Curvilinear body movement

Curvilinear body movement definition:

Curvilinear motion is a type of mechanical motion in which the direction of speed changes. The speed module may change.

Uniform body movement

Uniform body movement definition:

If a body travels equal distances in equal periods of time, then such motion is called. With uniform motion, the velocity module is a constant value. Or it can change.

Uneven body movement

Uneven body movement definition:

If a body travels different distances in equal periods of time, then such motion is called uneven. With uneven movement, the velocity module is a variable quantity. The direction of speed may change.

Equally alternating body movement

Equally alternating motion of a body definition:

There is a constant quantity with uniformly alternating motion. If the direction of speed does not change, then we obtain rectilinear uniform motion.

Uniformly accelerated motion of a body

Uniformly accelerated motion of a body definition:

Equally slow body movement

Uniformly slow motion of a body definition:

When we talk about the mechanical movement of a body, we can consider the concept of translational motion of the body.

Mechanical movement of a body (point) is the change in its position in space relative to other bodies over time.

Types of movements:

A) Uniform rectilinear motion of a material point: Initial conditions

. Initial conditions

G) Harmonic oscillatory motion. An important case of mechanical motion is oscillations, in which the parameters of a point’s movement (coordinates, speed, acceleration) are repeated at certain intervals.

ABOUT scriptures of the movement . There are various ways to describe the movement of bodies. With the coordinate method specifying the position of a body in a Cartesian coordinate system, the movement of a material point is determined by three functions expressing the dependence of coordinates on time:

x= x(t), y=y(t) And z= z(t) .

This dependence of coordinates on time is called the law of motion (or equation of motion).

With the vector method the position of a point in space is determined at any time by the radius vector r= r(t) , drawn from the origin to a point.

There is another way to determine the position of a material point in space for a given trajectory of its movement: using a curvilinear coordinate l(t) .

All three methods of describing the motion of a material point are equivalent; the choice of any of them is determined by considerations of the simplicity of the resulting equations of motion and the clarity of the description.

Under reference system understand a reference body, which is conventionally considered motionless, a coordinate system associated with the reference body, and a clock, also associated with the reference body. In kinematics, the reference system is selected in accordance with the specific conditions of the problem of describing the motion of a body.

2. Trajectory of movement. Distance traveled. Kinematic law of motion.

The line along which a certain point of the body moves is called trajectorymovement this point.

The length of the trajectory section traversed by a point during its movement is called the path traveled .

The change in radius vector over time is called kinematic law :
In this case, the coordinates of the points will be coordinates in time: x= x(t), y= y(t) Andz= z(t).

In curvilinear motion, the path is greater than the displacement modulus, since the length of the arc is always greater than the length of the chord contracting it

The vector drawn from the initial position of the moving point to its position at a given time (increment of the radius vector of the point over the considered period of time) is called moving. The resulting displacement is equal to the vector sum of successive displacements.

During rectilinear movement, the displacement vector coincides with the corresponding section of the trajectory, and the displacement module is equal to the distance traveled.

3. Speed. Average speed. Velocity projections.

Speed - speed of change of coordinates. When a body (material point) moves, we are interested not only in its position in the chosen reference system, but also in the law of motion, i.e., the dependence of the radius vector on time. Let the moment in time corresponds to the radius vector a moving point, and a close moment in time - radius vector . Then in a short period of time
the point will make a small displacement equal to

To characterize the movement of a body, the concept is introduced average speed his movements:
This quantity is a vector quantity, coinciding in direction with the vector
. With unlimited reduction Δt the average speed tends to a limiting value called instantaneous speed :

Velocity projections.

A) Uniform linear motion of a material point:
Initial conditions

B) Uniformly accelerated linear motion of a material point:
. Initial conditions

B) Movement of a body along a circular arc with a constant absolute speed:

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 system of measurements 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 of 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 start 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: along the toe of a skater’s skate, at the protruding point of a sprinter’s chest, or along 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.

In order to characterize how quickly the position of a moving body changes in space, the special concept of speed is used.

Speed – This is the ratio of the distance traveled to the time it takes to complete it. It shows how quickly the position of a body in space changes. Since velocity is a vector, it also indicates in which direction the body or point on the body is moving.

