Useful book

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USEFUL BOOK

The versatile study of outer space, the exploration of Lunya and the planets, the planning, preparation and implementation of space flights for various scientific purposes - all this will require many thousands of people to be successful in the near future. Quite a few of these schoolchildren will be occupied with these questions.

Those high school students who are fascinated by physical, mathematical and technical problems of astronautics will benefit greatly from the recently published book by V. I. Levantovsky on the mechanics of space flight *). Let us note right away that this book is not for busy bedtime reading. Working on it will require a certain direction from the reader; it must be read not by memorizing, but in moderate doses, with paper and pencil in hand. But there is no doubt that such an investment of time and effort will pay off handsomely.

They say that two monsters threaten every popular science book: the Scylla of inaccessibility and the Charybdis of inaccessibility.

It is pleasant to note that this book, despite the complexity of the problems discussed in it, is not only written in a fairly accessible way, but also at a good scientific level. Many issues raised in the book require the use of rather cumbersome (and significant for schoolchildren) mathematical apparatus for a detailed presentation. However, the author

*) V. I. Levantovsky. Mechanics of space flight in an elementary presentation. M., “Science”, 1970. Circulation 15,000 copies.

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In each case, it was possible to present the essence of the issue without getting carried away by mathematical calculations. As reported in the preface, the author “strove not to overuse formulas, bearing in mind that it is often difficult to see the forest of ideas behind the trees of the formula.” And the “Indian line” is really missing in this work.

The main subject is trajectory problems of space trajectories, their choice and their evolution. Let us dwell in more detail on containing books.

A small “Introduction” includes revealing the meaning of the concepts “cosmos-synamka” and contains an overview of the most necessary information from the school mechanics course for further presentation.

In the first part (“Fundamentals of Rocket and Space Science”) in Chapter 1, the reader will gain a meaningful understanding of how arbitrary calculations of the thrust and speed of a space rocket; he learned why it was necessary to use composite rockets and become familiar with various types of space propulsion systems. The other two chapters of the first part talk about the features of the flight of a spacecraft in the “active” section (the section of insertion into orbit) and in the “passive” section (free flight with the engine turned off only under the influence of gravity), about methods for calculating space trajectories in the gravitational field of one or more host bodies.

The remaining parts of the book correspond to the various stages of the conquest of space: “Near-Element Flights”, “Flights to the Moon”, “Interplanetary Flights”, “Interstellar Flights”.

During the first decade of the space age, more than 600 artificial Earth satellites were launched. The second part of the book will tell about the features of their trajectory, about the various “disturbing” factors influencing these trajectories. Here the reader learned how the continuity and non-sphericity of the Earth affects the nature of the satellite’s orbit. And after a few pages the reader was met with unexpected results of the influence of the atmosphere on the movement of the satellite. It was clearly explained why, due to the resistance of the atmosphere, the speed of the satellite does not decrease, but returns; why, in order to reduce the heating of the plovered satellite (when passing through the atmosphere), its front part should not be pointed, but blunt. Separate paragraphs are devoted to the unexpected results of the influence of the attraction of the Moon and the Sun, the pressure of sunlight and other factors on the trajectories and fate of satellites.

Drawing on rich factual material about already launched satellites, the author talks about space maneuvers (transition of a satellite from one orbit to another, encounters in orbit, etc.), and various practical and scientific applications of satellites.

The third part of the book covers the multi-Greek problems of flying to the Moon: various trajectories of getting to the Moon, the lunar landing, the launch of artificial lunar satellites, man landing on the Moon and the exploration of the Moon. The reader will find here a detailed description of the most important flights performed by Soviet and American spacecraft.

The study of the planets of the solar system using cosmos technology began in 1961-1962, when Soviet international spacecraft were launched.

The study of the planets of the solar system using cosmos technology began in 1961-1962, when Soviet international spacecraft were launched.

Flights outside the system to other star worlds; a phoron racket as a means to perform such a flight; the relativistic effects that galaxy burners will encounter are the subject of the last, fifth (and shortest) part of the book.

The book is written in a lively and expressive language, it contains a large number of interesting facts, figure, observations. We are confident that the reader will read it not only with benefit but also with convenience.

M. V. Balk

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To the article “On problems in physics”

1. Since the image is obtained at an infinite distance from the lens, its luminous intensity must obviously be considered infinite. But any area whose illumination may interest us is infinitely distant from the image of the light source. The illumination of any area parallel to the plane of the lens, if, of course, this area is located in the beam of light emerging from the lens, is equal to the illumination of the lens itself, regardless of the distance between the lens and the area.

2. $$\frac{1}{H^2(n+1)^2},$$ and we neglected the reflection of light at the boundary of water and air.

3. $$\frac{a \frac{b+c}{2}}{a+\frac{b+c}{2}} = \frac{a(b+c)}{2a+b+c}.$$

4. When the distance changed from 5 to 10 km, all the time the entire light from the laser minus the absorbed air fell on the lens (this is easy to verify by continuing the corresponding geometric calculations). Consequently, the total power of the laser beam over 5 km was 2 times greater. In 10 km (from 10 to 20) it settles 4 times. In addition, starting from 10 km, the receiver lens no longer intercepts the entire beam. According to the inverse square law, even in a completely free-air atmosphere, a signal at 20 km should be 4 times weaker than at 10 km. This means that, in general, the signal when changing the distance from 10 to 20 km is adjusted 16 times.

To the article “Entrance exams in mathematics at the Moscow Lekstilny Institute in 1970”

1. $$x = \frac{\pi}{9}(2k+1)$$ ($k=0, \pm 1, \dots$).

2. $$x_1 = \frac{1}{2}, x_2 = \frac{1}{4}$$.

3. $$\sqrt[6]{a}$$.

4. $$V = \frac{1}{24}a^3tg\varphi.$$

5. $$x = \frac{(-1)^n \arcsin(\sqrt{3}-1)}{2} + \frac{\pi n}{2}$$ ($n=0, \pm 1, \dots$).

6. $$x_1 = \frac{\pi}{15}(3k+1), x_2 = \frac{\pi}{6}(3n+2)$$ ($k,n=0, \pm 1, \dots$).

7. $$x = \pm \frac{\pi}{3} + 2k\pi$$ ($k=0, \pm 1, \dots$).

8. $$-\text{ctg}\frac{\alpha}{2}$$.

9. $$-2$$.

10. $$x_1=3, x_2=3\log_62$$.

11. $$x_1=10, x_2=0,1$$.

12. $$x_1=10^3, x_2=10^{-6}$$.

13. $$x=2$$.

14. $$x_1 = \log_3 \frac{28}{27}, x_2 = \log_3 10$$.

15. $$x>3$$.

16. $$-1 < x < 1$$.

17. $$0 < x < 1; x > 100$$.

18. $$1 < x < 2$$.

19. $$-3 < x < 2$$.

20. $$1 < x < 2; x > 3$$.

21. $$a_1 = 2, a_2 = \frac{2}{35}$$.

22. $$2\sqrt{\frac{a}{x^3}}$$.

23. $$\alpha = \frac{\pi}{3}, \beta = \frac{\pi}{6}$$.

To the article “Written examination in mathematics at the humanities faculties of Moscow State University in 1970”

1. $$a = \frac{8}{65}$$.

2. $$\frac{\sqrt{41}+1}{8} < x < \frac{\sqrt{5}}{2}$$.

3. $$a = -3 + \frac{\sqrt{2}}{2}$$.

4. $$1\frac{3}{8} < x < 2$$.