**P. 4849.** The total mass of an easily moveable cart and a hemisphere shaped container of radius \(\displaystyle R=0.3\) m attached to the cart is \(\displaystyle M=2\) kg. Initially a point-like object of mass \(\displaystyle m=1\) kg is held at the rim of the hemisphere (such that it touched the rim), and then it is released from rest.

\(\displaystyle a)\) What are the velocities of the small object and the cart when the object descended a height of \(\displaystyle h=R/2\)? (Give both directions and speeds.)

\(\displaystyle b)\) What is the radius of the curvature of the path of the motion of the object with respect to the ground, when it reaches its lowest position? (Friction is negligible everywhere.)

(5 points)

**P. 4850.** The mass of a small ball is 5 g, and its diameter is 4 cm. The ball is hit upward at a speed of \(\displaystyle 10~\frac{\rm m}{\rm s}\) in the air which has a temperature of \(\displaystyle 20~{}^\circ\)C and a pressure of 1 bar. Calculate with approximations the followings:

\(\displaystyle a)\) the height to which the ball ascends;

\(\displaystyle b)\) the time during which the ball arrives back to its initial position;

\(\displaystyle c)\) the speed of the ball when it gets back to its initial position.

(6 points)

**P. 4851.** In aerodynamics the *Mach number* is the ratio of the speed of the object moving in air and the local speed of sound.

During a fighter aircraft training one of the fighters ascends from a region of \(\displaystyle 0~{}^\circ\)C air at a height of some kilometres, while its speed increases from 1.5 Mach to 1.75 Mach. Due to the jet powered engine the fighter's kinetic energy increased by 20%, and its mass decreased by 1%.

\(\displaystyle a)\) What is the temperature of the air (in \(\displaystyle {}^\circ\)C) at the height to which the fighter moved?

\(\displaystyle b)\) How much did the fighter ascend, and what is its final speed if the temperature of the air decreases \(\displaystyle 0.65~{}^\circ\) in each 100 m increase in height?

(5 points)

**P. 4852.** There is a frictionlessly moveable piston in a thermally insulated vertical cylinder closed at both of its ends and of cross section \(\displaystyle A=1~{\rm dm}^2\). Two springs (which are both designed for compression and tension) are attached to the piston and to the two bases of the cylinder. The unstretched lengths of the springs are \(\displaystyle \ell_1=3~{\rm
dm}\) and \(\displaystyle \ell_2=5~{\rm dm}\), whilst their spring constants are \(\displaystyle D_1=1000\) N/m and \(\displaystyle D_2=1500\) N/m. Bellow the piston there is air, the pressure of which initially is \(\displaystyle p_1=4\cdot10^4\) Pa, whilst above the piston there is vacuum. Initially the distances between the piston and the bases of the cylinder are \(\displaystyle d_1=5~{\rm dm}\) and \(\displaystyle d_2=4~{\rm dm}\).

\(\displaystyle a)\) Determine the mass of the piston.

\(\displaystyle b)\) The air inside cylinder is slowly heated. By what factor should the absolute temperature of the air be raised in order that the piston moves up 10 cm?

\(\displaystyle c)\) How much heat is added to the gas during the heating process?

(5 points)