Medium speed of a body on a given section of the trajectory is called the ratio of the distance traveled to the time of movement, m/s:

If the average speed is the same in all parts of the trajectory, then the movement is called uniform.

The issue of running speed is important in sports biomechanics. It is known that the speed of running over a certain distance depends on the magnitude of this distance. A runner can only maintain maximum speed for a limited time (3-4 seconds, highly skilled sprinters up to 5 - 6 seconds). The average speed of stayers is much lower than that of sprinters. Below is the dependence of the average speed (V) on the length of the distance (S).

World sports records and the average speed shown in them

Type of competition and distance	Men		Women
		Average speed m/s	Time shown on the course	Average speed m/s
Run
100 m	9.83 s	10,16	10.49 s	9,53
400 m	43.29 s	9,24	47.60 s	8,40
1500 m	3 min 29.46 s	7,16	3 min 52.47 s	6,46
5000 m	12 min 58.39 s	6,42	14 min 37.33 s	5,70
10000 m	27 min 13.81 s	6,12	30 min 13.75 s	5,51
Marathon (42 km 195 m)	2 h 6 min 50 s	5,5	2 hours 21 minutes 0.6 s	5,0
Ice skating
500 m	36.45 s	13,72	39.10 s	12,78
1500 m	1 min 52.06 s	13,39	1 min 59.30 s	12,57
5000 m	6 min 43.59 s	12,38	7 min 14.13 s	11,35
10000 m	13 min 48.20 s	12,07
100 m (freestyle)	48.74 s	2,05	54.79 s	1,83
200 m (v/s)	1 min 47.25 s	1,86	1 min 57.79 s	1,70
400 m (v/s)	3 min 46.95 s	1,76	4 min 3.85 s	1,64

For convenience of calculations, the average speed can also be written through a change in the coordinates of the body. When moving in a straight line, the distance traveled is equal to the difference between the coordinates of the end and start points. So, if at time t0 the body was at a point with coordinate X0, and at time t1 - at a point with coordinate X1, then the distance traveled ∆Х = X1 - X0, and the time of movement ∆t = t1 - t0 (the symbol ∆ denotes difference of values ​​of the same type or to designate very small intervals). In this case:

The dimension of speed in SI is m/s. When covering long distances, speed is determined in km/h. If necessary, such values ​​can be converted to SI. For example, 54 km/h = 54000 m/3600 s = 15 m/s.

Average speeds on different sections of the path differ significantly even with a relatively uniform distance: starting acceleration, covering a distance with intra-cycle speed fluctuations (during take-off the speed increases, during free gliding in skating or the flight phase in speed skating it decreases) , finishing. As the interval over which the speed is calculated decreases, the speed at a given point on the trajectory can be determined, which is called instantaneous speed.

Or the speed at a given point of the trajectory is the limit to which the movement of a body in the vicinity of this point tends in time with an unlimited decrease in the interval:

Instantaneous speed is a vector quantity.

If the magnitude of the velocity (or the magnitude of the velocity vector) does not change, the movement is uniform; when the magnitude of the velocity changes, it is uneven.

Uniform called movement in which a body travels the same paths over any equal intervals of time. In this case, the magnitude of the speed remains unchanged (in the direction the speed can change if the movement is curvilinear).

Straightforward called movement in which the trajectory is a straight line. In this case, the direction of the speed remains unchanged (the magnitude of the speed can change if the movement is not uniform).

Uniform straight called movement that is both uniform and rectilinear. In this case, both magnitude and direction remain unchanged.

In the general case, when a body moves, both the magnitude and direction of the velocity vector change. In order to characterize how quickly these changes occur, a special quantity is used - acceleration.

Acceleration – this is a quantity equal to the ratio of the change in the speed of a body to the duration of the period of time during which this change in speed occurred. The average acceleration based on this definition is, m/s²:

Instant acceleration called physical quantity equal to the limit to which the average acceleration tends over an interval∆t → 0, m/s²:

Since the speed can change both in magnitude and direction along the trajectory, the acceleration vector has two components.

The component of the acceleration vector a, directed along the tangent to the trajectory at a given point, is called tangential acceleration, which characterizes the change in the velocity vector in magnitude.

