To perform calculations of velocities and accelerations, it is necessary to move from writing equations in vector form to writing equations in algebraic form.

Initial velocity and acceleration vectors may have different directions, so the transition from a vector representation of equations to an algebraic one can be very labor-intensive.

It is known that the projection of the sum of two vectors onto any coordinate axis is equal to the sum of the projections of the summands of the vectors onto the same axis.

	Therefore, to find the projection velocity vector on an arbitrary axis OX you need to find the algebraic sum of projections of vectors And on the same axis. The projection of a vector onto an axis is considered positive if it is necessary to go from the projection of the beginning to the projection of the end of the vector in the direction of the axis, and negative in the opposite case.

Speed ​​graph

From Eq.
it follows that the graph of the projection of the speed of uniformly accelerated motion versus time is a straight line. If the projection of the initial velocity onto the OX axis is zero, then the straight line passes through the origin.

Main types of movement

A n = 0, a  = 0 – rectilinear uniform motion;

A n = 0, a  = const– rectilinear uniform motion;

A n = 0, a   0 – rectilinear with variable acceleration;

A n = const, a  = 0 – uniform around the circumference

A n = const, a  = const– uniformly variable around the circumference

A n  const, a   const– curvilinear with variable acceleration.

Rotational motion of a rigid body.

Rotational motion of a rigid body relative to a fixed axis - a movement in which all points of a rigid body describe circles whose centers lie on the same straight line, called axis of rotation.

Uniform movement around a circle

Let's consider the simplest type of rotational motion, and pay special attention to centripetal acceleration.

With uniform motion in a circle, the speed value remains constant, and the direction of the speed vector changes during movement.

Over time interval ∆ t the body goes through the journey . This path is equal to the arc length AB. And Velocity vectors at points A And B  are directed tangent to the circle at these points, and the angle And between vectors equal to the angle between the radii A O.A. O.B. Let's find the vector difference ∆ t:

and determine the ratio of the change in speed to

From the similarity of triangles OAB and BCD it follows
If the time interval ∆t is small, then the angle  is also small. At small values ​​of the angle , the length of the chord AB is approximately equal to the length of the arc AB, i.e.
,
, then we get

.

Because the
, then we get

Period and frequency

The period of time during which a body makes a complete revolution when moving in a circle is called circulation periods (T). Because circumference is equal to 2  R, the period of revolution for uniform motion of a body with speed v in a circle of radius R equals:

The reciprocal of the period of revolution is called frequency. Frequency shows how many revolutions a body makes in a circle per unit time:

(s -1)

To perform calculations of velocities and accelerations, it is necessary to move from writing equations in vector form to writing equations in algebraic form.

The initial velocity and acceleration vectors can have different directions, so the transition from vector to algebraic writing of equations can be very labor-intensive.

It is known that the projection of the sum of two vectors onto any coordinate axis is equal to the sum of the projections of the summands of the vectors onto the same axis.

Speed ​​graph

From Eq. it follows that the graph of the projection of the speed of uniformly accelerated motion versus time is a straight line. If the projection of the initial velocity onto the OX axis is zero, then the straight line passes through the origin.

Main types of movement

1. a n = 0, a t = 0– rectilinear uniform motion;

2. a n = 0, a t = const– rectilinear uniform motion;

3. a n = 0, a t ¹ 0 – rectilinear with variable acceleration;

4. a n = const, a t = 0 – uniform around the circumference

5. a n = const, a t = const– uniformly variable around the circumference

6. a n ¹ const, a t ¹ const– curvilinear with variable acceleration.

Rotational motion of a rigid body.

Rotational motion of a rigid body relative to a fixed axis - a movement in which all points of a rigid body describe circles whose centers lie on the same straight line, called axis of rotation.

Uniform movement around a circle

Let's consider the simplest type of rotational motion, and pay special attention to centripetal acceleration.

With uniform motion in a circle, the speed value remains constant, and the direction of the speed vector changes during the movement.

