Mechanical movement. Material point

Basic concepts of kinematics and kinematic characteristics

Human movement is mechanical, that is, it is a change in the body or its parts relative to other bodies. Relative movement is described by kinematics.

Kinematics – a branch of mechanics in which mechanical motion is studied, but the causes of this motion are not considered. The description of the movement of both the human body (its parts) in various sports and various sports equipment is an integral part of sports biomechanics and, in particular, kinematics.

Whatever material object or phenomenon we consider, it turns out that nothing exists outside of space and outside of time. Any object has spatial dimensions and shape, and is located in some place in space in relation to another object. Any process in which material objects participate has a beginning and an end in time, how long it lasts in time, and can occur earlier or later than another process. This is precisely why there is a need to measure spatial and temporal extent.

Basic units of measurement of kinematic characteristics in the international measurement system SI.

Space. One forty-millionth of the length of the earth's meridian passing through Paris was called a meter. Therefore, length is measured in meters (m) and its multiple units: kilometers (km), centimeters (cm), etc.

Time– one of the fundamental concepts. We can say that this is what separates two successive events. One way to measure time is to use any regularly repeated process. One eighty-six thousandth of an earthly day was chosen as a unit of time and was called the second (s) and its multiple units (minutes, hours, etc.).

In sports, special time characteristics are used:

moment in time(t)- this is a temporary measure of the position of a material point, links of a body or system of bodies. Moments of time indicate the beginning and end of a movement or any part or phase of it.

Movement duration(∆t) – this is its temporary measure, which is measured by the difference between the moments of the end and the beginning of movement∆t = tcon. – tbeg.

Movement speed(N) – it is a temporal measure of the repetition of movements repeated per unit of time. N = 1/∆t; (1/s) or (cycle/s).

Rhythm of movements – this is a temporary measure of the relationship between parts (phases) of movements. It is determined by the ratio of the duration of the parts of the movement.

The position of a body in space is determined relative to a certain reference system, which includes a reference body (that is, relative to which the movement is considered) and a coordinate system necessary to describe at a qualitative level the position of the body in one or another part of space.

The beginning and direction of measurement are associated with the reference body. For example, in a number of competitions, the origin of coordinates can be chosen as the starting position. Various competitive distances in all cyclic sports are already calculated from it. Thus, in the selected “start-finish” coordinate system, the distance in space that the athlete will move when moving is determined. Any intermediate position of the athlete’s body during movement is characterized by the current coordinate within the selected distance interval.

To accurately determine a sports result, the competition rules stipulate at what point (reference point) the count is taken: at the toe of a skater’s skate, at the protruding point of a sprinter’s chest, or at the back edge of the landing long jumper’s track.

In some cases, to accurately describe the movement of the laws of biomechanics, the concept of a material point is introduced.

Material point – this is a body whose dimensions and internal structure can be neglected under given conditions.

The movement of bodies can be different in nature and intensity. To characterize these differences, a number of terms are introduced in kinematics, presented below.

Trajectory – a line described in space by a moving point of a body. When biomechanical analysis of movements, first of all, the trajectories of movements of characteristic points of a person are considered. As a rule, such points are the joints of the body. Based on the type of movement trajectories, they are divided into rectilinear (straight line) and curvilinear (any line other than a straight line).

Moving – is the vector difference between the final and initial position of the body. Therefore, displacement characterizes the final result of the movement.

Path – this is the length of the trajectory section traversed by a body or a point of the body during a selected period of time.

KINEMATICS OF A POINT

Introduction to Kinematics

Kinematics is a branch of theoretical mechanics that studies the motion of material bodies from a geometric point of view, regardless of the applied forces.

The position of a moving body in space is always determined in relation to any other unchanging body, called reference body. A coordinate system invariably associated with a reference body is called reference system. In Newtonian mechanics, time is considered absolute and not related to moving matter. In accordance with this, it proceeds identically in all reference systems, regardless of their motion. The basic unit of time is the second (s).

If the position of the body relative to the chosen frame of reference does not change over time, then it is said that body relative to a given frame of reference is at rest. If a body changes its position relative to the chosen reference system, then it is said to move relative to this system. A body can be at rest in relation to one reference system, but move (and in completely different ways) in relation to other reference systems. For example, a passenger sitting motionless on the bench of a moving train is at rest relative to the frame of reference associated with the car, but is moving with respect to the frame of reference associated with the Earth. A point lying on the rolling surface of the wheel moves in relation to the reference system associated with the car in a circle, and in relation to the reference system associated with the Earth, in a cycloid; the same point is at rest with respect to the coordinate system associated with the wheel pair.

Thus, the movement or rest of a body can only be considered in relation to any chosen frame of reference. Set the motion of a body relative to some reference system -means to give functional dependencies with the help of which one can determine the position of the body at any time relative to this system. Different points of the same body move differently in relation to the chosen reference system. For example, in relation to the system associated with the Earth, the tread point of the wheel moves along a cycloid, and the center of the wheel moves in a straight line. Therefore, the study of kinematics begins with the kinematics of a point.

§ 2. Methods for specifying the movement of a point

The movement of a point can be specified in three ways:natural, vector and coordinate.

With the natural way The movement assignment is given by a trajectory, i.e., a line along which the point moves (Fig. 2.1). On this trajectory, a certain point is selected, taken as the origin. The positive and negative directions of reference of the arc coordinate, which determines the position of the point on the trajectory, are selected. As the point moves, the distance will change. Therefore, to determine the position of a point at any time, it is enough to specify the arc coordinate as a function of time:

This equality is called equation of motion of a point along a given trajectory .