The component of the acceleration vector a, directed along the normal to the tangent at a given point on the trajectory, is called normal acceleration. It characterizes the change in direction of the velocity vector in the case of curvilinear motion. Naturally, when a body moves along a trajectory that is a straight line, the normal acceleration is zero.

Rectilinear motion is called uniformly variable if, over any period of time, the speed of the body changes by the same amount. In this case the relation

∆V/ ∆t is the same for any time intervals. Therefore, the magnitude and direction of acceleration remain unchanged: a = const.

For rectilinear motion, the acceleration vector is directed along the line of motion. If the direction of acceleration coincides with the direction of the velocity vector, then the magnitude of the velocity will increase. In this case, the movement is called uniformly accelerated. If the direction of acceleration is opposite to the direction of the velocity vector, then the magnitude of the velocity will decrease. In this case, the movement is called uniformly slow. In nature there is a natural uniformly accelerated movement - this is free fall.

Free fall- called the fall of a body if the only force acting on it is gravity. Experiments carried out by Galileo showed that during free fall, all bodies move with the same acceleration of gravity and are denoted by the letter ĝ. Near the Earth's surface ĝ = 9.8 m/s². The acceleration of free fall is caused by gravity from the Earth and is directed vertically downward. Strictly speaking, such movement is only possible in a vacuum. A fall in the air can be considered approximately free.

The trajectory of a freely falling body depends on the direction of the initial velocity vector. If a body is thrown vertically downwards, then the trajectory is a vertical segment, and the motion is called uniformly variable. If a body is thrown vertically upward, then the trajectory consists of two vertical segments. First, the body rises, moving equally slow. At the point of maximum ascent, the speed becomes zero, after which the body descends, moving uniformly accelerated.

If the initial velocity vector is directed at an angle to the horizon, then the movement occurs along a parabola. This is how a thrown ball, a disk, an athlete performing a long jump, a flying bullet, etc. move.

Depending on the form of representation of kinematic parameters, there are different types of laws of motion.

Law of motion is one of the forms of determining the position of a body in space, which can be expressed:

Analytically, that is, using formulas. This type of law of motion is specified using the equations of motion: x = x(t), y = y(t), z = z(t);

Graphically, that is, using graphs of changes in the coordinates of a point depending on time;

Tabular, that is, in the form of a data vector, when numerical time counts are entered in one column of the table, and in another, in comparison with the first, the coordinates of a point or points of the body.

In grade 7, you studied the mechanical motion of bodies occurring at a constant speed, i.e. uniform motion.

We now move on to consider uneven motion. Of all the types of non-uniform motion, we will study the simplest - rectilinear uniformly accelerated, in which the body moves along a straight line, and the projection of the body's velocity vector changes equally over any equal periods of time (in this case, the magnitude of the velocity vector can either increase or decrease).

For example, if the speed of an airplane moving along the runway increases by 15 m/s in any 10 s, by 7.5 m/s in any 5 s, by 1.5 m/s in every second, etc., then the plane moves with uniform acceleration.

In this case, the speed of an aircraft means its so-called instantaneous speed, i.e. the speed at each specific point of the trajectory at the corresponding moment in time (a more rigorous definition of instantaneous speed will be given in a high school physics course).

The instantaneous speed of bodies moving uniformly accelerated can change in different ways: in some cases faster, in others slower. For example, the speed of an ordinary passenger elevator of average power increases by 0.4 m/s for each second of acceleration, and by 1.2 m/s for a high-speed elevator. In such cases, they say that bodies move with different accelerations.

Let's consider what physical quantity is called acceleration.

Let the speed of some body moving uniformly accelerated change from v 0 to v over a period of time t. By v 0 we mean the initial speed of the body, i.e. the speed at the moment t 0 = O, taken as the beginning of time. And v is the speed that the body had at the end of the time period t, counted from t 0 = 0. Then for each unit of time the speed changed by an amount equal to

This ratio is denoted by the symbol a and is called acceleration:

• The acceleration of a body during rectilinear uniformly accelerated motion is a vector physical quantity equal to the ratio of the change in speed to the period of time during which this change occurred

Uniformly accelerated motion is motion with constant acceleration.