From the similarity of triangles OAB and BCD it follows

If the time interval ∆t is small, then the angle a is small. For small values ​​of the angle a, the length of the chord AB is approximately equal to the length of the arc AB, i.e. . Because , then we get

Since , we get

Period and frequency

The period of time during which a body makes a complete revolution when moving in a circle is called circulation periods (T). Because circumference is equal to 2pR, the period of revolution for uniform motion of a body with speed v in a circle of radius R equals:

The reciprocal of the period of revolution is called frequency. Frequency shows how many revolutions a body makes in a circle per unit time:

(s -1)

Kinematics of rotational motion

To indicate the direction of rotation, small angles of rotation are assigned a direction: directed along the axis of rotation so that the rotation viewed from its end occurs counterclockwise (right screw rule). If the body did N turns: . Average angular speed:

Instantaneous angular velocity:

(12)

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Rectilinear uniform motion - this is a movement in which, in equal periods of time, the body travels the same distance.

Uniform movement- this is the movement of a body in which its speed remains constant (), that is, it moves at the same speed all the time, and acceleration or deceleration does not occur ().

Straight-line movement- this is the movement of a body in a straight line, that is, the trajectory we get is straight.

The speed of uniform rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the velocity vector coincides with the displacement vector. With all this, the average speed in any period of time is equal to the initial and instantaneous speed:

Speed ​​of uniform rectilinear motion is a physical vector quantity equal to the ratio of the movement of a body over any period of time to the value of this interval t:

From this formula. we can easily express body movement with uniform motion:

Let's consider the dependence of speed and displacement on time

Since our body moves rectilinearly and uniformly accelerated (), the graph with the dependence of speed on time will look like a parallel straight line to the time axis.

Depending projections of body velocity versus time there is nothing complicated. The projection of the body's movement is numerically equal to the area of ​​the rectangle AOBC, since the magnitude of the movement vector is equal to the product of the velocity vector and the time during which the movement was made.

On the graph we see dependence of movement on time.

The graph shows that the projection of the velocity is equal to:

Size: px

Start showing from the page:

Transcript

1 Training option in Physics 2013, Option 03 A1 The dependence of the x coordinate of a body on time t has the form:. What is the projection of the body's velocity onto the Ox axis at the moment of motion? with this 1) 2) 3) 4) A2 What is the direction of the acceleration vector of the ball on the string at the moment of passing its equilibrium position during its free oscillations like a pendulum? 1) vertically upward 2) vertically downward 3) in the direction of the velocity vector 4) against the direction of the velocity vector A3 The figure shows four force vectors. With the elimination of which of the four vectors is the resultant of the remaining three vectors equal to zero? 1) 2) 3) 4)

2 A4 The system consists of two bodies a and b. In the figure, arrows on a given scale indicate the momenta of these bodies. What is the absolute value of the momentum of the entire system? 1) 2) 3) 4) A5 A ball of mass m is thrown vertically upward. If we neglect air resistance, then when the ball rises to a height h relative to the initial position, its total mechanical energy 1) will increase by the amount mgh 2) will decrease by the amount mgh 3) will not change 4) will be unknown, since the initial speed is not specified A6 In the figure a graph of the potential energy of a mathematical pendulum (relative to its equilibrium position) versus time is presented. At the moment of time 1) 0 J 2) 8 J the kinetic energy of the pendulum is equal to:

3 3) 16 J 4) 32 J A7 When the gas temperature in a sealed container increases, the gas pressure increases. This change in pressure is explained by the fact that 1) the volume of the vessel increases due to heating of its walls 2) the energy of thermal motion of gas molecules increases 3) the size of gas molecules increases when it is heated 4) the energy of interaction of gas molecules with each other increases A8 What relationship is true for pressure in vessels with hydrogen and oxygen, if the gas concentrations and root-mean-square velocities are the same? 1) 2) 3) 4) A9 Bodies A and B have different temperatures, higher than that of body B. Bodies A and B were brought into thermal contact with each other and waited until thermal equilibrium was established. If you then bring body A into thermal contact with body B, then body B 1) will receive heat 2) will give off heat 3) can both receive and give off heat 4) will immediately be in a state of thermal equilibrium with body A A10 First the body was given the amount of heat 1, and the second - 2. The masses of the bodies are the same. As a result, the temperature of the first body increased by 100, and the second - increased by 50. It can be argued that the specific heat capacity of the substance of the first body is 1) greater than that of the substance of the second body 2) less than that of the substance of the second body 3) the same as for the substance of the second body 4) cannot be correlated with the specific heat capacity of the substance of the second body A11 The distance between two point electric charges was reduced by 3 times, and one of the charges was increased by 3 times. The forces of interaction between them 1) did not change 2) decreased by 3 times 3) increased by 3 times 4) increased by 27 times