So, the movement of a point in the case under consideration is determined by a combination of the following data: the trajectory of the point, the position of the origin of the arc coordinate, the positive and negative directions of the reference and the function .

With the vector method of specifying the movement of a point, the position of the point is determined by the magnitude and direction of the radius vector drawn from the fixed center to a given point (Fig. 2.2). When a point moves, its radius vector changes in magnitude and direction. Therefore, to determine the position of a point at any time, it is enough to specify its radius vector as a function of time:

This equality is called vector equation of motion of a point .

With the coordinate method to specify the motion, the position of the point in relation to the selected reference system is determined using a rectangular Cartesian coordinate system (Fig. 2.3). When a point moves, its coordinates change over time. Therefore, to determine the position of a point at any time, it is enough to specify the coordinates , , as a function of time:

These equalities are called equations of motion of a point in rectangular Cartesian coordinates . The motion of a point in a plane is determined by two equations of system (2.3), rectilinear motion by one.

There is a mutual connection between the three described methods of specifying movement, which allows you to move from one method of specifying movement to another. This is easy to verify, for example, when considering the transition from the coordinate method of specifying movement to vector.

Let us assume that the motion of a point is given in the form of equations (2.3). Bearing in mind that

can be written down

And this is an equation of the form (2.2).

Task 2.1. Find the equation of motion and the trajectory of the middle point of the connecting rod, as well as the equation of motion of the slider of the crank-slider mechanism (Fig. 2.4), if ; .

Solution. The position of a point is determined by two coordinates and . From Fig. 2.4 it is clear that

, .

Then from and:

; ; .

Substituting values , and , we obtain the equations of motion of the point:

; .

To find the equation of the trajectory of a point in explicit form, it is necessary to exclude time from the equations of motion. For this purpose, we will carry out the necessary transformations in the equations of motion obtained above:

; .

By squaring and adding the left and right sides of these equations, we obtain the trajectory equation in the form

.

Therefore, the trajectory of the point is an ellipse.

The slider moves in a straight line. The coordinate , which determines the position of the point, can be written in the form

.

Speed ​​and acceleration

Point speed

In the previous article, the movement of a body or point is defined as a change in position in space over time. In order to more fully characterize the qualitative and quantitative aspects of movement, the concepts of speed and acceleration were introduced.

Velocity is a kinematic measure of the movement of a point, characterizing the speed of change of its position in space.
Velocity is a vector quantity, that is, it is characterized not only by its magnitude (scalar component), but also by its direction in space.

As is known from physics, with uniform motion, speed can be determined by the length of the path traveled per unit time: v = s/t = const (it is assumed that the origin of the path and time are the same).
During rectilinear motion, the speed is constant both in magnitude and direction, and its vector coincides with the trajectory.

Unit of speed in the system SI is determined by the length/time ratio, i.e. m/s .

Obviously, with curvilinear movement, the speed of the point will change in direction.
In order to establish the direction of the velocity vector at each moment of time during curvilinear motion, we divide the trajectory into infinitesimal sections of the path, which can be considered (due to their smallness) rectilinear. Then at each section the conditional speed v p such a rectilinear motion will be directed along the chord, and the chord, in turn, with an infinite decrease in the length of the arc ( Δs tends to zero) will coincide with the tangent to this arc.
It follows from this that during curvilinear motion the velocity vector at each moment of time coincides with the tangent to the trajectory (Fig. 1a). Rectilinear motion can be represented as a special case of curvilinear motion along an arc whose radius tends to infinity (trajectory coincides with tangent).

When a point moves unevenly, the magnitude of its velocity changes over time.
Let us imagine a point whose movement is given in a natural way by the equation s = f(t) .

If in a short period of time Δt the point has passed the way Δs , then its average speed is:

vav = Δs/Δt.

Average speed does not give an idea of ​​the true speed at any given moment in time (true speed is also called instantaneous speed). Obviously, the shorter the time period for which the average speed is determined, the closer its value will be to the instantaneous speed.

True (instantaneous) speed is the limit to which the average speed tends as Δt tends to zero:

v = lim v av at t→0 or v = lim (Δs/Δt) = ds/dt.

Thus, the numerical value of the true speed is v = ds/dt .
The true (instantaneous) speed for any movement of a point is equal to the first derivative of the coordinate (i.e., the distance from the origin of the movement) with respect to time.

At Δt tending to zero, Δs also tends to zero, and, as we have already found out, the velocity vector will be directed tangentially (i.e., it coincides with the true velocity vector v ). It follows from this that the limit of the conditional speed vector v p , equal to the limit of the ratio of the point's displacement vector to an infinitesimal period of time, is equal to the vector of the point's true speed.

Fig.1

Let's look at an example. If a disk, without rotating, can slide along an axis fixed in a given reference system (Fig. 1, A), then in a given reference frame it obviously has only one degree of freedom - the position of the disk is uniquely determined, say, by the x coordinate of its center, measured along the axis. But if the disk, in addition, can also rotate (Fig. 1, b), then it acquires one more degree of freedom - to the coordinate x the rotation angle φ of the disk around the axis is added. If the axis with the disk is clamped in a frame that can rotate around a vertical axis (Fig. 1, V), then the number of degrees of freedom becomes equal to three - to x and φ the frame rotation angle is added ϕ .

