Impact force - impulse, speed, equipment and explosive force exercises for fighters from Pavel Badyrov. Impact of shock

Take a look at the dictionary of foreign words: "impulse" - from lat. impulsus - push, strike, motivation. " The effect produced by the blow has always been surprising in humans. Why is a heavy hammer laid on a piece of metal on the anvil only presses it against the support, and the same hammer crushes the metal with a hammer blow? And what is the secret of the old circus trick, when the crushing blow of a hammer on a massive anvil does no harm to the person on whose chest this anvil is installed? What is the mistake in the question that one student once asked: “What is the impact force when a load of 20 kg falls from a height of 10 m?” And what does the expression “impact force” mean?

Galileo was also interested in the problem of “amazing impact force”. He describes the witty experience with which he tried to determine the "power of the blow." The experiment consisted of the following: two buckets were suspended from one end, and a load (stone), balancing them, from one end to a solid beam mounted horizontally on an axis like a beam of a balance (Fig. 39). The upper bucket was filled with water; a hole closed by a cork was made in the bottom of this bucket.

If you remove the cork, then the water will pour into the lower bucket and the force of the impact of the jet on the bottom of this bucket, it would seem, will cause the right side of the rocker to fall. The addition of the corresponding cargo on the left will restore equilibrium, and its mass will allow you to assess what the impact force of the jet is.

However, to Galileo's surprise, experience showed a completely different thing. At first, as soon as the cork was removed and the water began to pour out, not the right but the left part of the rocker arm fell. And only when the jet reached the bottom of the lower bucket, the balance was restored and was no longer disturbed until the end of the experiment.

How to explain this “strange” result? Is it wrong with Galileo's first assumption that a jet striking the bottom of the lower bucket will cause it to sink? To understand this rather complex issue, you need to know the law of conservation of momentum, which, together with the law of conservation of energy, refers to the greatest laws of nature.

The term “quantity of movement” was introduced by Galileo's contemporary, the French philosopher and mathematician Descartes, but it was introduced far from a scientific basis, but from metaphysical (not based on experience) religious ideas of the philosopher. The indefinite, foggy term “momentum” is now replaced by the term “momentum”.

In the previous conversation, we cited Newton’s second law in the form that Newton himself gave him: "The change in momentum is proportional to the moving force and occurs in the direction of the straight line along which this force acts."

Newton first introduced the concept of mass into mechanics and, using it, gave an exact definition of the quantity of motion as the product of the mass of a body and its speed (mv).

If the initial speed v 0 of a body of mass m under the influence of any force during time t increases to v 1, then the change in momentum per unit time will be:

  This change is proportional to the applied force F:

mv 1 - mv 0 \u003d Ft

This is Newton’s second law. It follows from it that the same change in the momentum can occur both with the prolonged action of a small force and with the short-term action of a large force. The product Ft can be considered as a measure of the force. It is called the impulse of power. Do not mix only the impulse of force with the force itself, as well as with the impulse. It can be seen from the above formula that the momentum of a force is not equal to the momentum itself, but to the change in momentum. In other words, the momentum of the force over time t is equal to the change in the momentum of the body during this time. Momentum is usually denoted by the letter p:

In the general case, it should be taken into account that the momentum is a vector physical quantity:

  We have already mentioned the two greatest laws of nature: the law of conservation of momentum and the law of conservation of energy. These laws are conveniently demonstrated by the example of a blow. The phenomenon of shock is of great importance in science and technology. Consider this phenomenon more carefully.

We distinguish between elastic and inelastic materials. For example, a rubber ball is resilient; this means that after the termination of the deforming force (compression or tension), it again returns to its original form. On the contrary, a piece of clay crumpled by hand does not return to its original form. Rubber, steel, marble, bone are elastic materials. You can easily see the elasticity of the steel ball by dropping it from a certain height onto the elastic support. If the ball was previously smoked, then the trace will remain on the support not in the form of a point, but in the form of a sufficiently distinguishable speck, since the ball crumpled upon impact, although then, having rebounded, it regained its shape. The support is also deformed. The elastic force arising in this case acts on the ball from the support side and gradually reduces its speed, informing it of the acceleration directed upward. In this case, the direction of the speed of the ball changes to the opposite and it takes off above the support to the same height from which it fell (ideal case with perfect elasticity of the colliding bodies). The support itself, as connected with the Earth having a huge mass, practically remains motionless.

