# Kinematics

Irodov I.E. - Problems in General Physics (1981)

## PHYSICAL FUNDAMENTALS OF MECHANICS | Kinematics

### Problem 1.1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?