A free material point in space has three degrees of freedom: for example, Cartesian coordinates x, y And z. The coordinates of a point can also be determined in cylindrical ( r, 𝜑, z) and spherical ( r, 𝜑, 𝜙) reference systems, but the number of parameters that uniquely determine the position of a point in space is always three.

A material point on a plane has two degrees of freedom. If we select a coordinate system in the plane xOy, then the coordinates x And y determine the position of a point on the plane, coordinate z is identically equal to zero.

A free material point on a surface of any kind has two degrees of freedom. For example: the position of a point on the Earth's surface is determined by two parameters: latitude and longitude.

A material point on a curve of any kind has one degree of freedom. The parameter that determines the position of a point on a curve can be, for example, the distance along the curve from the origin.

Consider two material points in space connected by a rigid rod of length l(Fig. 2). The position of each point is determined by three parameters, but a connection is imposed on them.

Fig.2

Equation l 2 =(x 2 -x 1) 2 +(y 2 -y 1) 2 +(z 2 -z 1) 2 is the coupling equation. From this equation, any one coordinate can be expressed in terms of the other five coordinates (five independent parameters). Therefore, these two points have (2∙3-1=5) five degrees of freedom.

Let us consider three material points in space that do not lie on the same straight line, connected by three rigid rods. The number of degrees of freedom of these points is (3∙3-3=6) six.

A free rigid body generally has 6 degrees of freedom. Indeed, the position of a body in space relative to any reference system is determined by specifying three of its points that do not lie on the same straight line, and the distances between points in a rigid body remain unchanged during any of its movements. According to the above, the number of degrees of freedom should be six.

Forward movement

In kinematics, as in statistics, we will consider all rigid bodies as absolutely rigid.

Absolutely solid body is a material body whose geometric shape and dimensions do not change under any mechanical influences from other bodies, and the distance between any two of its points remains constant.

Kinematics of a rigid body, as well as the dynamics of a rigid body, is one of the most difficult sections of the course in theoretical mechanics.

Rigid body kinematics problems fall into two parts:

1) setting the movement and determining the kinematic characteristics of the movement of the body as a whole;

2) determination of the kinematic characteristics of the movement of individual points of the body.

There are five types of rigid body motion:

1) forward movement;

2) rotation around a fixed axis;

3) flat movement;

4) rotation around a fixed point;

5) free movement.

The first two are called the simplest motions of a rigid body.

Let's start by considering the translational motion of a rigid body.

Progressive is the movement of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.

Translational motion should not be confused with rectilinear motion. When a body moves forward, the trajectories of its points can be any curved lines. Let's give examples.

1. The car body on a straight horizontal section of the road moves forward. In this case, the trajectories of its points will be straight lines.

2. Sparnik AB(Fig. 3) when the cranks O 1 A and O 2 B rotate, they also move translationally (any straight line drawn in it remains parallel to its initial direction). The points of the partner move in circles.

Fig.3

The pedals of a bicycle move progressively relative to its frame during movement, the pistons in the cylinders of an internal combustion engine move relative to the cylinders, and the cabins of Ferris wheels in parks (Fig. 4) relative to the Earth.

Fig.4

The properties of translational motion are determined by the following theorem: during translational motion, all points of the body describe identical (overlapping, coinciding) trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.

To prove this, consider a rigid body undergoing translational motion relative to the reference frame Oxyz. Let's take two arbitrary points in the body A And IN, whose positions at the moment of time t are determined by radius vectors and (Fig. 5).

Fig.5

Let's draw a vector connecting these points.

In this case, the length AB constant, like the distance between points of a rigid body, and the direction AB remains unchanged as the body moves forward. So the vector AB remains constant throughout the body's movement ( AB=const). As a result, the trajectory of point B is obtained from the trajectory of point A by parallel displacement of all its points by a constant vector. Therefore, the trajectories of the points A And IN will really be the same (when superimposed, coinciding) curves.

To find the velocities of points A And IN Let us differentiate both sides of the equality with respect to time. We get

But the derivative of a constant vector AB equal to zero. Derivatives of vectors and with respect to time give the velocities of points A And IN. As a result, we find that

those. what are the speeds of the points A And IN bodies at any moment of time are identical both in magnitude and direction. Taking derivatives with respect to time from both sides of the resulting equality:

Therefore, the accelerations of the points A And IN bodies at any moment of time are also identical in magnitude and direction.

Since the points A And IN were chosen arbitrarily, then from the results found it follows that for all points of the body their trajectories, as well as velocities and accelerations at any time, will be the same. Thus, the theorem is proven.

It follows from the theorem that the translational motion of a rigid body is determined by the movement of any one of its points. Consequently, the study of the translational motion of a body comes down to the problem of the kinematics of a point, which we have already considered.

During translational motion, the speed common to all points of the body is called the speed of translational motion of the body, and acceleration is called the acceleration of translational motion of the body. Vectors and can be depicted as applied at any point of the body.

Note that the concept of speed and acceleration of a body makes sense only in translational motion. In all other cases, the points of the body, as we will see, move with different speeds and accelerations, and the terms<<скорость тела>> or<<ускорение тела>> these movements lose their meaning.

Fig.6

During the time ∆t, the body, moving from point A to point B, makes a displacement equal to the chord AB and covers a path equal to the length of the arc l.

The radius vector rotates through an angle ∆φ. The angle is expressed in radians.