Successive changes in the shape of the ball and the surface of the support for different times are shown in Figure 40. The ball falls from a height h and at the time of landing (position in the figure) has a speed directed vertically downward. In position B, the deformation of the ball is maximum; at this moment, its speed is zero, and the force F acting on the ball from the side of the support plane is maximum: F \u003d F max. Then the force F begins to decrease, and the speed of the ball grows; point C corresponds to the moment when the speed value. In contrast to state A, now the speed is directed vertically upward, as a result of which the ball takes off (jumps) to a height h.

Suppose that an elastic ball moving at a certain speed collides with a stationary ball of the same mass. The action of a stationary ball is again reduced to a decrease in the speed of the first ball and its stopping. At the same time, the first ball, acting on the second, tells it the acceleration and increases its speed to its original speed. Describing this phenomenon, they say that the first ball transmitted its momentum to the second. You can easily verify this experimentally with two balls suspended on threads (Fig. 41). Measuring the speed with which the balls move is, of course, difficult. But you can use the well-known position that the speed acquired by the falling body depends on the height of incidence (). Except for small energy losses due to the incomplete elasticity of the balls, ball 2 will take off from the collision with ball 1 to the same height as ball 1 fell. Moreover, ball 1 will stop. The sum of the pulses of both balls, therefore, remains constant all the time.


  It can be proved that the law of conservation of momentum is observed in the interaction of many bodies. If external bodies do not act on the system of bodies, then the interaction of bodies inside such a closed system cannot change its total momentum. You can now “scientifically” refute the boastful tales of Baron Munchausen, who claimed that he managed to pull himself out of the swamp by his own hair.

Returning to the famous Galileo experiment with which we began our conversation, we will not be surprised at the result of the experiment: in the absence of external forces, the momentum of the whole system could not change and therefore the beam remained in balance, despite the impact of the jet on the bottom of the second bucket. A detailed mathematical analysis of the experiment is rather complicated: it is necessary to calculate the decrease in mass of the upper bucket, from which a stream of water is poured, the reaction of the leaky jet and, finally, the impulse communicated to the bottom of the lower bucket by the impact of the jet. The calculation shows that the sum of all impulses, taking into account their signs, is equal to zero, as it was before the cork was pulled out, and the whole system — a beam, buckets, counterweight — remains in equilibrium.

The law of conservation of momentum and the law of conservation of energy are the basic laws of nature. Note, however, that the conservation of momentum in mechanical processes is always and unconditionally valid, while applying the law of conservation of energy in mechanics, one must be careful (it requires certain conditions to be satisfied). "Can not be! “You will exclaim indignantly,“ the law of conservation of energy is valid always and everywhere! ” And I don’t argue, read on. Let us consider an example of a collision of elastic and inelastic balls.

Bounce. Let a ball weighing 2 kg move at a speed of 10 m / s to hit a second (motionless) ball of the same mass. As we already know, after the impact, the first ball will stop, and the second will move at the speed of the first ball before the collision.

Check the law of conservation of momentum:

  Law of energy conservation:

  Both laws are observed.

Inelastic impact (balls made of soft clay or putty). After the impact, the stuck together balls continue to move together, but at a speed half that of the first ball before the impact.

The law of conservation of momentum:

  The law is respected.

Law of energy conservation:

  Before the impact, the energy was 100 J, and after the impact, 50 J! Where did half the energy go? You probably guessed: the mechanical energy equal to 50 J turned into internal energy: after the impact, the molecules began to move more briskly - the balls heated up. If we could take into account all types of energy before and after the impact, we would be convinced that even in the case of an inelastic impact, the energy conservation law is not violated. The law of conservation of energy is always true, but one must take into account the possibility of converting energy from one type to another. In practical cases, the application of the laws of conservation of energy and momentum is especially important. Consider a few examples of the application of these laws.

