A rigid ingot is pressed between two parallel guides moving in horizontal directions at opposite velocities $v_1$ and $v_2$. At a certain instant of time, the points of contact between the ingot and the guides lie on a straight line perpendicular to the directions of velocities $v_1$ and $v_2$.

A block lying on a long horizontal conveyer belt moving at a constant velocity receives a velocity $v_0=5 \mathrm{m} {\mathrm{s}}^{-1}$ relative to the ground in the direction opposite to the direction of motion of the conveyor. After $t=4 \mathrm{s}$, the velocity of the block becomes equal to the velocity of the belt. The coefficient of friction between the block and the belt is $\mu =0.2$. Determine the velocity, $v$ of the conveyer belt.

A loaded sledge moving over ice gets into a region covered with sand and comes to rest before it passes half its length without turning. Then it acquires an initial velocity by a jerk. Determine the ratio of the braking lengths and braking times before the first stop and after the jerk.

A rope is passed round a stationary horizontal log fixed at a certain height above the ground. In order to keep a load of mass $m=6 \mathrm{kg}$ suspended on one end of the rope, the maximum force $T_1=40 \mathrm{N}$ should be applied to the other end of the rope. Determine the minimum force $T_2$ which must be applied to the rope to lift the load.

A certain constant force starts acting on a body moving at a constant velocity $v$. After a time interval $\mathrm{\Delta }t$, the velocity of the body is reduced by half, and after the same time interval, the velocity is again reduced by half. Determine the velocity $v_f$ of the body after a time interval $3\mathrm{\Delta }t$ from the moment, 

A person carrying a spring balance and a stopwatch is in a closed earriage standing on a horizontal segment of the railway. When the carriage starts moving, the person sitting with his face in the direction of motion (along the rails) and fixing a load of mass $m$ to the spring balance watches the direction of the deflection of the load and the readings of the balance, marking the instants of time when the readings change with the help of the stopwatch. When the carriage starts moving and the load is deflected during the first time interval $t_1=4$ stowards the observer, the balance indicates a weight $1.25mg$. During the next time interval $t_2=3 \mathrm{s}$, the load hangs in the vertical position, and the balance indicates a weight $mg$. Then the load is deflected to the left (across the carriage), and during an interval $t_3=25.12 \mathrm{s}$, the balance again indicates a weight $1.25\mathrm{mg}$. Finally, during the last time interval $t_4=4 \mathrm{s}$, the load is deflected from the observer, the reading of the balance remaining the same. Determine the position of the carriage relative to its initial position and its velocity by this instant of time, assuming that the observer suppresses by his hand the oscillations resulting from a change in the direction of deflection and in the readings of the balance.

Two identical weightless rods are hinged to each other and to a horizontal beam (Figure). The rigidity of each rod is $k_0$, and the angle between them is $2\alpha$. Determine the rigidity $k$ of the system of rods relative to the vertical displacement, 

of a hinge $A$ under the action of a certain force $F$, assuming that displacements are small in comparison with the length of the rods.