**G. 593.** The driver of a car travelling at a speed of 18 km/h catches the sight of a van, 250 m ahead of the car, travelling at a constant speed of 54 km/h. Then the car starts to accelerate at a constant rate, and it reaches the van in 20 seconds.

\(\displaystyle a)\) What is the acceleration of the car?

\(\displaystyle b)\) By what distance does the car overtake the van during the next 20 seconds, if it stops accelerating at the moment when it reaches the van?

(3 points)

This problem is for grade 1 - 9 students only.

**G. 595.** The mass of the load hanging on a wire rope, attached to the crab of a bridge crane (hoist trolley) is 500 kg. The trolley is moving horizontally at an acceleration of \(\displaystyle 0.1~{\rm m/s}^2\).

What is the (constant) angle enclosed by the wire rope and the vertical?

(3 points)

This problem is for grade 1 - 9 students only.

**P. 4910.** We would like to go from point \(\displaystyle B\) to point \(\displaystyle A\), which points are both in a forest. In any direction among the trees we can walk at a speed of \(\displaystyle u\). There is however exactly one straight road through the forest along which it is easy to go, at a speed of \(\displaystyle ku\), \(\displaystyle k>1\). Point \(\displaystyle A\) is on this road, but point \(\displaystyle B\) is not, and the angle between the road and the line segment \(\displaystyle AB\) is \(\displaystyle \alpha\). How should we walk in order to reach point \(\displaystyle A\) from \(\displaystyle B\) in the shortest time?

(5 points)

**P. 4913.** A small ball of mass \(\displaystyle m=0.3~\)g and of charge \(\displaystyle Q=2\cdot 10^{-7}\) C is attached to a thin, negligible-mass thread of length \(\displaystyle \ell=20\) cm. The thread with the ball is displaced horizontally, and released without initial speed.

\(\displaystyle a)\) What is the greatest speed of the ball if the whole pendulum is in a uniform horizontal electric field of \(\displaystyle E=10^4\) N/C?

\(\displaystyle b)\) What is the angle between the thread and the vertical when the pendulum is in its other extreme position?

(Air drag is negligible.)

(4 points)

**P. 4914.** In a series \(\displaystyle RL\) circuit \(\displaystyle (a)\) the phase difference between the voltage and the current is \(\displaystyle 45^\circ\). This phase difference is \(\displaystyle 65^\circ\) and \(\displaystyle 70^\circ\), when another inductor of inductance \(\displaystyle L\) (same as the inductance of the coil in the circuit) is connected in series into the circuit, once next to the resistor as shown in figure \(\displaystyle (b)\) and then next to the original inductor as shown in figure \(\displaystyle (c)\), respectively. How can this happen?

What is the ratio of the impedances in the three cases?

(5 points)

**P. 4915.** One of the research space ships of the titanium-devouring little green people found a spherical-shaped small asteroid, which has no atmosphere and does not rotate. The scientists bored a tunnel through the planet along one of its diameter, and found that the whole planet consists of titanium of uniform density.

They celebrated the opening of the tunnel with fireworks. By means of a cannon a projectile was shot exactly vertically downward through the tunnel such that the projectile emerged at the other end of the tunnel, and rose to a height which was same as the diameter of the asteroid measured from the surface of the asteroid, and there it exploded spectacularly. The experts of the Examining Institute for Cosmic Accidents (EXINCA) timed the explosion such that it occurred exactly at a time of \(\displaystyle T\) elapsed after shooting the projectile.

Find (both the formula and the numerical value of) this elapsed time of \(\displaystyle T\).

(6 points)