School Encyclopedia. Basic concepts of kinematics and kinematic characteristics
Characteristics of the mechanical movement of the body:
- trajectory (line along which the body moves),
- displacement (directed line segment connecting the initial position of the body M1 with its subsequent position M2),
- speed (the ratio of movement to movement time - for uniform movement) .
The main types of mechanical movement
Depending on the trajectory, the movement of the body is divided into:
Rectilinear;
Curvilinear.
Depending on the speed of movement are divided into:
Uniform,
Uniformly accelerated
Uniformly slow
Depending on the method of movement, movements are:
Translational
rotational
vibrational
Compound motions (For example: a screw motion in which the body rotates uniformly around some axis and at the same time performs uniform translational motion along this axis)
translational movement - This is the movement of a body in which all its points move in the same way. 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 an axis. With such a movement, all points of the body move along circles, the center of which is this axis.
An oscillatory motion is a periodic motion that occurs alternately in two opposite directions.
For example, the pendulum in a clock makes an oscillatory motion.
Translational and rotational motion are the simplest types of mechanical motion.
Rectilinear and uniform movement is called such a movement when for any arbitrarily small equal intervals of time the body makes the same displacement . Let's write down the mathematical expression of this definition s = υ? t. This means that the displacement is determined by the formula, and the coordinate - by the formula .
Uniformly accelerated motion called the movement of a body in which its speed for any equal intervals of time increases equally . To characterize this movement, you need to know the speed of the body at a given point in time or at a given point in the trajectory, t . e . instantaneous speed and acceleration .
Instant Speed- this is the ratio of a sufficiently small movement in the section of the trajectory adjacent to this point to a small period of time during which this movement takes place .
υ = S/t. The SI unit of measure is m/s.
Acceleration - a value equal to the ratio of the change in speed to the period of time during which this change occurred . α = ?υ/t(SI m/s2) Otherwise, acceleration is the rate of change of speed or increment of speed in every second α . t . Hence the formula for instantaneous speed: υ = υ 0 + α.t.
The movement during this movement is determined by the formula: S = υ 0 t + α . t2/2.
Equally slow motion movement is called when the acceleration has a negative value, the speed at the same time slows down uniformly.
With uniform circular motion the angles of rotation of the radius for any equal intervals of time will be the same . Therefore, the angular velocity ω = 2πn, or ω = πN/30 ≈ 0.1N , where ω - angular velocity n is the number of revolutions per second, N is the number of revolutions per minute. ω in the SI system is measured in rad / s . (1/c)/ It represents the angular velocity at which each point of the body travels in one second a path equal to its distance from the axis of rotation. During this movement, the velocity modulus is constant, it is directed tangentially to the trajectory and constantly changes direction (see . rice . ), so there is a centripetal acceleration .
Rotation period T \u003d 1 / n - this time , for which the body makes one complete revolution, therefore ω = 2π/T.
The 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 the points lying on the circumference of the shaft or pulley is called the circumferential speed of the shaft or pulley (in the SI system, m/s)
In 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. An 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 \u003d υ 2 / R or α p \u003d ω 2 R(as 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, displacement and speed on the choice reference systems.
The position of a body (point) in space can be determined relative to any other body chosen as reference body A . The body of reference, the coordinate system associated with it, and the clock constitute the frame of reference . Characteristics of mechanical movement are relative, t . e . they can be different in different reference systems .
Example: two observers are following the movement of the boat: one on the shore at point O, the other on the raft at point O1 (see . rice . ). Let's mentally draw through the point O the coordinate system XOY is a fixed frame of reference . Let's connect another X"O"Y" system with a raft - this is a moving coordinate system . Relative to the system X"O"Y" (raft), the boat moves in time t and will move at a speed υ = s boats relative to the raft /t v = (s boats- s raft )/t. Relative to the XOY (shore) system, the boat will move in the same time s boats where s boats moving the raft relative to the shore . The 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 a 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 frame of reference, a moving frame of reference, an inertial frame of reference, a non-inertial frame of reference.
1. The concept of uniformly accelerated motion. Its characteristics.2. The concept of a reference system. Examples of different reference systems. Uniformly slow motion, its characteristics.
3. The concept of a material point. Uniform rectilinear motion, its characteristics
4. The concept of a reference system. Examples of different reference systems. Uniformly accelerated motion, its characteristics.