The speed of movement of a body along a trajectory (circle) is directed tangent to the trajectory. It is called linear speed. The modulus of linear velocity is equal to the ratio of the length of the circular arc l to the time interval ∆t during which this arc is passed:

A scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred, is called angular velocity:

The SI unit of angular velocity is radian per second.

With uniform motion in a circle, the angular velocity and the linear velocity module are constant values: ω=const; v=const.

The position of the body can be determined if the modulus of the radius vector and the angle φ it makes with the Ox axis (angular coordinate) are known. If at the initial moment of time t 0 =0 the angular coordinate is equal to φ 0, and at the moment of time t it is equal to φ, then the angle of rotation ∆φ of the radius vector during the time ∆t=t-t 0 is equal to ∆φ=φ-φ 0. Then from the last formula we can obtain the kinematic equation of motion of a material point in a circle:

It allows you to determine the position of the body at any time t.

Considering that , we get:

Formula for the relationship between linear and angular velocity.

The time period T during which the body makes one full revolution is called the period of rotation:

Where N is the number of revolutions made by the body during time Δt.

During the time ∆t=T the body travels the path l=2πR. Hence,

At ∆t→0, the angle is ∆φ→0 and, therefore, β→90°. The perpendicular to the tangent to the circle is the radius. Therefore, it is directed radially towards the center and is therefore called centripetal acceleration:

Module , direction changes continuously (Fig. 8). Therefore, this movement is not uniformly accelerated.

Fig.8

Fig.9

Then the position of the body at any moment of time is uniquely determined by the angle φ between these half-planes taken with the appropriate sign, which we will call the angle of rotation of the body. We will consider the angle φ positive if it is plotted from the fixed plane in a counterclockwise direction (for an observer looking from the positive end of the Az axis), and negative if it is clockwise. We will always measure the angle φ in radians. To know the position of the body at any moment in time, you need to know the dependence of the angle φ on time t, i.e.

The equation expresses the law of rotational motion of a rigid body around a fixed axis.

During rotational motion of an absolutely rigid body around a fixed axis the angles of rotation of the radius vector of different points of the body are the same.

The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity ω and angular acceleration ε.

If during a period of time ∆t=t 1 -t the body rotates through an angle ∆φ=φ 1 -φ, then the numerically average angular velocity of the body during this period of time will be . In the limit at ∆t→0 we find that

Thus, the numerical value of the angular velocity of a body at a given time is equal to the first derivative of the angle of rotation with respect to time. The sign of ω determines the direction of rotation of the body. It is easy to see that when rotation occurs counterclockwise, ω>0, and when clockwise, then ω<0.

The dimension of angular velocity is 1/T (i.e. 1/time); the unit of measurement is usually rad/s or, which is the same, 1/s (s -1), since the radian is a dimensionless quantity.

The angular velocity of a body can be represented as a vector whose modulus is equal to | | and which is directed along the axis of rotation of the body in the direction from which the rotation can be seen occurring counterclockwise (Fig. 10). Such a vector immediately determines the magnitude of the angular velocity, the axis of rotation, and the direction of rotation around this axis.

Fig.10

The angle of rotation and angular velocity characterize the motion of the entire absolutely rigid body as a whole. The linear velocity of any point of an absolutely rigid body is proportional to the distance of the point from the axis of rotation:

With uniform rotation of an absolutely rigid body, the angles of rotation of the body for any equal periods of time are the same, there are no tangential accelerations at various points of the body, and the normal acceleration of a point of the body depends on its distance to the axis of rotation:

The vector is directed along the radius of the point's trajectory towards the axis of rotation.

Angular acceleration characterizes the change in the angular velocity of a body over time. If over a period of time ∆t=t 1 -t the angular velocity of a body changes by the amount ∆ω=ω 1 -ω, then the numerical value of the average angular acceleration of the body over this period of time will be . In the limit at ∆t→0 we find,

Thus, the numerical value of the angular acceleration of a body at a given time is equal to the first derivative of the angular velocity or the second derivative of the angle of rotation of the body with respect to time.

The dimension of angular acceleration is 1/T 2 (1/time 2); the unit of measurement is usually rad/s 2 or, what is the same, 1/s 2 (s-2).

If the module of angular velocity increases with time, the rotation of the body is called accelerated, and if it decreases, it is called slow. It is easy to see that the rotation will be accelerated when the quantities ω and ε have the same signs, and slowed down when they are different.

The angular acceleration of a body (by analogy with angular velocity) can also be represented as a vector ε directed along the axis of rotation. At the same time

The direction of ε coincides with the direction of ω when the body rotates at an accelerated rate (Fig. 10, a), and is opposite to ω when the body rotates at a slow speed (Fig. 10, b).

Fig.11 Fig. 12

2. Acceleration of body points. To find the acceleration of a point M let's use the formulas

In our case ρ=h. Substituting the value v into the expressions a τ and a n, we get:

or finally:

The tangential component of acceleration a τ is directed tangentially to the trajectory (in the direction of motion during accelerated rotation of the body and in the opposite direction during slow rotation); the normal component a n is always directed along the radius MS to the axis of rotation (Fig. 12). Total point acceleration M will

The deviation of the total acceleration vector from the radius of the circle described by the point is determined by the angle μ, which is calculated by the formula

Substituting the values ​​of a τ and a n here, we get

Since ω and ε have the same value for all points of the body at a given moment in time, the accelerations of all points of a rotating rigid body are proportional to their distances from the axis of rotation and form at a given moment in time the same angle μ with the radii of the circles they describe . The acceleration field of points of a rotating rigid body has the form shown in Fig. 14.

