Energy and momentum of fast particles

Kvant 1970, No. 3 p. 34, 35, 36, 37, 38, 40, 41 Tag ZJWE

Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →

ENERGY AND MOMENTUM OF FAST PARTICLES

When presenting something new, one must rely on
already known. Let us assume that the formula
E=Mc² the reader knows.

This formula was discovered by Einstein. This
we owe to him what we know now,
how to calculate the energy of the fast parts, and we know that even in a supine position
than the stone is drawn to countless non-sharp
required energy reserves. He you-
introduced this formula long before
she got caught for the first time practically
(long before 1919, when
vye was replaced by nuclear transformation). Back in 1905, Einstein
said that energy and momentum are very
fast body cannot be calculated by
grafting formulas E=mv²/2 or
P=mv. He proved many other things,
he literally turned our principles upside down
familiar ideas about in all chapters -
nal things: movement, space,
time, light, mass. But for now we
the only thing that matters is what he said about
energy and momentum.

The essence of Einstein’s discovery can be
put it roughly like this.

MASS AND SPEED

There is nothing in the world faster than light. And one light cannot be faster than the other. Any light (in emptiness) always moves equally quickly. Therefore, it is convenient to take the speed of light as unity. Any other movement, for example the movement of any body, cannot occur faster than the propagation of light, that is, the speed of any body is always less than unity. But what then to do with a body that some force accelerates for a very long time? After all, any force causes acceleration, and acceleration increases speed, and won’t there come a time when the speed of the accelerated body exceeds this amount of one? But this is impossible; This means that with increasing speed, the acceleration should gradually decrease - decrease so quickly that it does not have time to bring the body’s speed to unity. But what does it mean that when building up the action of our force, the acceleration decreases? How can this be?

Another property of motion is known: acceleration is inversely proportional to the mass of a body - the heavier the body, the more difficult it is to accelerate it with the same force. This means that we can conclude that the acceleration decreases due to the fact that the mass increases. Then the ends meet: as the speed increases, the body becomes heavier, and the previous force can no longer give it the same acceleration. The acceleration drops and the speed remains almost unchanged. Eischstein derived a formula for the increase in mass as the speed of the body $v$ approaches unity:

$M = \frac{m}{\sqrt{1-v^2}}.$

Here $m$ denotes the mass of the body when it is motionless, that is, when $v=0$ . At speed $v$ , they approach unity, the substitute will become less and less, and the fraction itself will become more and more.

MASS AND ENERGY

Now let’s approach the issue from the other side. After all, the force that acted on the body for so long must have been applied by some person or some kind of engine. Let, for example, an engine. He worked as long as

For some time, I spent fuel and energy on this. And energy, as we know, cannot disappear for free. It is apparently transmitted to the accelerating body, and the more the engine must act, the more energy the body absorbs. But where to absorb it if the speed of the body still cannot exceed one? The answer is simple: very much, energy is spent on increasing body weight. Mass growth is a reflection of σταenergy. Everything comes together again: the force does work on the body, mutilating its energy; energy accumulates, accumulates in the body, mutilating its mass. It becomes clear where the famous formula $E=Mc^2$ could have come from, which we write in the form $E=M$ , because we took the speed of light $c$ as one. Just don’t think that we derived the formula $E=Mc^2$ . It was obtained on the basis of completely different messages, and we used the simplest method of its meaning.

Let us now summarize what has been said, but express it differently. Why do we need new formulas for mass and energy when moving quickly? If the mass of a body did not increase as it accelerated, then its speed would increase, and in the end the body would be overtaken by light, and this contradicts experience. If the energy of a body did not increase during acceleration, then where would the work spent on acceleration go?

ANSWERS TO QUESTIONS

“This is all good,” you say, “but why has no one noticed that bodies, as they accelerate, become heavier?”

This is really difficult to notice: everything that surrounds us moves too slowly. slow compared to the speed of light, with one. After all, the accumulation of mass in a body becomes noticeable only as the body’s speed approaches the maximum, and the speed of the fastest speed is less than 0.0001 - that’s how high the speed of light is. Now, if the speed of light were, say, 10 km/sec, then the designer would have to use Einstein’s formulas in his calculations, taking into account the deterioration of inertia as the speed approaches this speed. And if the speed of light were even lower, for example 1 km/sec, then quite a number of phenomena in the world would flow completely differently, and Einstein’s mechanics would seem to us as natural as Newton’s mechanics now.