Forging products in the forge shop. The purpose of the forging is to change the shape of the product using hammer blows. For the best use of the kinetic energy of the falling hammer, it is necessary to put the product on a large anvil. Such an anvil will receive a negligible speed, and most of the energy upon impact will turn into deformation energy (the shape of the product will change).

Pile driving. In this case, it is advisable to transfer most of the kinetic energy to the pile so that it can do the job of overcoming the soil resistance and go deeper into the soil. The mass of the pile driver, i.e., the load that falls on the pile, must be greater than the mass of the pile. In accordance with the law of momentum, the pile speed will be higher in this case and the pile will go deeper into the ground.

On the power of impact. The task set at the beginning of our conversation does not indicate the duration of the strike, but the latter depends on the nature of the support. With rigid support, the duration of the impact will be less, and the average force of the impact is longer; with soft support, vice versa. The net, stretched under the trapezoid in the circus, protects the air gymnast from a strong blow when falling. A footballer, taking a hit of the ball, should be fed back, thereby increasing the duration of the strike - this will soften the kick. There are many such examples. In conclusion, we will examine another interesting problem, which after all of the above will be clear to you.

“Two boats move by inertia in the calm water of the lake towards each other in a parallel course at a speed of v 1 \u003d 6 m / s. When they caught up, the cargo was quickly transferred from the first boat to the second. After that, the second boat continued to move in the same direction, but with a speed of v 2 \u003d 4 m / s.

Determine the mass M 2 of the second boat if the mass M 1 of the first without a load is 500 kg and the mass m of the load is 60 kg. Calculate the energy reserve of boats and cargo before and after shifting the cargo. Explain why this energy reserve has changed. ”

Decision. Before meeting, the momentum of the first boat is: (M 1 + m) v 1, and the momentum of the second boat: M 2 v 1.

When transferring cargo from the first boat to the second, the speed of the first boat does not change, since it experiences a push in the lateral direction (recoil), which cannot overcome the resistance of water. The speed of the second boat changes, since the transferred cargo must sharply change the direction of its speed to the opposite, which can be considered as a push.

Applying the law of conservation of momentum, we write:


  Energy decreased by 3500 J. Where did the energy go? The lost part of the mechanical energy turned into internal energy (heat) when the speeds of the load and the second boat were aligned.

MECHANICAL SHOCK

Nizhny Novgorod
  year 2013

Laboratory work No. 1-21

Mechanical shock

purpose of work: Get familiar with the elements of the theory of mechanical shock and experimentally determine the impact time, the average impact force Frecovery factor E, as well as study the basic characteristics of the impact and familiarize yourself with digital instruments for measuring time intervals.

Theoretical part

  Impact is called a change in the state of movement of the body, due to its short-term interaction with another body. During impact, both bodies undergo shape changes (deformation). The essence of elastic shock lies in the fact that the kinetic energy of the relative motion of colliding bodies, in a short time, is converted into the energy of elastic deformation or, to one degree or another, into the energy of molecular motion. In the process of impact, energy is redistributed between colliding bodies.

Let a ball with a certain speed V 1 fall on a flat surface of a massive plate and bounce off it with a speed V 2.

We denote   Are the normal and tangential components of the velocities and, and, and are the angles of incidence and reflection, respectively. In the ideal case, with absolutely elastic impact, the normal components of the rates of incidence and reflection and their tangential components would be equal; . Upon impact, a partial loss of mechanical energy always occurs. The ratio of both normal and tangential velocity components after impact to velocity components before impact is a physical characteristic that depends on the nature of the colliding bodies.

This characteristic Ecalled the recovery coefficient. Its numerical value lies between 0 and 1.

Determination of average impact force,

Initial and final ball velocities upon impact

  The experimental setup consists of a steel ball A suspended on conductive threads and a motionless body B of a larger mass with which the ball collides. Suspension angle α is measured on a scale. At the moment of impact, a ball of mass m is affected by gravity from the side of the Earth, the reaction force from the side of the thread, and the average force of impact from the side of body B (see Fig. 2.).