Acceleration is a vector quantity that is characterized not only by its magnitude, but also by its direction.

The magnitude of the acceleration vector shows how much the magnitude of the velocity vector changes in each unit of time. The greater the acceleration, the faster the speed of the body changes.

The SI unit of acceleration is the acceleration of such uniformly accelerated motion, in which the speed of the body changes by 1 m/s in 1 s:

Thus, the SI unit of acceleration is meter per second squared (m/s2).

Other units of acceleration are also used, for example 1 cm/s 2 .

You can calculate the acceleration of a body moving rectilinearly and uniformly accelerated using the following equation, which includes projections of the acceleration and velocity vectors:

Let us show with specific examples how acceleration is found. Figure 8, a shows a sled that is rolling down a mountain with uniform acceleration.

Rice. 8. Uniformly accelerated motion of a sled rolling down a mountain (AB) and continuing to move along the plain (CD)

It is known that the sled covered part of the path AB in 4 s. Moreover, at point A they had a speed of 0.4 m/s, and at point B they had a speed of 2 m/s (the sled is taken as a material point).

Let us determine with what acceleration the sled moved in section AB.

In this case, the beginning of the time count should be taken as the moment the sled passes point A, since according to the condition, it is from this moment that the period of time during which the magnitude of the velocity vector changed from 0.4 to 2 m/s is counted.

Now let’s draw the X axis parallel to the sled’s speed vector and directed in the same direction. Let us project the beginnings and ends of the vectors v 0 and v onto it. The resulting segments v 0x and v x are projections of the vectors v 0 and v onto the X axis. Both of these projections are positive and equal to the modules of the corresponding vectors: v 0x = 0.4 m/s, v x = 2 m/s.

Let's write down the conditions of the problem and solve it.

The projection of the acceleration vector onto the X axis turned out to be positive, which means that the acceleration vector is aligned with the X axis and with the speed of the sled.

If the velocity and acceleration vectors are directed in the same direction, then the speed increases.

Now let's consider another example, in which a sled, having rolled down a mountain, moves along a horizontal section CD (Fig. 8, b).

As a result of the friction force acting on the sled, its speed continuously decreases, and at point D the sled stops, i.e., its speed is zero. It is known that at point C the sled had a speed of 1.2 m/s, and they covered section CD in 6 s.

Let's calculate the acceleration of the sled in this case, i.e., determine how much the speed of the sled changed for each unit of time.

Let's draw the X axis parallel to the segment CD and align it with the speed of the sled, as shown in the figure. In this case, the projection of the sled's velocity vector onto the X axis at any moment of their movement will be positive and equal to the magnitude of the velocity vector. In particular, at t 0 = 0 v 0x = 1.2 m/s, and at t = 6 s v x = 0.

Let's record the data and calculate the acceleration.

The acceleration projection onto the X axis is negative. This means that the acceleration vector a is directed opposite to the X axis and, accordingly, opposite to the speed of movement. At the same time, the speed of the sled decreased.

Thus, if the velocity and acceleration vectors of a moving body are directed in one direction, then the magnitude of the body’s velocity vector increases, and if in the opposite direction, it decreases.

Questions

1. What type of motion - uniform or non-uniform - does rectilinear uniformly accelerated motion belong to?
2. What is meant by instantaneous speed of uneven motion?
3. Give the definition of acceleration of uniformly accelerated motion. What is the unit of acceleration?
4. What is uniformly accelerated motion?
5. What does the magnitude of the acceleration vector show?
6. Under what condition does the magnitude of the velocity vector of a moving body increase; is it decreasing?

Exercise 5

Characteristics of mechanical body movement:

- trajectory (the line along which the body moves),

- displacement (directed straight line segment connecting the initial position of the body M1 with its subsequent position M2),

- speed (ratio of movement to movement time - for uniform movement) .

Main types of mechanical movement

Depending on the trajectory, body movement is divided into:

Straight-line;

Curvilinear.

Depending on the speed, movements are divided into:

Uniform,

Uniformly accelerated

Equally slow

Depending on the method of movement, movements are:

Progressive

Rotational

Oscillatory

Complex movements (For example: a screw movement in which the body rotates uniformly around a certain axis and at the same time makes a uniform translational movement along this axis)

Forward movement - This is the movement of a body in which all its points move equally. In translational motion, any straight line connecting any two points of the body remains parallel to itself.