“But then,” you ask more questions, “don’t they contradict each other given our thoughts and grafting speeds?” No, it doesn’t contradict. Einstein conducted his deliberations in such a way that at low speeds he did not require Newton’s repeatedly tested laws. If the speed $v$ is very small, the fraction $\frac{1}{\sqrt{1-v^2}}$ turns into $1+\frac{v^2}{2}$ with good accuracy (check by substituting, for example, $v=0,0001$ ), and the mass growth formula turns into the formula

$M = m + \frac{1}{2} mv^2.$

When a package flies even at a speed of 30 km/sec, this means that $v = 0,0001$ , that is, the mass increases by approximately one two hundred thousandth. It is impossible to notice this.

Instead of the last formula, you can write an equivalent formula, if you remember that the mass of a body and its energy reserve are one and the same:

$E = m + \frac{1}{2} mv^2.$

This means that at low speeds the energy of any free moving body consists of two parts: the $m$ part, which does not depend on the speed, and the $\frac{mv^2}{2}$ part, which grows as the square of the speed… Wait, but $\frac{mv^2}{2}$ is the kinetic energy of the body! This means that Einstein discovered that kinetic energy (which we usually think of as the energy of a body moving freely, without the influence of any forces) is only part of the total energy reserve that a body has. And a very small part. The main energy is contained in the term $m$ - in that mass that is not affected by speed and is present in the body even when it stands still. This is what can be called the energies of existence.

If a new grain of matter arose somewhere, then some work was expended on its creation, some other body had to equalize or from some source the energy reserve for the construction of this grain of matter, and this supply will already be in it, even if the grain does not move. For ordinary, large bodies made of atoms, this sounds incredible: after all, we always create them from ready-made building materials (atoms) and do not waste energy on creating atoms. This means that in this case the energy of existence does not seem to be of particular importance: everything that is needed already exists. There, the question of the interaction of substances simply does not arise.

Transformations of the smallest particles are another matter. There is actual creation of new varieties of particles from particles of the previous variety and from the energy they have accumulated, or even from light alone. Like Wibsen’s button maker, all the old stuff is melted down without a trace, and no one will allow us to neglect the energies $m$ .

HOW TO NOT SPOKE AN ELEPHANT

The presence of the term $m$ in the formula for table energy is important, so it’s worth talking about it in more detail. Why don’t we notice this term? Why didn’t Einstein notice such reserves “under our feet”, millions and billions of times greater than all the energy available at that time? Doesn’t this mean that Einstein is wrong? No, he doesn’t. The thing is that we notice not the energy itself, but its changes. Converter kinetic energy into potential - we will immediately notice this, because the speed will drop. It turns into warm - again, we note: the body will heat up. And if the energy does not change, how can you notice it? Let’s say Earth. Her kinetic

The chemical energy is enormous; It rushes around the Sun at a speed of 30 km/sec, its mass is 6·10²⁷ e - this is an amazing supply of energy that exceeds our imagination. But who is replacing her? How does it manifest itself? Should it be taken into account and included in the balance of transformations occurring with earthly bodies? Of course not: it does not change during such transformations, it is dead capital; the way it enters the balance is the way it comes out.

The same situation applies to energy m; it does not change in mechanical, electrical, chemical transformations, it is silently present in both parts of the energy balance equation, and it is neither hot nor cold from it. Now, if it were possible to find some forces capable of “prying off” at least a piece from m, then m would immediately make itself felt. But at first they knew nothing about such forces. The good thing is that the formula

$E = m + \frac{mv^2}{2}$

suggested that it is worth looking for such powers. And they were found many years later; these were nuclear forces. In nuclear power plants or ships, such forces are engaged in the fact that small parts float away from m and convert them into electrical or mechanical energy.

In the transformations of elementary particles, forces of similar nature, but incomparably greater in magnitude, no longer float away from the rest mass m piece by piece. Their activity radially rearranges one brick of matter into others, sometimes in no way - neither properties nor purpose - similar to the first.

IMPULSE AND SPEED

But we were distracted from our direct goal. So, we know how body mass depends on speed:

$M = \frac{m}{\sqrt{1-v^2}}.$

The energy of a body also depends on speed:

$E = M = \frac{m}{\sqrt{1-v^2}}.$

What now to do with the value P=Mv, called the momentum of the body? maybe it also needs to be replaced with something?

It turns out not. The impulse to the mode is given by the formula

$P = Mv,$

but only M is now a magnitude dependent on speed. This means that momentum, like the mass of a body and its energy, can become somewhat larger as the body accelerates. And it remains due to Newton’s assertion that the growth of momentum under the influence of a force is proportional to the magnitude of the force itself and the duration of its action. If the force acts long enough (and in the right direction), then the impulse can reach any value.