4 A12 How will the resistance of the circuit section AB shown in the figure change if key K is opened? The resistance of each resistor is 4 ohms. 1) will decrease by 4 Ohms 2) will decrease by 2 Ohms 3) will increase by 2 Ohms 4) will increase by 4 Ohms A13 The figure shows a horizontal conductor through which electric current flows in the direction “away from us”. At point A, the magnetic field induction vector is directed 1) vertically down 2) vertically up 3) left 4) right A14 An electric current flows through the coil, the strength of which depends on time as shown in the graph. At time c, the energy stored in the coil is 40 mJ. What is the inductance of the coil? 1) 40 mg 2) 10 mg 3) 20 mg 4) 160 mg

5 A15 The actual image of an object in a converging lens is at a distance of double focus from the lens. The object is located 1) behind the triple focus 2) at double the focal distance from the lens 3) between the focus and double focus 4) between the focus and the lens A16 For visible light, the angle of refraction of light rays at a certain interface between two media decreases with increasing wavelength of radiation. The path of rays for three colors when white light falls from air onto the interface is shown in the figure. The numbers correspond to the colors 1) 1 blue, 2 green, 3 red 2) 1 blue, 2 red, 3 green 3) 1 red, 2 green, 3 blue 4) 1 red, 2 blue, 3 v green A17 Which of the following values Is photon energy proportional? 1) square of the photon speed 2) photon speed 3) radiation frequency 4) wavelength A18 A 2 mm thick layer of cardboard is placed between the source of radioactive radiation and the detector. What radiation can pass through it? 1) only 2) and 3) and 4) and A19 Can the nucleus of an atom of one chemical element spontaneously transform into the nucleus of an atom of another chemical element? 1) any nucleus can 2) no nucleus can 3) only the nuclei of atoms of radioactive isotopes can 4) only the nuclei of atoms behind uranium in D.I. Mendeleev’s table can

6 A20 There is a set of weights of 20g, 40g, 60g and 80g and a spring attached to a support in a vertical position. The weights are carefully suspended one by one from the spring (see Figure 1). The dependence of the elongation of the spring on the mass of the load attached to the spring is shown in Figure 2. What mass of load, being attached to this spring, can perform small oscillations along the axis with an angular frequency? 1) 20 g 2) 40 g 3) 50 g 4) 80 g A21 In the circuit shown in the figure, key K is closed at the moment of time. The ammeter readings at successive times are shown in the table. t, ms I, ma Determine the emf of the source if the resistance of the resistor. Neglect the resistance of the wires and ammeter, the active resistance of the inductor and the internal resistance of the source. 1) 1.5 V 2) 3 V 3) 6 V 4) 7 V

7 A22 A massive block moves translationally along a horizontal plane under the action of a constant force directed at an angle to the horizontal. Modulus of this force. The coefficient of friction between the block and the plane. The modulus of the friction force acting on the block is 2.8 N. What is the mass of the block? 1) 1.4 kg 2) 2 kg 3) 2.4 kg 4) 2.6 kg A23 A piece of ice at a temperature of 0 C is placed in a calorimeter with an electric heater. To turn this ice into water at a temperature of 10 C, an amount of heat of 200 kJ is required. What temperature will be established inside the calorimeter if the ice receives an amount of heat of 120 kJ from the heater? Neglect the heat capacity of the calorimeter and heat exchange with the external environment. 1) 4 C 2) 6 C 3) 2 C 4) 0 C A24 The figure shows a graph of temperature versus pressure for a constant mass of an ideal monatomic gas. The gas performed work equal to 5 kJ. The amount of heat received by the gas is 1) 0 kJ 2) 3 kJ 3) 3.5 kJ 4) 5 kJ A25 An ideal electromagnetic oscillating circuit consists of a capacitor with a capacity of 20 μF and an inductor. At the initial moment of time, the capacitor is charged to a voltage of 4 V, no current flows through the coil. At the moment when the voltage on the capacitor becomes equal to 2 V, the energy of the magnetic field of the coil will be equal to 1) 0.12 mJ 2) 120 J 3) 20 J 4) 40 μJ