Fig.13 Fig.14

3. Vectors of velocity and acceleration of body points. To find expressions directly for vectors v and a, let’s draw from an arbitrary point ABOUT axes AB radius vector of a point M(Fig. 13). Then h=r∙sinα and by the formula

So I can

Mechanical movement is considered for material point and For solid body.

Motion of a material point

Forward movement of an absolutely rigid body is a mechanical movement during which any straight line segment associated with this body is always parallel to itself at any moment in time.

If you mentally connect any two points of a rigid body with a straight line, then the resulting segment will always be parallel to itself in the process of translational motion.

During translational motion, all points of the body move equally. That is, they travel the same distance in the same amount of time and move in the same direction.

Examples of translational motion: the movement of an elevator car, mechanical scales, a sled rushing down a mountain, bicycle pedals, a train platform, engine pistons relative to the cylinders.

Rotational movement

During rotational motion, all points of the physical body move in circles. All these circles lie in planes parallel to each other. And the centers of rotation of all points are located on one fixed straight line, which is called axis of rotation. Circles that are described by points lie in parallel planes. And these planes are perpendicular to the axis of rotation.

Rotational movement is very common. Thus, the movement of points on the rim of a wheel is an example of rotational movement. Rotational motion is described by a fan propeller, etc.

Rotational motion is characterized by the following physical quantities: angular velocity of rotation, period of rotation, frequency of rotation, linear speed of a point.

Angular velocity A body rotating uniformly is called a value equal to the ratio of the angle of rotation to the period of time during which this rotation occurred.

The time it takes a body to complete one full revolution is called rotation period (T).

The number of revolutions a body makes per unit time is called speed (f).

Rotation frequency and period are related to each other by the relation T = 1/f.

If a point is located at a distance R from the center of rotation, then its linear speed is determined by the formula:

Section 1 MECHANICS

Chapter 1: BASIC KINEMATICS

Mechanical movement. Trajectory. Path and movement. Speed ​​addition

Mechanical body movement is called the change in its position in space relative to other bodies over time.

Mechanical movement of bodies studies mechanics. The section of mechanics that describes the geometric properties of motion without taking into account the masses of bodies and acting forces is called kinematics .

Mechanical motion is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates of a material point, you must first select a reference body and associate a coordinate system with it.

Body of referencecalled a body relative to which the position of other bodies is determined. The reference body is chosen arbitrarily. It can be anything: Land, building, car, ship, etc.

The coordinate system, the reference body with which it is associated, and the indication of the time reference form frame of reference , relative to which the movement of the body is considered (Fig. 1.1).

A body whose dimensions, shape and structure can be neglected when studying a given mechanical movement is called material point . A material point can be considered a body whose dimensions are much smaller than the distances characteristic of the motion considered in the problem.

Trajectoryit is the line along which the body moves.

Depending on the type of trajectory, movements are divided into rectilinear and curvilinear

Pathis the length of the trajectory ℓ(m) ( fig.1.2)

The vector drawn from the initial position of the particle to its final position is called moving of this particle for a given time.

Unlike a path, displacement is not a scalar, but a vector quantity, since it shows not only how far, but also in what direction the body has moved during a given time.

Motion vector module(that is, the length of the segment that connects the starting and ending points of the movement) can be equal to the distance traveled or less than the distance traveled. But the displacement module can never be greater than the distance traveled. For example, if a car moves from point A to point B along a curved path, then the magnitude of the displacement vector is less than the distance traveled ℓ. The path and the modulus of displacement are equal only in one single case, when the body moves in a straight line.

Speedis a vector quantitative characteristic of body movement

Average speed– this is a physical quantity equal to the ratio of the vector of movement of a point to the period of time

The direction of the average speed vector coincides with the direction of the displacement vector.

Instant speed, that is, the speed at a given moment in time is a vector physical quantity equal to the limit to which the average speed tends as the time interval Δt decreases infinitely.

Basic level

Option 1

A1. The trajectory of a moving material point in a finite time is

line segment

part of the plane

finite set of points

among answers 1,2,3 there is no correct one

A2. The chair was moved first by 6 m, and then by another 8 m. What is the modulus of total displacement?

1) 2 m 2) 6 m 3) 10 m 4) cannot be determined

A3. A swimmer swims against the current of the river. The speed of the river is 0.5 m/s, the speed of the swimmer relative to the water is 1.5 m/s. The speed modulus of the swimmer relative to the shore is equal to

1) 2 m/s 2) 1.5 m/s 3) 1 m/s 4) 0.5 m/s

A4. Moving in a straight line, one body covers a distance of 5 m every second. Another body, moving in a straight line in one direction, covers a distance of 10 m every second. The movements of these bodies

A5. The graph shows the dependence of the coordinate X of a body moving along the OX axis on time. What is the initial coordinate of the body?

3) -1 m 4) - 2 m

A6. What function v(t) describes the dependence of the velocity modulus on time for uniform rectilinear motion? (length is measured in meters, time in seconds)

1) v= 5t2)v= 5/t3)v= 5 4)v= -5

A7. The modulus of the body's velocity has doubled over some time. Which statement would be correct?

body acceleration doubled

acceleration decreased by 2 times

acceleration hasn't changed

body moves with acceleration

A8. The body, moving rectilinearly and uniformly accelerated, increased its speed from 2 to 8 m/s in 6 s. What is the acceleration of the body?

