Problem database

The figure shows a "blurred photo" of a flying jet airplane. The length of the airplane is $30$ m, the length of its nose part is $10$ m. Determine from this "photography" the speed of the airplane. Shutter exposure time is $0.1$ s. The shape of the airplane is shown in the picture with a dashed line.

The radar measures the angle between the North Pole and the plane's direction and the distance to the aircraft. At a certain point in time the position of the plane was determined by the following coordinates: angle $\alpha_1 = 44^{\circ}$, distance $R_1 = 100$ km. In a time interval of $5$ seconds after this moment the coordinates of the aircraft were: angle $\alpha_2 = 46^{\circ}$, distance $R_2 = 100$ km. In a Cartesian coordinate system with the y-axis pointing north and the radar at the origin, determine the position of the plane at both points in time and the modulus and direction of the aircraft's velocity. Read the angle in a clockwise direction.

A bug flew into the room through an open window. The distance from the bug to the ceiling changed with a speed of $1$ m/s, the distance to the wall opposite to the window changed with a speed of $2$ m/s, to the side wall - with a speed of $2$ m/s. After $1$ s of flight the bug hit the corner between the ceiling and the side wall of the room. Determine the speed of the bug's flight and the position in the window through which it flew into the room. The room is $2.5$ m high, $4$ m wide, and $4$ m long.

Counters $A$ and $B$, which register the moment of the arrival of a $\gamma$-quantum, are located at a distance of $2$ m from each other. At a point between them a $\pi_0$ meson decayed into two $\gamma$-quanta. Find the position of this point if counter $A$ detected the $\gamma$-quantum $10^{-9}$ s later than counter $B$. The speed of light is $3 \cdot 10^8$ m/s.

Three microphones located on the same straight line at points $A$, $B$, $C$ recorded successively at the moments $t_A > t_B > t_C$ the sound of an explosion that occurred at point $O$, which lies on the segment $AC$. Find the length of the segment $AO$ if $AB = BC = L$. At what point in time did the explosion occur?

Athletes run in a column of length $l$ at speed $v$. The coach runs towards them with speed $u < v$. Each athlete, when he reaches the coach, turns around and starts running back with the same speed. What is the length of the column when all the athletes turn around?

A submarine diving vertically and uniformly emits sound pulses of duration $\tau_0$. The duration of the pulse reflected from the bottom is $\tau$. The speed of sound in water is $c$. At what speed does the submarine dive?

The conveyor belt has speed $w$. Above the belt moves a machine, throwing $\nu$ balls per unit time. The balls stick to the belt. A ball counter with a photocell counts only the balls that have passed directly under it. How many balls will the counter count in a unit of time if the speed of the machine is $v < w$, the speed of the counter is $u < w$?

a. A rod of length $l$ is made of explosive material. The detonation velocity (the rate of involvement in the explosion of new parts of the explosive) is equal to $v$, and the rate of spreading of the products of the explosion is $u < v$. How does the region occupied by the products of the explosion change with time if the rod is detonated at one end? Make a drawing.

b. From the same explosive material it is necessary to make such a thin-walled conical shell so that when detonating it from the top, the products of the explosion simultaneously hit the rod on the axis of the cone. What angle between the axis of the cone and the generatrix should be chosen?

A bus is driving along a straight highway at constant speed $v$. You have noticed the bus when it was at some point $A$. From what area near the highway can you catch up with this bus if your running speed is $u < v$? Draw this area for $u = v/2$.

A supersonic airplane is flying horizontally. Two microphones on the same vertical at a distance $l$ from each other register the arrival of sound from an airplane flying over the microphones with a time lag $\Delta t$. The speed of sound in air is $c$. What is the speed of the plane?

Two rods intersect at an angle $2 \alpha$ and move with equal velocities $v$ perpendicular to themselves. What is the velocity of the intersection point of the rods? Image 1.1.12

From the graph of the dependence of the coordinate on time, graph the dependence of the velocity on time. Image 1.1.13

Using coordinate-time graphs, find the point in time and place of collision of particles moving along one straight line. The speed of the first particle is $v$, the speed of the second particle is $v/2$. The first particle at time $t = 0$ had coordinate $x = 0$, the second particle at time $t_1$ had coordinate $x = a$.

Using the velocity-time graphs, plot the coordinate-time graphs. In cases b and c find the average velocity over a large time.

A particle moves in one plane. From the graphs of the time dependence of the velocity projections $v_x$ and $v_y$ plot the trajectory of the particle if $x_{(0)} = 2$ m, $y_{(0)} = 1$ m.

The motion of the beam on the oscilloscope screen is described by plots of $x$ and $y$ coordinates versus time. What picture will appear on the screen when $\tau_y = \tau_x$, $\tau_x/3$, $3 \tau_x$? Consider the two cases (see figure 1.1.17). In case a, the horizontal lines are almost invisible on the screen. Why? At what ratio of $\tau_x$ and $\tau_y$ in case b is the trajectory of the beam on the screen closed?

The car moves with speed $v$ away from a long wall, moving at an angle $\alpha$ to it. At the moment when the distance to the wall equals $l$, the driver gives a short beep. How far will the car travel before the chauffeur hears the echo? The speed of sound in the air is $c$.

By what angle will the direction of velocity of the ball change after two elastic impacts on the walls, the angle between which is equal to $\alpha$? How will the ball fly if the angle $\alpha = \pi/2$? The motion occurs in a plane perpendicular to the walls. In an elastic collision with a smooth stationary wall, the angle of incidence of the ball is equal to the angle of reflection.

A ball is launched along a pool table with sides $a$ and $b$ from the middle of side $b$. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?