8 B1 A mass of mass suspended from a long inextensible thread of length oscillates with a period. The angle of maximum deviation is equal to. What will happen to the period of oscillation, maximum kinetic energy and frequency of oscillation of a thread pendulum if, with a constant maximum angle of deflection of the load, the length of the thread is reduced? For each element of the first column, select an element from the second and enter the selected numbers under the corresponding letters in the answer line. Numbers may be repeated. PHYSICAL QUANTITIES A. Period of oscillation B. Maximum kinetic energy C. Frequency of oscillation CHANGE IN QUANTITY 1) Increases 2) Decreases 3) Does not change A B C??? B2 A block of wood was pushed up a smooth inclined plane and it began to slide without friction. What happens to its speed, potential energy, and reaction force of the inclined plane? For each element of the first column, select the corresponding element from the second and enter the selected numbers under the corresponding letters in the answer line. PHYSICAL QUANTITIES A. Velocity B. Potential energy C. Reaction force of an inclined plane CHANGE IN QUANTITY 1) Increases 2) Decreases 3) Does not change A B C???

9 B3 The figure shows a DC electrical circuit. Designations in the figure: EMF of the current source, R resistance of the resistor. To the key. The internal resistance of the current source and the resistance of the supply conductors can be neglected. Establish a correspondence between physical quantities and formulas by which they can be calculated. For each position in the first column, select the corresponding position in the second and write the selected numbers in the table under the corresponding letters. PHYSICAL QUANTITIES A. Current power in the circuit with the key open B. Current power in the circuit with the key closed FORMULAS 1) 2) 3) 4) A B?? Transfer the resulting sequence of numbers to the answer form (without spaces or any symbols).

10 B4 The oscillating circuit capacitor is connected to a constant voltage source. Graphs A and B represent the dependence on time t of physical quantities characterizing oscillations in the circuit after switch K is moved to position 2 at the moment. Establish a correspondence between the graphs and physical quantities, the dependence of which on time these graphs can represent. For each position in the first column, select the corresponding position in the second and write down the selected numbers in the table under the corresponding letters. Physical quantities: 1) charge of the left plate of the capacitor; 2) current strength in the coil; 3) the energy of the magnetic field of the coil; 4) capacitor capacity. A B??

11 C1 The experiment established that at the air temperature in the room, condensation of water vapor from the air begins on the wall of a glass of cold water if the temperature of the glass is reduced to. Based on the results of these experiments, determine the relative humidity of the air. Use the table to solve the problem. Will the relative humidity change as the air temperature in the room increases if the condensation of water vapor from the air begins at the same temperature of the glass? The pressure and density of saturated water vapor at different temperatures are shown in the table: ,7 8.8 10.0 10.7 11.4 12.11 12.8 13.6 16.3 18.4 20.6 23.0 25, 8 28.7 51.2 130.5 C2 A schoolboy in the summer at his dacha lived near a military airfield, on which military transport planes constantly landed, which always flew along the same trajectory (“glide path”), the projection of which on the ground was a straight line spaced at a distance from the student’s dacha. He armed himself with a stopwatch and a precision angular measuring instrument, carried out repeated measurements of certain times and angles and averaged them for the same types of aircraft. It turned out that when the plane was at a minimum distance from the student, the angle between the horizontal and the direction towards the plane was a, and the sound of its engines was heard at the student’s location after a while. During this time, the plane managed to move away from the point of maximum approach to the student by an angular distance. Based on these data, the student determined the speed of the plane. What was it equal to? C3 The diagram shows changes in pressure and volume of an ideal monatomic gas. How much heat was received or given off by the gas during the transition from state 1 to state 3?