1) 1m/s 2 2) 1.2m/s 2 3) 2.0m/s 2 4) 2.4m/s 2

A9. When a body falls in free fall, its speed (take g = 10 m/s 2)

in the first second it increases by 5 m/s, in the second – by 10 m/s;

in the first second it increases by 10 m/s, in the second – by 20 m/s;

in the first second it increases by 10 m/s, in the second – by 10 m/s;

in the first second it increases by 10 m/s, and in the second – by 0 m/s.

A10. The speed of rotation of the body in a circle increased by 2 times. Centripetal acceleration of a body

1) increased by 2 times 2) increased by 4 times

3) decreased by 2 times 4) decreased by 4 times

Option 2

A1. Two problems are solved:

A. the docking maneuver of two spacecraft is calculated;

b. The period of revolution of spacecraft around the Earth is calculated.

In what case can spaceships be considered as material points?

only in the first case

only in the second case

in both cases

neither in the first nor in the second case

A2. The car drove around Moscow twice along the ring road, which is 109 km long. The distance traveled by the car is

1) 0 km 2) 109 km 3) 218 ​​km 4) 436 km

A3. When they say that the change of day and night on Earth is explained by the rising and setting of the Sun, they mean a reference system associated

1) with the Sun 2) with the Earth

3) with the center of the galaxy 4) with any body

A4. When measuring the characteristics of the rectilinear movements of two material points, the values ​​of the coordinates of the first point and the speed of the second point were recorded at the moments of time indicated in Tables 1 and 2, respectively:

What can be said about the nature of these movements, assuming that he hasn't changed in the time intervals between the moments of measurements?

1) both are uniform

2) the first is uneven, the second is uniform

3) the first is uniform, the second is uneven

4)both are uneven

A5. Using the graph of the distance traveled versus time, determine the speed of the cyclist at time t = 2 s. 1) 2 m/s 2) 3 m/s

3) 6 m/s4) 18 m/s

A6. The figure shows graphs of the distance traveled in one direction versus time for three bodies. Which body was moving with greater speed? 1) 1 2) 2 3) 34) the speeds of all bodies are the same

A7. The speed of a body moving rectilinearly and uniformly accelerated changed when moving from point 1 to point 2 as shown in the figure. What direction does the acceleration vector have in this section?

A8. Using the graph of the velocity modulus versus time presented in the figure, determine the acceleration of a rectilinearly moving body at time t=2s.

1) 2 m/s 2 2) 3 m/s 2 3) 9 m/s 2 4) 27 m/s 2

A9. In a tube from which the air has been evacuated, a pellet, a cork and a bird feather are simultaneously dropped from the same height. Which body will reach the bottom of the tube faster?

1) pellet 2) cork 3) bird feather 4) all three bodies at the same time.

A10. A car on a turn moves along a circular path of radius 50 m with a constant absolute speed of 10 m/s. What is the acceleration of the car?

1) 1 m/s 2 2) 2 m/s 2 3) 5 m/s 2 4) 0 m/s 2

Answers.

Basic level
Option 1

A1. The trajectory of a moving material point in a finite time is

1. 
   line segment
2. 
   part of the plane
3. 
   finite set of points
4. 
   among answers 1,2,3 there is no correct one

A3. A swimmer swims against the current of the river. The speed of the river is 0.5 m/s, the speed of the swimmer relative to the water is 1.5 m/s. The speed modulus of the swimmer relative to the shore is equal to

1) 2 m/s 2) 1.5 m/s 3) 1 m/s 4) 0.5 m/s

A4. Moving in a straight line, one body covers a distance of 5 m every second. Another body, moving in a straight line in one direction, covers a distance of 10 m every second. The movements of these bodies

A5. The graph shows the dependence of the X coordinate of a body moving along the OX axis on time. What is the initial coordinate of the body?

3) -1 m 4) - 2 m

A6. What function v(t) describes the dependence of the velocity modulus on time for uniform rectilinear motion? (length is measured in meters, time in seconds)

1) v = 5t 2) v = 5/t 3) v = 5 4) v = -5

A7. The modulus of the body's velocity has doubled over some time. Which statement would be correct?

1. 
   body acceleration doubled
2. 
   acceleration decreased by 2 times
3. 
   acceleration hasn't changed
4. 
   body moves with acceleration

1) 1m/s 2 2) 1.2m/s 2 3) 2.0m/s 2 4) 2.4m/s 2

A9. When a body is in free fall, its speed (take g=10m/s 2)

1. 
   in the first second it increases by 5 m/s, in the second – by 10 m/s;
2. 
   in the first second it increases by 10 m/s, in the second – by 20 m/s;
3. 
   in the first second it increases by 10 m/s, in the second – by 10 m/s;
4. 
   in the first second it increases by 10 m/s, and in the second – by 0 m/s.

1) increased by 2 times 2) increased by 4 times

3) decreased by 2 times 4) decreased by 4 times
Option 2

A1. Two problems are solved:

A. the docking maneuver of two spacecraft is calculated;

b. the orbital period of spacecraft is calculated
around the Earth.

In what case can spaceships be considered as material points?

