Problem database

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?

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?

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 $c$.

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?

The shooter tries to hit a disk of radius $R$, which moves from one wall to another at constant modulo velocity so fast that it cannot be tracked. Draw a graph of the probability of hitting the disk as a function of the distance between the aiming point and the left wall.

The shots are fired at height $R$ from the floor perpendicular to the direction of motion of the disc. At what point is the least and most probable shot? What is their value? Consider the cases $L > 4R$, $4R > L > 2R$, where $L$ is the distance between the walls.

. The particle, after leaving the source, flies at a constant speed for a distance $L$, and then decelerates with acceleration $a$. At what speed will the particle have the shortest travel time from its departure to its stop?

a. In a conical vessel, the water level rises at a constant rate $v_0$. How does the rate of water entering a vessel through a tube of section $s$ depend on time? At time zero, the vessel is empty.

b. A jet of oil hitting the surface of the water spreads over it in a round spot of thickness $h$. How does the speed of movement of the spot boundary depend on time, if the volume of oil $q$ enters per unit of time? At the initial time, the spot radius is zero.

The body starts moving from point $A$ and moves first equidistant for time $t_0$, then with the same modulo acceleration — equidistant. After what time from the beginning of the movement, the body will return to point $A$?

The scheduled departure time of the train is $12.00$. It's $12.00$ on your watch, but the penultimate car that moves past you during time $t_1$ is already starting to pass by you. The last car passes you during $t_2$. The train left on time and is moving equidistant. How far behind is your watch?

At what angle to the vertical should a smooth chute be directed from point $A$ so that the ball slides down it to the inclined plane in the shortest time?

From a hose lying on the ground, water shoots at an angle of $45^\circ$ to the horizon with an initial velocity of $10$ $\frac{m}{s}$. The cross-sectional area of the hose opening is $5$ $cm^2$ . Determine the mass of the jet in the air.

The projectile flew out of the gun and hit a point with $x$ coordinates horizontally and $y$ coordinates vertically. Initial velocity of the projectile $v$. Find: a) the tangent of the angle formed by the gun barrel with the horizon; b) the boundary of the area of possible hit of the projectile; c) the lowest initial velocity of the projectile at which it can hit the point with coordinates $x$, $y$. Note. For the solution, use the trigonometric identity $\frac{1}{\cos^2 \varphi} = \operatorname{tg}^2 \varphi + 1$.

A ball flies into a tube of length $l$, inclined at an angle $\alpha$ to the horizon, with a horizontal velocity $v$. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic.

In a spherical hole, a ball jumps, elastically hitting its walls at two points located on the same horizontal line. The time interval between strokes when moving the ball from left to right is always equal to $T_1$, and when moving from right to left-$T_2 \ne T_1$. Determine the radius of the hole.

What is the minimum speed required for a stone thrown by a boy to fly between height $H$ and length $L$, if the throw is made from height $h$ and the boy can choose any place to throw?

Planes fly in a straight line towards each other at the same speed $v$. The maximum range of detection of each other by them $l$. One plane, after detecting the other, makes a $U$-turn without changing the speed module, and flies parallel to the second plane. At what constant acceleration will the planes lose sight of each other at the end of the turn?

A spherical tank standing on the ground has a radius of $R$. What is the lowest speed at which a rock thrown from the ground can fly over the reservoir just by touching its top?

The projectile leaves the cannon at a velocity $V$ at an angle $\alpha$ to the horizon. What time does the projectile approach the cannon?

The buoy is a sailing sled. He can only move along the line along which his skates are directed. The wind blows at a speed $v$ perpendicular to the direction of movement of the buoy. The sail is $30^\circ$ with the direction of travel. What speed can not exceed the buoy in this wind?