Based on the theorem on the change in momentum of a material point:

where and are the velocity vectors of the ball before and after the impact; τ is the duration of the impact.

After designing equation (2) on the horizontal axis, we determine the average impact force:

(3)

Ball speeds V 1 and V 2 are determined on the basis of the law of conservation and conversion of energy. The change in the mechanical energy of the system formed by the ball and the stationary body B in the Earth's gravitational field is determined by the total work of all external and internal non-potential forces. Since the external force is perpendicular to the movement and the thread is inextensible, this force does not work, the external force and the internal force of the elastic interaction are potential. If these forces are much larger than other non-potential forces, then the total mechanical energy of the selected system does not change. Therefore, the energy balance equation can be written as:

(4)

From the drawing (Fig. 2) it follows that   , then from equation (4) we obtain the values \u200b\u200bof the initial V 1 and final V 2 ball speeds:

(5)

where and are the angles of deflection of the ball before and after the collision.

Method for determining the duration of an impact

In this paper, the duration of a ball hitting a plate is determined by the frequency meter Ch3-54, whose functional diagram is shown in Fig. 3. From the generator, pulses with a period T are fed to the input of the control system of the control system. When during the collision of the metal plate B, the electric circuit formed by the control system, the conductive threads of the ball suspension, the ball, the plate B and the pulse counter C h turns out to be closed, and the control system passes at the counter input C h, electric current pulses only in the time interval equal to the duration of the shock. The number of pulses recorded during the time is equal to where.

To determine the duration of the impact, it is necessary to multiply the number of pulses recorded by the counter by the period of pulses taken from generator G.

experimental part

Initial data:

1. The mass of the ball m \u003d (16.7 ± 0.1) * 10 -3 kg.

2. Thread length l \u003d 0.31 ± 0.01 m.

3. Acceleration of gravity g \u003d (9.81 ± 0.005) m / s 2.

4. Experience for each corner is performed 5 times.

The results of the experiment are listed in the table:

Calculations

=20 0   μs

=30 0   μs

=40 0   μs

Pulse - health, life expectancy, aging and immortality.

Pulse is a shock in the blood vessels from strokes.of our heart, and the size and nature of the work,  all our life depends on them, as on the main pendulum, life expectancy, health, aging and immortality are determined. Heart rate and heart size givelife speed its duration  and aging. The heart of living organisms, perfect and accuratetime mechanisms  and meters speed of life.For thousands of years, people have tried to reproduce the unique accuracy and capabilities of the heart in the form of a water, hourglass, or mechanical clock.Information encoded andbuilt into genes chromosomes, organisms and populations, on the intensity and level of work on which prosperity depends,life expectancy and  their service life.

3 the dependence of the nature of the pulse and the work of the heart on the impulse, stimulus, or conditions formed the basispulse diagnostics,determining and managing the state of the body, sports prospects, reproductive properties, tone depth and possible life expectancy.

Normal heart rate  a healthy person should be 65-75 beats. in min., its level for average weight should not change, the aging rate and life expectancy of 25 and 100 years, depend on the optimal and harmonious pulse. The heart rate of a person at rest, it happens  from 30 to 200 beats. in minutes and more, changes weight, age, time of day, fitness, habits  and lifestyle. The heartbeat frequency and the size of the heart, change the disease of a person and the body, a lower pulse with bradycardia, increases the heart, and an increased pulse with tachycardia, reduces the size.

Heart rate and nature indicate the amount of health  physical   state and size - this is strength, speed, endurance and weight - growth characteristics of the body. Birds and animals at home live much more than their free counterparts in nature, sometimes this difference is different at times, their metabolic rate changes and decreases and their size grows.

Flight Caliber Pulse  eg it is 1,200 beats per minute, alone 500 beats, and in the cordon only 50 beats. And the crocodile pulse normally makes 25-40 beats per minute, and in a state of stupor 1-5 beats, depending on the mass.Calibri live 1–2 years, some species up to 9 years old, crocodiles 5–8 years old, some species can live up to 100 years, and whales live 30–50 years, some species of whales up to 200 years or more.