1. 
   only in the first case
2. 
   only in the second case
3. 
   in both cases
4. 
   neither in the first nor in the second case

1) 0 km 2) 109 km 3) 218 ​​km 4) 436 km

A3. When they say that the change of day and night on Earth is explained by the rising and setting of the Sun, they mean a reference system associated

1) with the Sun 2) with the Earth

3) with the center of the galaxy 4) with any body

A4. When measuring the characteristics of the rectilinear movements of two material points, the values ​​of the coordinates of the first point and the speed of the second point were recorded at the moments of time indicated in Tables 1 and 2, respectively:

What can be said about the nature of these movements, assuming that he hasn't changed in the time intervals between the moments of measurements?

1) both are uniform

2) the first is uneven, the second is uniform

3) the first is uniform, the second is uneven

4)both are uneven

A5. Using the graph of the distance traveled versus time, determine the speed
cyclist at time t = 2 s.
1) 2 m/s 2) 3 m/s

3) 6 m/s 4) 18 m/s

A6. The figure shows graphs of the distance traveled in one direction as a function of time for three bodies. Which body was moving with greater speed?
1) 1 2) 2 3) 3 4) the speeds of all bodies are the same
A7. The speed of a body moving rectilinearly and uniformly accelerated changed when moving from point 1 to point 2 as shown in the figure. What direction does the acceleration vector have in this section?

A8. Using the graph of the velocity modulus versus time shown in the figure, determine the acceleration of a rectilinearly moving body at the time t=2s.

1) 2 m/s 2 2) 3 m/s 2 3) 9 m/s 2 4) 27 m/s 2
A9. In a tube from which the air has been evacuated, a pellet, a cork and a bird feather are simultaneously dropped from the same height. Which body will reach the bottom of the tube faster?

1) pellet 2) cork 3) bird feather 4) all three bodies at the same time.

A10. A car on a turn moves along a circular path of radius 50 m with a constant absolute speed of 10 m/s. What is the acceleration of the car?

1) 1 m/s 2 2) 2 m/s 2 3) 5 m/s 2 4) 0 m/s 2
Answers.

Profile level
Option 1

A1. A body thrown vertically upward reached a maximum height of 10 m and fell to the ground. The displacement module is equal to

1) 20m 2) 10m 3) 5m 4) 0m

A2. A body thrown vertically upward reached a maximum height of 5 m and fell to the ground. The distance traveled by the body is

1) 2.5m 2) 10m 3) 5m 4) 0m

A3. Two cars are moving along a straight highway: the first at a speed V, the second at a speed 4 V. What is the speed of the first car relative to the second?

1) 5V 2) 3V 3) -3V 4) -5V

A4. A small object comes off from an airplane flying horizontally at a speed V at point A. What line is the trajectory of this object in the reference frame associated with the airplane, if air resistance is neglected?

A5. Two material points move along the OX axis according to the laws:

x 1 = 5 + 5t, x 2 = 5 - 5t (x - in meters, t - in seconds). What is the distance between them after 2 s?

1) 5m 2) 10m 3) 15m 4) 20m

A6. The dependence of the X coordinate on time during uniformly accelerated motion along the OX axis is given by the expression: X(t)= -5 + 15t 2 (X is measured in meters, time in seconds). The initial velocity module is equal to

A7. Two material points move in circles of radii R, = R and R 2 = 2R with the same speeds. Compare their centripetal accelerations.

1) a 1 = a 2 2)a 1 =2a 2 3)a 1 =a 2 /2 4)a 1 =4a 2
Part 2.

B1. The graph shows the dependence of movement speed on time. What is the average speed during the first five seconds?

B2. A small stone thrown from a flat horizontal surface of the earth at an angle to the horizon reached a maximum height of 4.05 m. How much time passed from the throw to the moment when its speed became directed horizontally?
Part 3.

C1. The coordinates of a moving body change according to the law X=3t+2, Y=-3+7t 2. Find the speed of the body 0.5 s after the start of movement.
Option 2

A1. A ball thrown vertically down from a height of 3 m bounces off the floor vertically and rises to a height of 3 m. The path of the ball is

1) -6m 2) 0m 3) 3m 4) 6m

A2. A stone thrown from a second floor window from a height of 4 m falls to the ground at a distance of 3 m from the wall of the house. What is the modulus of movement of the stone?

1) 3m 2) 4m 3) 5m 4) 7m

A3. A raft floats uniformly down the river at a speed of 6 km/h. A person moves across a raft at a speed of 8 km/h. What is the speed of a person in the reference frame associated with the shore?

1) 2 km/h 2) 7 km/h 3) 10 km/h 4) 14 km/h

A4. The helicopter rises vertically upward evenly. What is the trajectory of a point at the end of a helicopter rotor blade in the reference frame associated with the helicopter body?

3) point 4) helix

A5. A material point moves in a plane uniformly and rectilinearly according to the law: X = 4 + 3t, ​​Y = 3 - 4t, where X,Y are the coordinates of the body, m; t - time, s. What is the speed of the body?
1) 1m/s 2) 3 m/s 3) 5 m/s 4) 7 m/s

A6. The dependence of the X coordinate on time during uniformly accelerated motion along the OX axis is given by the expression: X(t)= -5t+ 15t 2 (X is measured in meters, time in seconds).

The initial velocity module is equal to

1)0m/s 2) 5m/s 3) 7.5m/s 4) 15m/s

A7. The period of uniform motion of a material point along a circle is 2 s. After what minimum time does the direction of velocity change to the opposite?