Rotational motion is the movement of a body around a certain axis. With such a movement, all points of the body move in circles, the center of which is this axis.

Oscillatory motion is a periodic motion that occurs alternately in two opposite directions.

For example, a pendulum in a clock performs an oscillatory motion.

Translational and rotational movements are the simplest types of mechanical movement.

Straight and uniform movement is called such a movement when, for any arbitrarily small equal intervals of time, the body makes identical movements . Let us write down the mathematical expression of this definition s = v? t. This means that the displacement is determined by the formula, and the coordinate - by the formula .

Uniformly accelerated motion is the movement of a body in which its speed increases equally over any equal intervals of time . To characterize this movement, you need to know the speed of the body at a given moment in time or at a given point in the trajectory, t . e . instantaneous speed and acceleration .

Instantaneous speed- this is the ratio of a sufficiently small movement on the section of the trajectory adjacent to this point to the small period of time during which this movement occurs .

υ = S/t. The SI unit is m/s.

Acceleration is a quantity equal to the ratio of the change in speed to the period of time during which this change occurred . α = ?υ/t(SI system m/s2) Otherwise, acceleration is the rate of change of speed or the increase in speed for each second α. t. Hence the formula for instantaneous speed: υ = υ 0 + α.t.

The displacement during this movement is determined by the formula: S = υ 0 t + α . t 2 /2.

Equally slow motion motion is called when the acceleration is negative and the speed uniformly slows down.

With uniform circular motion the angles of rotation of the radius for any equal periods of time will be the same . Therefore the angular speed ω = 2πn, or ω = πN/30 ≈ 0.1N, Where ω - angular speed n - number of revolutions per second, N - number of revolutions per minute. ω in the SI system it is measured in rad/s . (1/c)/ It represents the angular velocity at which each point of the body in one second travels a path equal to its distance from the axis of rotation. During this movement, the velocity module is constant, it is directed tangentially to the trajectory and constantly changes direction (see . rice . ), therefore centripetal acceleration occurs .

Rotation period T = 1/n - this time , during which the body makes one full revolution, therefore ω = 2π/T.

Linear speed during rotational motion is expressed by the formulas:

υ = ωr, υ = 2πrn, υ = 2πr/T, where r is the distance of the point from the axis of rotation. The linear speed of points lying on the circumference of a shaft or pulley is called the peripheral speed of the shaft or pulley (in SI m/s)

With uniform motion in a circle, the speed remains constant in magnitude but changes in direction all the time. Any change in speed is associated with acceleration. Acceleration that changes speed in direction is called normal or centripetal, this acceleration is perpendicular to the trajectory and directed to the center of its curvature (to the center of the circle, if the trajectory is a circle)

α p = υ 2 /R or α p = ω 2 R(because υ = ωR Where R circle radius , υ - point movement speed)

Relativity of mechanical motion- this is the dependence of the trajectory of the body, the distance traveled, movement and speed on the choice reference systems.

The position of a body (point) in space can be determined relative to some other body chosen as the reference body A . The reference body, the coordinate system associated with it, and the clock constitute the reference system . The characteristics of mechanical movement are relative, t . e . they can be different in different reference systems .

Example: the movement of a boat is monitored by two observers: one on the shore at point O, the other on the raft at point O1 (see . rice . ). Let us mentally draw through the point O the XOY coordinate system - this is a fixed reference system . We will connect another X"O"Y" system to the raft - this is a moving coordinate system . Relative to the X"O"Y" system (raft), the boat moves in time t and will move at speed υ = s boats relative to raft /t v = (s boats- s raft )/t. Relative to the XOY (shore) system, the boat will move during the same time s boats where s boatsmoving the raft relative to the shore . Speed ​​of the boat relative to the shore or . The speed of a body relative to a fixed coordinate system is equal to the geometric sum of the speed of the body relative to a moving system and the speed of this system relative to a fixed one .

Types of reference systems can be different, for example, a fixed reference system, a moving reference system, an inertial reference system, a non-inertial reference system.