5. The concept of a material point. Description of the laws of motion of a body along a parabola.
6. Description of the movement of the body in a circle. Its characteristics.
7. The concept of uniformly accelerated motion. Its characteristics.
8. Description of the motion of a body in a plane at an angle to the horizon. Its characteristics.
9. Newton's first law, its application in life and natural phenomena.
10. Newton's second law. Applying it to calculate acceleration.
11. Newton's third law. Force types. Graphic representation of the forces applied to the body.
12. Statics. The condition of static equilibrium, by examples.
13. Law of conservation of momentum by examples.
14. The concept of energy, classification. Kinetic energy.
15. The concept of energy, classification. Potential energy of tension of a spring.
16. The concept of energy, classification. Potential energy of gravity.
17. The concept of total mechanical energy. Law of energy conservation.
18. MKT - postulates. Characteristics of the three states of matter.
19. Gas - the movement of molecules. Stern's experiment, velocity distribution of molecules.
20. The concept of an ideal gas. Klaiperon-Mendeleev equation. Isoprocesses - isobar.
21. Equation of ideal gas, execution conditions. Isoprocesses - isotherm.
22. The concept of an ideal gas. Klaiperon-Mendeleev equation. Isoprocesses - isochore.
23. MKT. The concept of real gas, its comparison with the ideal.
24. The first law of thermodynamics, the concept of heat transfer.
25. First law of thermodynamics for isochoric process.
26. First law of thermodynamics for isobaric process.
27. The first law of thermodynamics for an isothermal process.
28. The concept of the internal energy of an ideal gas for isoprocesses.
29. The second law of thermodynamics. Its application to cyclic processes on the example of a steam engine.
30. The second law of thermodynamics. Its application to cyclic processes on the example of an internal combustion engine.
31. The concept of heat engines. Jet engines.
32. The concept of heat engines. Refrigeration machines.
33. The third law of thermodynamics.
34. Adiobatic process. The concept of heat capacity.
49 m long? Which waves (long, medium or short) are these waves?
In the 7th grade, you studied the mechanical movement of bodies that occurs at a constant speed, that is, uniform movement.
We now turn to the consideration of non-uniform motion. Of all types of uneven 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 in the same way for any equal time intervals (in this case, the modulus of the velocity vector can both increase and decrease).
For example, if the speed of an aircraft 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., the plane is moving with uniform acceleration.
In this case, the speed of the 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 the 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 a conventional passenger elevator of medium power increases by 0.4 m/s for every second of acceleration, and of a high-speed one by 1.2 m/s. In such cases, the bodies are said to move with different accelerations.
Consider what physical quantity is called acceleration.
Let the speed of a body moving uniformly accelerated change from v 0 to v over a period of time t. Under v 0 is meant the initial speed of the body, i.e., the speed at the moment t 0 \u003d O, taken as the origin of time. And v is the speed that the body had by the end of the time interval t, counted from t 0 \u003d 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 in a rectilinear uniformly accelerated motion is a vector physical quantity equal to the ratio of the change in speed to the time interval during which this change occurred
Uniformly accelerated motion is motion with constant acceleration.
Acceleration is a vector quantity, which is characterized not only by the module, but also by the direction.
The module of the acceleration vector shows how much the module of the velocity vector changes in each unit of time. The greater the acceleration, the faster the speed of the body changes.
The unit of acceleration in SI is the acceleration of such a uniformly accelerated movement, in which for 1 s the speed of the body changes by 1 m / s:
Thus, in SI, the unit of acceleration is meter per second squared (m/s 2).
Other units of acceleration are also used, such as 1 cm/s 2 .
You can calculate the acceleration of a body moving in a straight line and uniformly accelerated using the following equation, which includes the projections of the acceleration and velocity vectors:
Let's show on concrete examples how acceleration is found. Figure 8, a shows a sled that rolls down the mountain with uniform acceleration.
Rice. 8. Uniformly accelerated movement of a sled rolling down a mountain (AB) and continuing to move along the plain (CD)
It is known that the sled passed the section of the path AB in 4 s. At the same time, at point A, they had a speed equal to 0.4 m / s, and at point B - a speed equal to 2 m / s (the sled was taken as a material point).
Let us determine with what acceleration the sled moved in the section AB.
In this case, the moment when the sled passes point A should be taken as the beginning of the time reference, since, according to the condition, it is from this moment that the time interval is measured, during which the velocity vector module changed from 0.4 to 2 m/s.
Now let's draw the X axis, parallel to the speed vector of the sled and directed in the same direction. We 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 condition of the problem and solve it.