12 C4 Inside an uncharged metal ball of radius r 1 = 40 cm there are two spherical cavities with radii located in such a way that their surfaces almost touch in the center of the ball. An ncl charge was placed in the center of one cavity, and then an ncl charge was placed in the center of the other cavity (see figure). Find the magnitude and direction of the electrostatic field strength vector at a point located at a distance = 1 m from the center of the ball on a perpendicular to the segment connecting the centers of the cavities. C5 In the electrical circuit shown in the figure, the emf of the current source is 12 V, the capacitance of the capacitor is 2 mF, the inductance of the coil is 5 mg, the lamp resistance is 5 ohms and the resistor is 3 ohms. At the initial moment of time, key K is closed. What energy will be released in the lamp after the key is opened? Neglect the internal resistance of the current source and wires. C6 The patient was given an intravenous dose of a solution containing an isotope. The activity of this solution disintegrates per second. The half-life of the isotope is equal to hours. After = 3 hours 50 minutes, the patient’s blood activity began to disintegrate per second. What is the volume of the injected solution if the total volume of the patient’s blood is 1? Neglect the transfer of isotope nuclei from the blood to other tissues of the body.

13 Answers: A1 2 A2 1 A3 1 A4 4 A5 3 A6 1 A7 2 A8 4 A9 1 A10 2 A11 4 A12 3 A13 3 A14 3 A15 2 A16 3 A17 3 A18 4 A19 3 A20 2 A21 3 A22 2 A23 4 A24 4 A25 1 В1 221 В2 213 В3 34 В4 14

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Speed ​​is one of the main characteristics. It expresses the very essence of the movement, i.e. determines the difference that exists between a stationary body and a moving body.

The SI unit of speed is m/s.

It is important to remember that speed is a vector quantity. The direction of the velocity vector is determined by the movement. The velocity vector is always directed tangentially to the trajectory at the point through which the moving body passes (Fig. 1).

For example, consider the wheel of a moving car. The wheel rotates and all points of the wheel move in circles. The splashes flying from the wheel will fly along tangents to these circles, indicating the directions of the velocity vectors of individual points of the wheel.

Thus, speed characterizes the direction of movement of a body (direction of the velocity vector) and the speed of its movement (modulus of the velocity vector).

Negative speed

Can the speed of a body be negative? Yes maybe. If the speed of a body is negative, this means that the body is moving in the direction opposite to the direction of the coordinate axis in the chosen reference system. Figure 2 shows the movement of a bus and a car. The speed of the car is negative and the speed of the bus is positive. It should be remembered that when we talk about the sign of velocity, we mean the projection of the velocity vector onto the coordinate axis.

Uniform and uneven movement

In general, speed depends on time. According to the nature of the dependence of speed on time, movement can be uniform or uneven.

DEFINITION

Uniform movement– this is movement with a constant modulus speed.

In case of uneven movement we speak of:

Examples of solving problems on the topic “Speed”

EXAMPLE 1

Exercise	The car covered the first half of the journey between two settlements at a speed of 90 km/h, and the second half at a speed of 54 km/h. Determine the average speed of the car.
Solution	It would be incorrect to calculate the average speed of a car as the arithmetic mean of the two indicated speeds. Let's use the definition of average speed: Since rectilinear uniform motion is assumed, the signs of the vectors can be omitted. Time spent by the car to cover the entire distance: where is the time spent on completing the first half of the path, and is the time spent on completing the second half of the path. The total movement is equal to the distance between populated areas, i.e. . Substituting these ratios into the formula for average speed, we get: Let's convert the speeds in individual sections to the SI system: Then the average speed of the car is: (m/s)
Answer	The average speed of the car is 18.8 m/s

EXAMPLE 2

Exercise	A car travels for 10 seconds at a speed of 10 m/s and then drives for another 2 minutes at a speed of 25 m/s. Determine the average speed of the car.
Solution	Let's make a drawing.