In everyday life, the epithet “impulsive” is sometimes used to characterize a person committing spontaneous acts. However, some people do not even remember, and a significant portion do not even know what physical quantity this word is associated with. What is hidden under the concept of "body impulse" and what properties does it have? Answers to these questions were sought by such great scientists as Rene Descartes and Isaac Newton.

## Body impulse: definition

Like any science, physics operates with clearly defined concepts. At the moment, the following definition is adopted for a quantity bearing the name of a body impulse: it is a vector quantity that is a measure (quantity) of a mechanical body movement.

Suppose that a question is considered within the framework of classical mechanics, that is, it is considered that the body moves at a normal rather than relativistic speed, which means that it is at least an order of magnitude smaller than the speed of light in a vacuum. Then the impulse module of the body is calculated by the formula 1 (see photo below).

Thus, by definition, this quantity is equal to the product of body mass and its speed, with which its vector is co-directed.

The impulse measurement unit in the SI (International System of Units) is taken to be 1 kg / m / s.

## Where does the term "impulse"

A few centuries before the concept of the amount of mechanical movement of a body appeared in physics, it was believed that the cause of any movement in space was a special force - impetos.

In the 14th century, Jean Buridan introduced amendments to this concept. He suggested that a flying cobblestone has an impetus that is directly proportional to speed, which would be unchanged if there were no air resistance. At the same time, in the opinion of this philosopher, bodies with greater weight had the ability to “contain” more such a driving force.

Further development of the concept, later called the impulse, gave Rene Descartes, who designated it with the words "the amount of motion". However, he did not take into account that speed has a direction. That is why the theory put forward by him in some cases contradicted experience and did not find recognition.

The fact that the amount of motion should have more direction, the English scientist John Wallis was the first to guess. It happened in 1668. However, it took another couple of years to formulate a known law of conservation of momentum. The theoretical proof of this fact, established empirically, was given by Isaac Newton, who used the third and second laws of classical mechanics discovered by him, named after him.

## The momentum of a system of material points

Consider first the case when it comes to speeds that are much lower than the speed of light. Then, according to the laws of classical mechanics, the total momentum of a system of material points is a vector quantity. It is equal to the sum of the products of their masses at a speed (see formula 2 in the picture above).

At the same time, for the impulse of one material point, they take a vector quantity (formula 3), which is co-directed with the velocity of the particle.

If we are talking about a body of finite size, then at first it is mentally broken into small parts. Thus, the system of material points is considered again; however, its momentum is calculated not by ordinary summation, but by integration (see formula 4).

As you can see, time dependence, so the momentum of the system, which is not affected by external forces (or their effects are mutually compensated), remains unchanged over time.

## Proof of the law of conservation

We continue to consider a body of finite size as a system of material points. For each of them, Newton's Second Law is formulated according to formula 5.

Pay attention to the fact that the system is closed. Then, summing over all points and applying Newton's Third Law, we obtain the expression 6.

Thus, the impulse of a closed system is constant.

The law of conservation is also valid in those cases where the total amount of forces that act on the system from the outside is zero. From here follows one important particular statement. It states that the impulse of a body is constant if there is no external influence or the influence of several forces is compensated. For example, in the absence of friction after a stick strike, the puck must maintain its momentum. Such a situation will be observed even though gravity and the reactions of the support (ice) act on this body, since, although they are equal in magnitude, they are directed in opposite directions, i.e. they compensate each other.

The impulse of a body or a material point is an additive quantity. What does it mean? Everything is simple: the impulse of the mechanical system of material points consists of the impulses of all the material points included in the system.

The second property of this quantity is that it remains unchanged during interactions that change only the mechanical characteristics of the system.

In addition, the impulse is invariant with respect to any rotation of the reference system.

## Relativistic case

Suppose that we are talking about noninteracting material points having speeds of the order of 10 to the 8th power or slightly less in the SI system. The three-dimensional impulse is calculated by the formula 7, where by c we mean the speed of light in a vacuum.

In the case when it is closed, the law of conservation of momentum is true. At the same time, the three-dimensional momentum is not a relativistically invariant quantity, since its dependence on the reference system is present. There is also a four-dimensional option. For one material point it is determined by the formula 8.

## Momentum and energy

These quantities, as well as mass, are closely related to each other. In practical problems, relations (9) and (10) are usually used.

## Determination through de Broglie waves

From the last relation, we find that the modulus of the pulse and the wavelength, denoted by the letter “lambda”, are inversely proportional to each other (13).

If a particle with a relatively low energy is considered, which moves at a speed that is not comparable to the speed of light, then the pulse modulus is calculated in the same way as in classical mechanics (see formula 1). Consequently, the wavelength is calculated according to expression 14. In other words, it is inversely proportional to the product of the mass and velocity of the particle, that is, its momentum.

Now you know that the impulse of a body is a measure of mechanical movement, and you have become acquainted with its properties. Among them, in practical terms, conservation law is particularly important. Even people far from physics observe it in everyday life. For example, we all know that firearms and artillery give feedback when firing. The law of conservation of impulse clearly demonstrates the game of billiards. With it, you can predict the direction of expansion of the balls after the strike.

The law has found application in the calculations necessary for studying the effects of possible explosions, in the field of developing jet apparatus, in designing firearms, and in many other areas of life.