### Problem 1.1

#### Task:

A motorboat going downstream overcame a raft at a point \(A\); \(\tau = 60\) min later it turned back and after some time passed the raft at a distance \(l = 6.0\) km from the point \(A\). Find the flow velocity assuming the duty of the engine to be constant.

### Problem 1.2

#### Task:

A point traversed half the distance with a velocity \(v_0\). The remaining part of the distance was covered with velocity \(v_1\) for half the time, and with velocity \(v_2\) for the other half of the time. Find the mean velocity of the point averaged over the whole time of motion.

### Problem 1.3

#### Task:

A car starts moving rectilinearly, first with acceleration \( \omega = 5.0 \text{ m/s}^2 \) (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate \( \omega \), comes to a stop. The total time of motion equals \(t = 25\) s. The average velocity during that time is equal to \(\langle v \rangle = 72\) km per hour. How long does the car move uniformly?

### Problem 1.4

#### Task:

A point moves rectilinearly in one direction. Fig.1.1 shows the distance \( s \) traversed by the point as a function of the time \( t \).

Using the plot find:

(a) the average velocity of the point during the time of motion;

(b) the maximum velocity;

(c) the time moment \( t_0 \) at which the instantaneous velocity is equal to the mean velocity averaged over the first \( t_0 \) seconds.

### Problem 1.5

#### Task:

Two particles, \( 1 \) and \( 2 \), move with constant velocities \( \vec{v_1} \) and \( \vec{v_2} \). At the initial moment their radius vectors are equal to \( \vec{r_1} \) and \( \vec{r_2} \). How must these four vectors be interrelated for the particles to collide?

### Problem 1.6

#### Task:

A ship moves along the equator to the east with velocity \( v_0 = 30 \) km/hour. The southeastern wind blows at an angle \( \varphi = 60^{\circ} \) to the equator with velocity \( v = 15 \) km/hour. Find the wind velocity \( v' \) relative to the ship and the angle \( \varphi' \) between the equator and the wind direction in the reference frame fixed to the ship.

### Problem 1.7

#### Task:

Two swimmers leave point \( A \) on one bank of the river to reach point \( B \) lying right across on the other bank. One of them crosses the river along the straight line \( AB \) while the other swims at right angles to the stream and then walks the distance that he has been carried away by the stream to get to point \( B \). What was the velocity \( u \) of his walking if both swimmers reached the destination simultaneously? The stream velocity \( v_0 = 2.0 \) km/hour and the velocity \( v' \) of each swimmer with respect to water equals \( 2.5 \) km per hour.

### Problem 1.8

#### Task:

Two boats, \( A \) and \( B \), move away from a buoy anchored at the middle of a river along the mutually perpendicular straight lines: the boat \( A \) along the river, and the boat \( B \) across the river. Having moved off an equal distance from the buoy the boats returned. Find the ratio of times of motion of boats \( \tau_A / \tau_B \) if the velocity of each boat with respect to water is \( \eta = 1.2 \) times greater than the stream velocity.

### Problem 1.9

#### Task:

A boat moves relative to water with a velocity which is \( n = 2.0 \) times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?

### Problem 1.10

#### Task:

Two bodies were thrown simultaneously from the same point: one, straight up, and the other, at an angle of \( \theta = 60^{\circ} \) to the horizontal. The initial velocity of each body is equal to \( v_0 = 25 \) m/s. Neglecting the air drag, find the distance between the bodies \( t = 1.70 \) s later.

### Problem 1.11

#### Task:

Two particles move in a uniform gravitational field with an acceleration \( g \). At the initial moment the particles were located at one point and moved with velocities \( v_1 = 3.0 \) m/s and \( v_2 = 4.0 \) m/s horizontally in opposite directions. Find the distance between the particles at the moment when their velocity vectors become mutually perpendicular.

### Problem 1.12

#### Task:

Three points are located at the vertices of an equilateral triangle whose side equals \( a \). They all start moving simultaneously with velocity \( v \) constant in modulus, with the first point heading continually for the second, the second for the third, and the third for the first. How soon will the points converge?

### Problem 1.13

#### Task:

Point \( A \) moves uniformly with velocity \( v \) so that the vector \( \vec{v} \) is continually "aimed" at point \( B \) which in its turn moves rectilinearly and uniformly with velocity \( u < v \). At the initial moment of time \( \vec{v} \perp \vec{u} \) and the points are separated by a distance \( l \). How soon will the points converge?

### Problem 1.14

#### Task:

A train of length \( l = 350 \) m starts moving rectilinearly with constant acceleration \( \omega = 3.0 \cdot 10^{-2} \) m/s\( ^2 \); \( t = 30 \) s after the start the locomotive headlight is switched on (event \( 1 \)), and \( \tau = 60 \) s after that event the tail signal light is switched on (event \( 2 \)). Find the distance between these events in the reference frames fixed to the train and to the Earth. How and at what constant velocity \( v \) relative to the Earth must a certain reference frame \( K \) move for the two events to occur in it at the same point?

### Problem 1.15

#### Task:

An elevator car whose floor-to-ceiling distance is equal to \( 2.7 \) m starts ascending with constant acceleration \( 1.2 \) m/s\( ^2 \); \( 2.0 \) s after the start a bolt begins falling from the ceiling of the car. Find:

(a) the bolt's free fall time;

(b) the displacement and the distance covered by the bolt during the free fall in the reference frame fixed to the elevator shaft.

### Problem 1.16

#### Task:

Two particles, \( 1 \) and \( 2 \), move with constant velocities \( v_1 \) and \( v_2 \) along two mutually perpendicular straight lines toward the intersection point \( O \). At the moment \( t = 0 \) the particles were located at the distances \( l_1 \) and \( l_2 \) from the point \( O \). How soon will the distance between the particles become the smallest? What is it equal to?