The biochemistry of the body and the work of organs change already seconds after exposure, and the pulse changes its work after a fraction of a second, changing  proportions of substances and health, priorities and  nature of adaptation,level of aging and futurelife expectancy or immortality.

Due to changes in the so-called variability, different species can reduce energy expenditures when changing external conditions and environments, showing records of endurance and speed in the struggle for survival. A crocodile can do without food for a year or more, and young antelopes and gazelles compete in speed with a cheetah in a few days and even hours after birth.

Of course, a person could not do without food for months and even more so a year, like a crocodile, but reaction and adaptation can also vary widely, asheart rate fluctuationswherein. So, when cooling, the pulse slows down, and when doing work or a disease, it increases sharply. The stronger these fluctuations, the usually the higher the depth of the body tone and the level of metabolism.

Life expectancy depends on the genes of a particular organism, heart rate, and metabolic rate. The greater the mass of the type of organism, the higher the life expectancy; it is noticed that the lower the natural temperature of the organism, the higher it is. It is enough to lower the temperature by one and a half to two degrees, from the natural temperature of 36.6 degrees, to a person with an optimal weight, this will reduce aging and increase life expectancy by tens of years or more. It is worth making a reservation, each species has its own optimal mass. For peopledepending on gender and height,it is from 55 to 85 kilograms, going beyond these limits reduces life expectancy.

Objectively, any excess over 60 kilograms is already a drawback, and the difference in average weight, which depends on gender, should not exceed 20 - 25 kg. It is noticed that people whose weight and height are lower, they have less background of nerve diseases, cancer, diabetes and so on, which is associated with better functioning of the immune system and higher quality of tissues and the level of regeneration that fall with increasing weight.

High human life expectancy at an average of 70 - 80 years, and in other cases up to 100 years or more. The slow pace of aging compared to animals is a payment for the loss of metabolic rate. As a result, we suffer from diseases, many of which are not in the animal kingdom and must be put up with a long period of restoration of the functions of organs and the body after diseases, injuries and work. For example, some insects will repair damage incompatible with life in half an hour, and a torn flower of a plant can go through a full cycle to form full seeds, which is not given to humans. A person is forced to care for his children up to 18-20 years or more until they are fully adapted to an independent life, this is the period by which all the main animal species are already completing their life cycle.

You need to understand that the main regulators are in our brain, these are small departments - the thymus, pineal gland and the most important hypothalamus, from which all our functions, including the pulse, depend. These are the organs on the work of which the production of hormones of youth and life depends, especially the most important of them is the gonadotropin hormone, known as growth hormone.The pineal gland produces melatonin and serotonin. Melotonin sets the sleep, rest and life span, and serotonin is responsible for physical growth and good mood. The more hormones per unit mass, the higher the level of health, and a drop in their values \u200b\u200bleads to illness, impairing the management of organs and tissues. This is a common situation, the occurrence and development of cancer, a decrease in tissue quality, when the health of the body is measured by the weakest or worst organ.

Known for the production of hormones, during sleep the temperature of the human body falls,and the pulse rate in the REM stage is growing, we can conclude - life expectancy depends on the quantity and quality of sleep. By increasing the duration and quality of sleep, you can control the production of hormones, the growth of life expectancy and other processes and functions of the body.

In nature, animals fall into a stupor and prolonged sleep, having found complete safety, stable and comfortable conditions, deep in the ground or on the ceiling of caves and  far from the action of the sun.In extreme cases, due to the shadow high on the tree, providing the body with extreme relaxation and the prototype of the necessary biochemistry, reducing the pulse. It turns out that animals turn the worst environmental conditions into the greatest advantage, that is, into the production of harmonies, turning into a stupor or a prolonged sleep and losing weight.

