**P. 4662.** The goal-keeper is just about to kick off the ball. The kick lasts for \(\displaystyle \Delta
t=0.01\) s, during which the goalkeeper exerts an average force of \(\displaystyle F=850\) N on the ball lying on the ground. If there is no air drag, the kicked ball would reach the ground in \(\displaystyle t=3\) s and it would be \(\displaystyle \ell=45\) m further.

\(\displaystyle a)\) What is the greatest height that the ball could reach?

\(\displaystyle b)\) According to the official rules of association football the mass of the ball must be between 410 g and 450 g. Is the match played according to the rules?

(4 points)

**P. 4663.** A rigid rod of mass \(\displaystyle M\) can rotate about a horizontal axle which is attached to the rod at its trisecting point. A point-like object of mass \(\displaystyle m\) is attached to the end of the rod closer to the axle, and a pan of negligible mass is attached to the other end. The rod stays at rest and is horizontal.

What is the initial speed of the object of mass \(\displaystyle m\) if another object of mass \(\displaystyle m\) is dropped into the pan from a height of \(\displaystyle h\)? The dropped object stays in the pan, and friction is negligible.

(5 points)

**P. 4664.** Mr. Tompkins (The chief character of the book titled *Mr. Tompkins in Wonderland* written by George Gamow) in his dreams enters to Wonderland where the laws of physics are nearly the same as in our Universe, ,,only'' gravity differs from the usual Newton's laws. When he woke up in the morning he remembered that there are several planets revolving around the single Sun of Wonderland, and there are three ,,Kepler's laws'' which hold true:

1. The planets revolve along elliptical paths, and the Sun is at the *centre* of these paths.

2. During the motion of the planet the *ray* drawn to the planet ...(unfortunately Mr. Tompkins forgot the end of this law.)

3. The period of each planet is the *same *(independently of the lengths of the major and the minor axes of the ellipses).

What can the law of gravity be in the physics book of Wonderland, and what may the end of Kepler's second law be?

(5 points)

**P. 4669.** Uranium-238 is an alpha-decaying isotope, its half-life is 4.5-billion years. How many atoms are there in that Uranium block in which on average one atom decays in each

\(\displaystyle a)\) second;

\(\displaystyle b)\) hour;

\(\displaystyle c)\) day;

\(\displaystyle d)\) year?

(4 points)