually very clear. The force of air resistance consists of two quantities - the frictional resistance of the surface of the body and the resistance of the body's shape. In order to reduce and achieve a reduction in irregularities and roughness on external parts in the manufacture of cars and any other vehicles.

To do this, they are primed, painted, polished and varnished. Such processing of parts leads to the fact that the air resistance acting on the car decreases, the speed of the car increases and fuel consumption when driving decreases. The presence of a resistance force is explained by the fact that when a car moves, the air is compressed and a local area is created in front of it. high blood pressure, and behind it, accordingly, a region of rarefaction.

It should be noted that at increased vehicle speeds, the main contribution to resistance is made by the shape of the car. The resistance force, the calculation formula for which is given below, determines the factors on which it depends.

Resistance force = Cx*S*V2*r/2

where S is the front projection area of the machine;

Cx - coefficient taking into account ;

As is easy to see from the above, the resistance does not depend on the mass of the car. The main contribution comes from two components - the square of the speed and the shape of the car. Those. When the speed is doubled, the resistance will quadruple. Well, the cross section of the car has a significant influence. The more streamlined the car, the less air resistance.

And in the formula there is another parameter that simply requires paying close attention to it - air density. But its influence is already more noticeable during airplane flights. As you know, air density decreases with increasing altitude. This means that the force of its resistance will decrease accordingly. However, for an airplane, the same factors will continue to influence the amount of resistance provided - speed and shape.

No less interesting is the history of studying the influence of air on shooting accuracy. Work of this nature was carried out a long time ago; their first descriptions date back to 1742. Experiments were carried out in different countries, with various shapes bullets and shells. As a result of the research, the optimal shape of the bullet and the ratio of its head and tail were determined, and ballistic tables of the behavior of the bullet in flight were developed.

Subsequently, studies were carried out on the dependence of the flight of a bullet on its speed, the shape of the bullet continued to be worked out, and a special mathematical tool was developed and created - the ballistic coefficient. It shows the ratio of the aerodynamic drag forces acting on the bullet.

The article discusses what the force of air resistance is and gives a formula that allows you to determine the magnitude and degree of influence various factors on the magnitude of resistance, its impact in different fields of technology is considered.

All components of air resistance are difficult to determine analytically. Therefore, an empirical formula has been used in practice, which has the following form for the range of speeds characteristic of a real car:

Where With X – dimensionless air flow coefficient, depending on body shape; ρ in – air density ρ in = 1.202…1.225 kg/m 3 ; A– midsection area (transverse projection area) of the car, m2; V– vehicle speed, m/s.

Found in literature air resistance coefficient k V :

F V = k V AV 2 , Where k V =c X ρ V /2 , – air resistance coefficient, Ns 2 /m 4.

…and streamlining factorq V : q V = k V · A.

If instead With X substitute With z, then we get the aerodynamic lift force.

Midsection area for a car:

A=0.9 B max · N,

Where IN max – maximum vehicle track, m; N– vehicle height, m.

The force is applied at the metacenter, and moments are created.

Air flow resistance speed taking into account wind:

, where β is the angle between the directions of movement of the car and the wind.

WITH X some cars

VAZ 2101…07			Opel astra Sedan
VAZ 2108…15


			Land Rover Free Lander
VAZ 2102…04
VAZ 2121…214
			truck
			truck with trailer

Lifting resistance force

F n = G A sin α.

In road practice, the magnitude of the slope is usually estimated by the magnitude of the rise of the road surface, related to the magnitude of the horizontal projection of the road, i.e. tangent of the angle, and denote i, expressing the resulting value as a percentage. If the slope is relatively small, it is permissible not to use sinα., and the value i in relative terms. For large slope values, replace sinα by the tangent value ( i/100) unacceptable.

Acceleration resistance force

When accelerating a car, the forward moving mass of the car accelerates and the rotating masses accelerate, increasing the resistance to acceleration. This increase can be taken into account in the calculations if we assume that the masses of the car are moving translationally, but use a certain equivalent mass m uh, somewhat larger m a (in classical mechanics this is expressed by the Koenig equation)

We use the method of N.E. Zhukovsky, equating the kinetic energy of a translationally moving equivalent mass to the sum of energies:

Where J d– moment of inertia of the engine flywheel and associated parts, N s 2 m (kg m 2); ω d– angular speed of the engine, rad/s; J To– moment of inertia of one wheel.

Since ω k = V A / r k , ω d = V A · i kp · i o / r k , r k = r k 0 ,