The most interesting thing is that sometimes people in some situations also fall into a prolonged sleep, and even in a stupor ceasing to grow old, there are numerous cases of litargic sleep and even a case of stupor.Hamba lama   He entered this state in 1927, according to his will in 2002 he was pulled out of the grave when he was 160 years old and breathing, the horse beat with a frequency of 2 beats per minute, and the biological age, according to scientists, was 75 years. Now he most likely died due to the fact that there is no one to help him get out of suspended animation, because for various reasons none of hisstudents and followers.

Giving our body relaxation, comfort and ideal biochemistry, stimulating the production or introducing ready-made hormones, you can get an increase in life expectancy, changing the pulse in accordance with external influences in the phase and interests of the body, essentially reproducing the macropulos remedy.

Scientists have noticed that a high IQ - level of intelligence is the key to high life expectancy, so the ownersIQ - 85 live up to 80 years, and withIQ - 115 live more than 100 years, explain this by higher stress resistance of people with higher intelligence. But most likely he is tallIQ and high life expectancy are interconnected by a feature of genetics, a type of biochemistry, and characteristics of the heart and pulse.

Statistics show that it is precisely nervous and overexcited people who often get sick and shorten their lives due to depletion of reserves of the most valuable components of the body. For the population, the favorable environment is important, the harder the external conditions, the shorter the period between generations. So with the advent of comfortable conditions, the average life expectancy of people has tripled.

There is a clear relationship between performance, productivity, reproduction on the one hand, and life expectancy on the other. Higher any component of the first partand the higher the pulse or less body weight,the lower the life expectancy. Reproductivity occupies a special place in life expectancy, which may be why the gods, who lived in myths forever, but could not have children.

It is necessary to pay attention to the fact that each type of living organism, including ours, has its own optimal pulse and mass values, going beyond the pedals of which causes various diseases and a reduction in life expectancy. It’s not a secret that people whose height is more than 195 centimeters live 30–50 years, that is, significantly less than those whose height is less than 180 centimeters, who live 60–100 years, and sometimes more.

One of the innermost desires of any person to live forever, in connection with these aspirations, great minds, experienced specialists and alchemists have been searching for the elixir or code of immortality for thousands of years. Recently, this search has led to an inconspicuous microscopic subspecies of the jellyfish of the turinopsis nutricule with a size of only 5 millimeters. It turned out that they are truly immortal and able to live a thousand years. And the code of immortality or youth is contained in the biochemistry of their body. They are able to regain youth by injecting some substance after reproduction and reaching a certain limit of biorhythms. From this moment, rejuvenation begins, turning in the opposite direction from the adult state to the larval form, reaching the stage of the larval polyp, again towards the adult organism. This continues as many times as you like, but practically forever, if they are not physically destroyed, for example, by a predator.

To increase life expectancy and the necessary biochemistry with a pulse of one to two beats per minute, it is more correct to enter the body into a trance or numbness instead of freezing it and damaging the cells. Given that in a limited space you can create virtually any conditions that are thousands or millions of times different in magnitude from external influences, the nature of sleep or numbness can also be created quite comfortable and harmonious for a particular organism. This is extremely important when flying outside the solar system, where it is necessary to maintain the internal constancy of biochemistry, where the background of calcium and potassium is especially important, but there are also mass restrictions when cryostations are unacceptable luxury.

It is only necessary to recreate the conditions in order to achieve eternal youth and immortality.

Since time immemorial, people have been racking their brains for what megalithic dolmens were intended for. And all in similar features describe their structure, these are usually four stone stones, carefully adjusted to each other, one of which has a hole, and is covered with a fifth stone on top. All together sometimes with the sixth stone intended for the floor, it forms a room with a carefully fitted stopper covering the hole.

The conclusion is that the person who got inside, and the more so having closed the plug, was going to fence off something. From what? In this version, one is the most suitable conclusion from external influences, and first of all from the sun, as high-precision instruments are placed deep underground to raise their sensitivity.Dalmens most likely -it is a kind of sanctuary to achieve enlightenmentand a trance with a pulse of several beats per minute, where everyone, depending on what his brain was imprisoned for, could receive his innermost.

