V All-Union Physics Olympiad

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At these meetings, the guys could look at their notebooks and talk with the jury members, find out all the questions they were interested in regarding the assignments, and learn new tasks. For example, jury member N. Konstantinov conducted a two-hour circle in which schoolchildren proposed problems they had invented and solved them together. We could not solve only one problem proposed by Olympiad participant N. Romanov.

Task. Which point on the plane is colored in one of three colors: red, blue or yellow. Is it necessary to find three points of a single point that form a triangle similar to this one?

Team leaders did not stand aside either. The jury members gave several lectures for them.

We conclude with another research problem from the jury’s portfolio that was intended for Type II but was not included due to option overload.

Task a. Let $-1 \leq x_1 \leq x_2 \leq \leq x_n \leq 1$. If for some $a$, poppy, that $-1 \leq a \leq 0$, $|(a - x_1)(a - x_2) \dots (a - x_n)| > 1$, then for any $x$, poppy, that $0 \leq x \leq 1$, $|(x - x_1)(x - x_2) \dots (x - x_n)| \leq 1$.

Prove this statement:
a) at $n=2$; 6) at $n=3$;
b) for arbitrary $n$.

V All-Union Physical Olympiad

I. A. Zaitsev

From April 15 to 20, the 5th Weight-Union Physics Olympiad was held in Novosibirsk. According to the already established tradition, it took place in two rounds: theoretical and experimental. When selecting problems, the jury sought to ensure that when solving them there was enough knowledge assumed by a high school programmer, but it would require “physical thinking”, an understanding of what can be neglected, an understanding of how the process described in the task occurs, etc. The jury had the most difficulty with the problems of the experimental round. They should not take too much time to solve and they should be “olympiad”:

When making a decision, it was necessary to show resourcefulness and think about how best to set up the experiment in order to obtain the most accurate result possible. This, however, applied to both theoretical and experimental problems.

In total, the jury selected 14 theoretical and 10 experimental problems. Some tasks were repeated in different classes. Below we publish all the tasks that were offered at the Olympiad (some of them have already been published in «Kvant» No. 10, some we publish in this issue in “Problem «Kvant»”).

Each of the works of the Olympiad participants was carried out twice - first by one of the jury members, then

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Each of the tasks in the work was checked separately by “experts” on this task. This allowed the requirements to be successful and reduce the likelihood of errors. The best and worst works were also checked by the senior according to the variant.

The work on preparing the task and checking the work was led by: the chairman of the jury - academician S. T. Bellev, his deputies - E. I. Bichenkov and I. Sh. Slobodetsky.

Let us give, as an example, solutions to two characteristics of “Olympic” problems of a theoretical type.

1. Two protons and two positrons, initially ending up at the vertices of the square on the $a$ (Fig. 1) side, are separated. The ratio of their masses is $M/m=2000$, and the charges are the same. Find the ratio of the velocities of protons and positrons after decomposition (at infinity). (10th grade).

Initially, all particles are subject to equal forces. But the masses of protons are 2000 times greater than the masses of positrons. This means that the acceleration of positrons will be 2000 times greater than the acceleration of protons. Therefore, positrons quickly spread out to infinity, and then protons will spread out, interacting only with each other. This makes it possible to consider protons as stationary when calculating the velocities of positrons.

Let’s find the total potential energy of positrons before decay. If there were no protons, then the potential energy of interaction between two positrons would be equal to $\frac{e^2}{a\sqrt{2}}$ (i.e., the work that needs to be spent to merge two positrons). The field potential that each proton creates at the point where the positron is located is very high. Therefore, the total potential energy of positrons will be equal to

$$\Pi_1 = \frac{e^2}{a\sqrt{2}} + 2\frac{e}{a}e + 2\frac{e}{a}e = \frac{e^2}{a} \left( 4 + \frac{1}{\sqrt{2}} \right).$$

(We can ignore the potential energy of interaction of protons, since it does not change - protons are motionless.) All this energy will turn into the kinetic energy of positrons as they disperse. It is clear that the velocities of positrons at infinity are the same. That’s why

$$\frac{e^2}{a\sqrt{2}} = Mu^2. \tag{1}$$

Now let’s consider the expansion of protons. Their potential energy before decomposition, very high, is equal to $\frac{e^2}{a\sqrt{2}}$, and the total kinetic energy after decomposition is equal to $Mu^2$ (and is the speed of protons). According to the law of conservation of energy

$$\frac{e^2}{a\sqrt{2}} = Mu^2. \tag{2}$$

Now dividing equation (2) by equation (1), we get

$$\frac{1}{4\sqrt{2}} = \frac{M}{m} \left( \frac{u}{v} \right)^2,$$

where

$$\frac{u}{v} = \sqrt{\frac{m}{M} \frac{1}{4\sqrt{2}} + 1} \approx 0,007.$$

2. On the main table there is a weight system shown in figure 2. The friction coefficient between weights $M$ and $m$ is $k$. The lower right weight is pulled along the table with a force of $F$, as indicated at figure. Find the accelerations of all loads in the system. (8th grade).

