KöMaL Problems in Physics, January 2025

Problems with sign 'M' Deadline expired on February 17, 2025.

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M. 437. Ask one of your friends to walk slowly, walk at a normal speed or move at a fast pace. Take a slow-motion video of these movements (e.g., with a mobile phone) and analyse the video to see what percentage of the time both feet were on the ground during the motion.

(6 pont)

statistics

Problems with sign 'G' Deadline expired on February 17, 2025.

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G. 873. A ball is thrown up at speed \(\displaystyle v_0\). When it reaches the top, another ball is thrown up, also at \(\displaystyle v_0\). How does the relative speed of the two balls change over time? When and where do the balls meet?

(3 pont)

solution (in Hungarian), statistics

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G. 874. A truck needs to be loaded with \(\displaystyle M=3~\mathrm{ton}\) of coal. The coal is at a distance of \(\displaystyle L=20~\mathrm{m}\) from the truck. Two conveyor belts of length \(\displaystyle L=20~\mathrm{m}\) can be used for loading. One conveyor belt can transport \(\displaystyle m_1=10~\mathrm{kg}\) of coal in a second at a speed of \(\displaystyle v_1=1~\mathrm{m/s}\), the other can transport \(\displaystyle m_2=15~\mathrm{kg}\) of coal in each second at a speed of \(\displaystyle v_2=0.5~\mathrm{m/s}\). What is the minimum time to load the coal onto the truck?

(4 pont)

solution (in Hungarian), statistics

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G. 875. When the straight ruler is supported 15 cm from one of its end, the ruler can be balanced by two alike coins placed 5 cm from that end. When the support is placed 10 cm from the end, 6 coins are needed to balance the ruler at the former location. How long is the ruler?

(4 pont)

solution (in Hungarian), statistics

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G. 876. The voltmeter shown in the figure reads \(\displaystyle 5~\mathrm{V}\) less after closing switch \(\displaystyle K\) than in the case when the switch is open. What is the voltage \(\displaystyle U\) if \(\displaystyle R=2.4\,\Omega\)?

(3 pont)

solution (in Hungarian), statistics

Problems with sign 'P' Deadline expired on February 17, 2025.

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P. 5616. The fox runs along a straight line at a constant speed \(\displaystyle v_1\). He is chased by the dog, whose speed is constant \(\displaystyle v_2\) and whose direction is always towards the fox. At a given instant, when the distance between the fox and the dog is \(\displaystyle d\), the two velocities are exactly perpendicular to each other. What is the acceleration of the dog at this instant?

(5 pont)

solution (in Hungarian), statistics

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P. 5617. A small body rests at the bottom of a frictionless slope. The slope has an angle of inclination of \(\displaystyle 30^\circ\) and a height of \(\displaystyle 1~\mathrm{m}\). The slope is moved horizontally with an acceleration of \(\displaystyle 7~\mathrm{m/s^2}\). How long does it take for the body to reach the top of the slope?

(4 pont)

solution (in Hungarian), statistics

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P. 5618. A bead with weight \(\displaystyle m\) was strung on a long enough fixed, taut, vertical wire to which a piece of yarn was attached. The yarn was laid over a horizontal rod as shown in the figure, and a small weight of mass \(\displaystyle 2m\) was attached to its vertically hanging end. The rod is at a distance \(\displaystyle d\) from the wire. When the yarn attached to the bead is in the horizontal position, the bead is given an initial upward velocity of \(\displaystyle v_0=\sqrt{2gd}\).

a) Let the distance of the top point of the path of the bead from its starting point be \(\displaystyle f\), and the distance of the bottom point of the path from the starting point be \(\displaystyle \ell\). What is the quotient \(\displaystyle \ell/f\)?

b) What is the tension in the yarn when the bead is at the topmost position and when the bead passes the initial position?

(Friction is negligible everywhere.)

(5 pont)

solution (in Hungarian), statistics

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P. 5619. A space probe is launched vertically upwards from the Earth's surface at the orbital speed.

a) How high does the probe go?

b) How long will it take to fall back to Earth?

Neglect air resistance and the rotation of the Earth.

Hint: see the article titled Mesterséges égitestek mozgásával kapcsolatos problémák és feladatok on the website (only in Hungarian).

(5 pont)

solution (in Hungarian), statistics

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P. 5620. The specific heat capacity at constant volume of a sample of monatomic gas is \(\displaystyle 316~\tfrac{\mathrm{J}}{\mathrm{kg}\,\mathrm{K}}\), that of a sample of a diatomic gas is \(\displaystyle 741~\tfrac{\mathrm{J}}{\mathrm{kg}\,\mathrm{K}}\). What can these two gases be?

(3 pont)

solution (in Hungarian), statistics

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P. 5621. A concave-convex lens has a power of \(\displaystyle -1\) dioptre. What is the radius of curvature of the concave side of the lens if, by holding the lens horizontally with the convex side downwards and pouring water into the concave side, we obtain an optical system with a power of \(\displaystyle +1\) dioptre? What is the radius of curvature of the convex side if the lens is made of glass which has a refractive index of \(\displaystyle 1.6\)?

(4 pont)

solution (in Hungarian), statistics

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P. 5622. Two small insulating disks of mass \(\displaystyle m\) and of charge \(\displaystyle q\) are placed far apart on a horizontal, frictionless, insulating table top. They move initially in opposite directions, travelling at a speed of \(\displaystyle v\). If they had no charge, they would move along the lines at a distance of \(\displaystyle b\), as shown in the figure. What will the minimum velocity of the two discs be during their motion?

(5 pont)

solution (in Hungarian), statistics

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P. 5623. At what constant speed should a spacecraft travel to a distant star if the astronauts are to age the same number of years as the distance to the star in light years?

(4 pont)

solution (in Hungarian), statistics

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P. 5624. Two identical, thin rods are connected at one end by a hinge, around which they are free to turn. Initially the two rods are positioned along a straight line, one rod is stationary, and the free end of the other rod moves at some speed. The rods are free to move in a state of weightlessness, with no external forces acting on them and also friction is negligible. What is the minimum angle between the two rods during their further motions?

(6 pont)

solution (in Hungarian), statistics