The cells in the monasteries’s monasteries are designed for the same purpose, only 10,000 years ago people came to this, more thoroughly and monumental, given the interactions of nature, a living organism and the laws of physics. In this design, the buildings and the Krasnodar dolmens, without fail, made it possible to increase the sensitivity and prepare the brain for entering into a trance. For example, to communicate with the spirits of the dead, they were connected to the information field, which allowed proscopy and retroscopy to see the future and the past. In addition, they just turned offfrom earthly problems and past to fully relax and start a new life.

Our ancestors gave   dolmens, a method and a device for the shortest way, achieving harmony and perfection, and we need to restore the “technique” and “school” ourselves.

A shock is a mechanical phenomenon in which a short-term interaction of bodies causes a finite change in the velocity vector of all or some points of the material system with a negligible change in the position of the points of the system. The time interval during which a strike occurs is indicated by a letter and is called the stroke time.

Impact is a common phenomenon when considering the motion of both macroscopic bodies and microscopic particles, such as gas molecules. Thus, the impact phenomenon plays an essential role in a number of technical and physical problems. The nature of the impact substantially depends on the physical structure of the colliding bodies.

Instant power

Since the time during which the impact occurs is small, the final change in velocity during the impact corresponds to very large accelerations of the points of the system. Therefore, the forces acting in the process of impact are many times higher than conventional forces.

These forces are called instantaneous forces. The direct measurement of instantaneous forces is very difficult, since the impact time is usually expressed in thousandths or ten thousandths of a second. In addition, during this extremely short period of time, the instantaneous forces do not remain constant: they increase from zero to a certain maximum, and then decrease again to zero. Thanks to this, the forces causing the blow have to be characterized using some special concepts.

Shock impulse

Consider a point of mass moving under the action of some finite force. Then, at the moment, then apply an instantaneous force P to it, the action of which ceases at the moment. We denote the speeds of the point at the moments and, accordingly, applying the momentum theorem to these moments, we obtain:

The first of these integrals represents the momentum of a finite force in time and therefore is a small quantity of the same order as. Therefore, the speed of the point in question can receive a finite change only if the momentum of the instantaneous force P is finite, denoting by which we have:

where it is called shock, or instantaneous impulse, it characterizes the action of instantaneous force upon impact.

The basic equation of shock theory

Since the impulse of the final force is of the order of a small value, it can be neglected in comparison with the final impulse. Therefore, when studying the action of instantaneous forces during an impact, the actions of finite forces can be ignored, and the impulse theorem for a point during an impact has the form:

The velocities of the point corresponding to the beginning and end of the impact are called before the shock and after the shock speed. The resulting equality connecting the speed of the point before and after the impact with an instantaneous momentum is called the basic equation of the theory of impact. It in this theory plays the role of the basic law of dynamics.

Point offset on impact

The speed of the point during the impact remains finite, varying from to From here the movement of the point will be or it will be a small value of the order of m. Thus, during the impact the point does not have time to shift in any noticeable way. Neglecting this negligible movement, we can say that the only consequence of the action of instantaneous force is a sudden change in the speed of the point. Since the velocity vector can change in this case not only in magnitude, but also in direction, the trajectory of a point at the moment of impact can get a kink (an angular point forms on the trajectory) (Fig. 131).

Material system impact equations

Consider a mechanical system consisting of material points. Suppose that among the external and internal forces acting on the points of the system there are instantaneous forces, which we denote accordingly. For each point of the system, we can write the basic equation of impact:

We multiply each of these equalities by r, vector, where is the radius vector of the point corresponding to the moment of impact (or an infinitely small interval of time of impact). Then we get the equality:

To exclude internal instantaneous forces acting on the system, we add each group of these equalities term by term. As a result, we get:

as previously proved that for internal forces

where P is the momentum of the system.

Moreover,

where is the shock impulse of an external force acting on a point in the system. Therefore, the first of the obtained equalities can be written in the form:

Since they will be the amount of movement of the system before and after the impact, we have: the change in the amount of movement of the system during the time is equal to the sum of the instantaneous pulses of all external forces acting on the system.