1) 0.5 s 2) 1 s 3) 1.5 s 4) 2 s
Part 2.

B1. The graph shows the dependence of the speed V of the body on time t, describing the movement of the body along the OX axis. Determine the module of the average speed of movement in 2 seconds.
B2. A small stone was thrown from a flat horizontal surface of the earth at an angle to the horizon. What is the range of the stone if, 2 s after the throw, its speed was directed horizontally and equal to 5 m/s?
Part 3.

C1. A body emerging from a certain point moved with acceleration constant in magnitude and direction. Its speed at the end of the fourth second was 1.2 m/s, at the end of 7 seconds the body stopped. Find the path traveled by the body.
Answers.

Test on the topic “Newton’s Laws. Forces in mechanics."

Basic level
Option 1

A1. Which equality correctly expresses Hooke's law for an elastic spring?

1) F=kx 2) F x =kx 3) F x =-kx 4) F x =k | x |

A2. Which of the following bodies are associated with reference systems that cannot be considered inertial?

A . A skydiver descending at a steady speed.

B. A stone thrown vertically upward.

B. A satellite moving in orbit with a constant absolute velocity.

1) A 2) B 3) C 4) B and C

A3. Weight has a dimension

1) mass 2) acceleration 3) force 4) speed

A4. A body near the Earth's surface is in a state of weightlessness if it moves with an acceleration equal to the acceleration of gravity and directed

1) vertically down 2) vertically up

3) horizontally 4) at an acute angle to the horizontal.

A5. How will the sliding friction force change when the block moves along a horizontal plane if the normal pressure force is doubled?

1) will not change 2) will increase by 2 times

3) will decrease by 2 times 4) will increase by 4 times.

A6. What is the correct relationship between static friction force, sliding friction force and rolling friction force?

1) F tr.p =F tr >F tr.k 2) F tr.p >F tr >F tr.k 3) F tr.p F tr.k 4) F tr.p >F tr =F tr. .To

A7. A paratrooper launches uniformly at a speed of 6 m/s. The force of gravity acting on it is 800N. What is the mass of the skydiver?

1) 0 2) 60 kg 3) 80 kg 4) 140 kg.

A8. What is the measure of interaction between bodies?

1) Acceleration 2) Mass 3) Impulse. 4) Strength.

A9. How are changes in speed and inertia of a body related?

A . If the body is more inert, then the change in speed is greater.

B. If the body is more inert, then the change in speed is less.

B. A body that changes its speed faster is less inert.

G . The more inert body is the one that changes its speed faster.

1) A and B 2) B and D 3) A and D 4) B and C.
Option 2

A1. Which of the following formulas expresses the law of universal gravitation?
1) F=ma 2) F=μN 3) F x =-kx 4) F=Gm 1 m 2 /R 2

A2. When two cars collided, the buffer springs with a stiffness of 10 5 N/m were compressed by 10 cm. What is the maximum elastic force with which the springs acted on the car?

1) 10 4 N 2) 2*10 4 N 3) 10 6 N4) 2*10 6 N

A3. A body of mass 100 g lies on a horizontal stationary surface. Body weight is approximately

1) 0H 2) 1H 3) 100N 4) 1000 N.

A4. What is inertia?

2) the phenomenon of conservation of the speed of a body in the absence of the action of other bodies on it

3) change in speed under the influence of other bodies

4) movement without stopping.

A5. What is the dimension of the friction coefficient?
1) N/kg 2) kg/N 3) no dimension 4) N/s

A7. The student jumped to a certain height and sank to the ground. On what part of the trajectory did he experience the state of weightlessness?

1) when moving up 2) when moving down

3) only at the moment of reaching the top point 4) during the entire flight.

A8. What characteristics determine strength?

A. Module.

B. Direction.

B. Application point.

1) A, B, D 2) B and D 3) B, C, D 4) A, B, C.

A9. Which of the quantities (speed, force, acceleration, displacement) during mechanical motion always coincide in direction?

1) force and acceleration 2) force and speed

3) force and displacement 4) acceleration and displacement.
Answers.

Profile level
Option 1

A1. What forces in mechanics retain their significance during the transition from one inertial system to another?

1) forces of gravity, friction, elasticity.

2) only gravity

3) only friction force

4) only elastic force.

A2. How will the maximum static friction force change if the force of normal pressure of the block on the surface is doubled?

1) Will not change. 2) Will decrease by 2 times.

3) Will increase by 2 times. 4) Will increase 4 times.

A3. A block of mass 200 g slides on ice. Determine the sliding friction force acting on the block if the coefficient of sliding friction of the block on ice is 0.1.

1) 0.2N. 2) 2H. 3) 4H. 4) 20N

A4. How and by how many times should the distance between the bodies be changed so that the gravitational force decreases by 4 times?

1) Increase by 2 times. 2) Reduce by 2 times.

3) Increase by 4 times. 4) Reduce by 4 times

A5. A load of mass m lies on the floor of an elevator starting to move downward with acceleration g.

What is the weight of this load?

1) mg. 2) m (g+a). 3) m (g-a). 4) 0

A6. After the rocket engines are turned off, the spacecraft moves vertically upward, reaches the top of the trajectory and then descends. At what part of the trajectory is the astronaut in a state of weightlessness? Neglect air resistance.

1) Only during upward movement. 2) Only during downward movement.

3) During the entire flight with the engine not running.

4) During the entire flight with the engine running.