This is how Vadim Mirny (school No. 2, Moscow) solved this problem, and received the optimal grade II range.

Let’s renumber the weights as shown in figure 2. Select the reporting system associated with the table and the positive direction to the right. Then the lower one cannot have negative acceleration.

Let together weights 3 and 4 on the table the post to which the thread is attached. In this case, threading

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cut glass cubes $A$ and $B$ of length $l$ each (Fig. 6). The speed of light in air is $c$, in stationary glass - $v$. How much faster will light travel along the lower path if cube $B$ is set in motion at speed $u$? (77)

This problem had to be solved in the approximation $v \ll c$ and $u \ll c$. Many participants, however, solved the problem in the general case, using formulas from the theory of relative energy.

2. Three bodies with masses $m_1, m_2, m_3$ can slide along a horizontal line without friction (Fig. 7), and $m_1 \gg m_2$ and $m_3 \gg m_2$. Determine the maximum velocities of the two edges of these bodies, if at the initial moment of time they accumulated, and the middle body had a speed of $V$. The impacts are considered to be absolutely elastic. (43)

You should not solve this problem “head-on” by composing a large number of equations. Here it was necessary to obtain the consequences of the conditions $m_1 \gg m_2$ and $m_3 \gg m_2$.

3. A ball of radius $R$ slides down a staircase whose width and height of steps are $a \ll R$ (Fig. 8). The covariate sphere with steps is inelastic, there is no friction. What is the maximum speed the ball will reach when it reaches the big staircase? (16)

4. There is water in a cylinder with a piston, inside of which at the initial moment there is a cavity of volume $V$ (Fig. 9). The vapor pressure in the cavity is negligible. The piston exerts constant pressure $P$ on the water. What kinetic energy will the water acquire at the moment when the cavity disappears? The initial speed of water is zero. Strength can be ignored. (109)

5. The mass $m$ of hydrogen and the amount of oxygen required for complete combustion enters the jet propulsion chamber per second. The cross section of the output response of the motion nozzle is $S$, the pressure at this second is $P$, the temperature is $T^\circ$ K. Determine the traction force of the motion.

30 ninth-graders who participated in the Olympiad did not solve a single problem, 45 solved one problem, 51 solved two, 29 solved three, 13 solved it on Thursday. Only two completed all the tasks.

10th grade (249 participants)

1. figure 10 shows part of a circuit consisting of unknown resistances. How, having an ammeter, a voltmeter, a current source and connected wires, can you measure the value of one of the resistances without breaking a single contact in the circuit? (45)

The problem has a very short and interesting solution. Try to find it.

2. A pendulum $BC$ with a ball of mass $m$ (Fig. 11) is suspended from the pendulum $AB$ with a ball of mass $M$. Point $A$ oscillates in the horizontal direction with a period of $T$. Find the length of thread $BC$ if it is known that thread $AB$ remains vertical all the time. (35)

3. See task 5 for grade 9. (120)

4. Two protons and two positrons, initially rummaged in the ver-

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The square tires on the side $a$ (see Fig. 1 on page 44) fly apart. The ratio of their masses is $M/m=2000$, and the charges are the same. Find the ratio of the velocities of protons and positrons after decomposition (at infinity). (5)

The main thing in this task is to understand how the particles will disperse.

5. Light from source $S$ passes through a glass cube with edge $l$ (Fig. 12) on its way to the screen. How much faster does light reach the screen if the cube is moved at a speed of $v$? The speed of light is $c$, the refractive index of glass is $n$. (97)

See substitution for task 1 for 9th grade.

74 tenth-class Olympiad participants did not solve a single problem, 69 solved only one, 52 - two, 25 - third, 13 - quadruple. Only one Olympiad participant solved all the problems.

Experimental tour assignment

When performing experimental work, it was important not only to do it correctly, but also to do it simply, witty and original. It was necessary to consider whether it was possible to verify the required measurements using several methods.

8th grade

1. Determine the efficiency of an inclined plane using a stopwatch, a ruler and a block.

2. Find the ratio of the static and sliding friction coefficients, given a bar and two round wooden sticks.

3. There are two balls suspended on threads of equal length to one point, a piece of plasticine and a protractor. The mass of one of the balls is known. Determine the mass of the second.

4. There are two black boxes. One of them contains a light bulb, and the other contains a resistor. Find where what is, using a battery, an ammeter and a voltmeter.

9th grade

1. See task 2 for grade 8.

2. Determine the specific gravity of the liquid using a tube, vessels with liquid and a ruler.

3. Determine the dependence of the period of oscillation of the physical matrix (stick with answers) on the length. There are two identical sticks with responders and a shatter with a pin on which these sticks can be put on.

4. Determine the heat of melting of ice using a calorimeter, thermometer, ice, and beakers.

10th grade

1. Determine e. d.s. and internal resistance of the source, having a source with known e. d.s., micropermeter, two identical capacitors, known resistance.

2. Determine the refractive index of a plane-parallel glass plate.

3. Determine the moisture content of snow using a calorimeter, thermometer, beaker and water.

4. See task 3 for grade 9.

Try to do this work yourself.