Therefore, the formula for momentum can be written in three forms:

$P = Mv, P = Ev, P = \frac{mv}{\sqrt{1-v^2}},$

and use the immi of your choice.

This is the transformed form

was 0.94 GeV. This means that the proton becomes more than 10 times more massive in 3 seconds (that’s how long the acceleration lasts). The power consumed by the large electromagnet accelerator increases many times towards the end of acceleration. If you want to verify the validity of Einstein’s formula, look at the wattmeters of the distributed shield, how the reactive load returns to them.

Wait, wait! - the vigilant reader will exclaim. - What does this mean? At the power plant, electrical grains disappear, and wood-heavy protons appear in the accelerator. So, in your opinion, energy turned into mass?!
Why does this bother you so much?
Because this is an erroneous philosophical thesis… and you yourself said that energy and mass are just two different shades of the same physical concept.

— In physics, yes. But in everyday life, an increase in energy does not mean an increase in mass. Heating the kettle does not make it heavier. Therefore, in the everyday sense, the difference between energy and mass is enormous. And when suddenly you witness how the power supplied to the input of the accelerator turns into unusually heavy protons at the output, you have the right to say in surprise: “Electricity has passed into the mass of the proton!”

Or in his energy…
Or into its energy, if we want to emphasize not the “intransigence” of the proton, but its “reserve of strength”, “creative potency”… We must get used to the fact that the “intransigence” and “reserve of creative strength” of the particle are synonymous. And when we get used to this, we will even have an ineradicable desire to banish one of the words - “energy” or “mass” - and make do with only one. In modern service books on physics they try to do this. But in our book, where the lack of formulas will come to be compensated by verbal expression,

Dubensky synchrophasotron.

ness, we will use both si-
nonima: both energy and mass.

What about philosophers?
— Philosophers are different from philosophers.
Why look for problems where there are none, why monitor the use of words, where the true meaning is not in words, but in exact relationships…
People with a practical mindset are concerned about something else: is it true that the energy released in the decay of elementary particles is much greater than nuclear energy?
Yes. For example, one of the cycles of nuclear reactions that give energy to stars coincides with the transformation of four protons into a gel nucleus. Their mass is 0.94·4=3.76 GeV, and the mass of the gel is 3.73 GeV, therefore, 0.03 GeV is released - less than 1% of the total energy. And in the decay of the π° meson into photons, the entire mass of the meson (100%) is converted into energy.
So, we have before us a source of energy more powerful than a thermonuclear reaction?
Not at all. The rarity and instability of such mechanisms is a hindrance. Next to the accelerator there are at least 10¹⁹ times fewer of them than protons. In addition, they die instantly; they cannot be accumulated. And most importantly, they must be created, spending energy on this exactly equal to that released during their decay; protons are always ready - these are hydrogen nuclei. In a thermonuclear reaction, we transfat the energy reserves accumulated by nature; decays of π° mesons, at best, only return to us the energy spent on their creation.

“Then what’s the use of them?”
— Tolk from mezons, hyperons, etc. in another: they provide clues to the structure of the world…

TASKS

Calculate the momentum of the proton in MeV/c, where c is the speed of light, if it is known that the kinetic energy of the proton is 500 MeV.
Calculate the acceleration of an electron moving into an accelerator along the lines of direction of a uniform electric field with E = 25 kV/cm, at the moment when its kinetic energy is T = 0.5 m₀c², where m₀ is the rest mass of the electron.
What mass of hydrogen is converted into energy when a 20-megaton hydrogen bomb explodes? The equivalent of trinitrotoluene is 10³ cal/2.
How much mass does a 100-watt light bulb emit (in the form of tell and light) over the course of a year?
How to determine the rest mass of a system of moving particles? Is it equal to the sum of the masses of the particles at rest?
How long does a person have to ride on the magnitude to convert it into energy? When a person cranks the teacher as hard as he can, he produces 0.5 net horsepower (373 bhp). The efficiency of the human body is about 25%, that is, 75% of food is burned, turning into heat, and only 25% of food goes into useful work.
How much mass does the Earth receive from the Sun in the form of light during the year if 1.4 km of light energy arrives per square meter of surface perpendicular to the direction of the sun’s rays?
What is the phonon impulse? Show, based on the laws of conservation of energy and momentum, that a free electron cannot absorb a phonane.