The projection of the acceleration vector on the X-axis turned out to be positive, which means that the acceleration vector is co-directed with the X-axis and with the speed of the sledge.
If the velocity and acceleration vectors are directed in the same direction, then the velocity increases.
Now let's consider another example, in which the sled, having rolled down the mountain, moves along the horizontal section CD (Fig. 8, b).
As a result of the action of the friction force on the sledge, their speed continuously decreases, and at point D the sledge stops, i.e., their speed is zero. It is known that at point C, the sled had a speed of 1.2 m/s, and they covered the section CD in 6 s.
Let us calculate the acceleration of the sledge in this case, i.e., determine how much the speed of the sledge changed for each unit of time.
Let's draw the X axis parallel to the segment CD and direct it with the speed of the sledge, as shown in the figure. In this case, the projection of the velocity vector of the sledge on the X axis at any moment of their movement will be positive and equal to the modulus of the velocity vector. In particular, at t 0 = 0 v 0x = 1.2 m/s, and at t = 6 with v x = 0.
Let's write down the data and calculate the acceleration.
The acceleration projection on 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 modulus of the body's velocity vector increases, and if in the opposite direction, it decreases.
Questions
- Which type of motion - uniform or non-uniform - is rectilinear uniformly accelerated motion?
- What is meant by instantaneous speed of uneven motion?
- Define acceleration of uniformly accelerated motion. What is the unit of acceleration?
- What is uniformly accelerated motion?
- What does the modulus of the acceleration vector show?
- Under what condition does the modulus of the velocity vector of a moving body increase; decreases?
Exercise 5
The movement of a person 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 that studies mechanical motion, but does not consider the causes that cause this motion. Description of the movement of both the human body (its parts) in various sports, and various sports equipment are 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 time. Any object has spatial dimensions and shape, 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, it can be performed earlier or later than another process. That is why it becomes necessary to measure the spatial and temporal extent.
The main 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, the length is measured in meters (m) and multiple units of measurement: kilometers (km), centimeters (cm), etc.
Time is 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 Earth day was chosen as a unit of time and was called a second (s) and multiples of it (minutes, hours, etc.).
In sports, special temporal characteristics are used:
Moment of time(t)- it is a temporary measure of the position of a material point, links of a body or a system of bodies. Moments of time denote the beginning and end of a movement or any of its parts or phases.
Duration of movement(∆t) – this is its time measure, which is measured by the difference between the moments of the end and the beginning of the movement∆t = tcon. – tini.
Movement pace(N) - it is a temporary measure of repetition of movements repeated per unit of time. N = 1/∆t; (1/c) or (cycle/c).
Rhythm of movements – this is a temporary measure of the ratio of parts (phases) of movements. It is determined by the ratio of the duration of the parts of the movement.
The position of the body in space is determined relative to some reference system, which includes the reference body (that is, relative to which the movement is considered) and the coordinate system necessary to describe the position of the body in one or another part of space at a qualitative level.
The reference body is associated with the beginning and direction of measurement. For example, in a number of competitions, the start position can be chosen as the origin of coordinates. Various competitive distances are already calculated from it in all cyclic sports. Thus, in the chosen coordinate system "start - finish" determine the distance in space, which will move the athlete when moving. Any intermediate position of the athlete's body during movement is characterized by the current coordinate within the selected distance interval.
To accurately determine the sports result, the rules of the competition provide for which point (reference point) is counted: along the toe of the skater's skate, along the protruding point of the sprinter's chest, or along the trailing edge of the footprint of the landing jumper in length.
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, the dimensions and internal structure of which, under given conditions, can be neglected.
The movement of bodies can be different in nature and intensity. To characterize these differences, a number of terms are introduced in kinematics, which are presented below.
Trajectory – a line described in space by a moving point of a body. In the biomechanical analysis of movements, first of all, the trajectories of movements of the characteristic points of a person are considered. As a rule, these points are the joints of the body. According to the type of trajectory of movements, 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, the displacement characterizes the final result of the movement.
Way – this is the length of the trajectory section traversed by the body or a point of the body for a selected period of time.
In order to characterize how quickly the position of a moving body changes in space, a special concept of speed is used.
Speed – is the ratio of the distance traveled to the time it took to travel. It shows how quickly the position of the body in space changes.. Since speed is a vector, it also indicates in which direction the body or point of the body is moving.
medium speed body in a given section of the trajectory is the ratio of the distance traveled to the time of movement, m / s:
If the average speed is the same on all parts of the trajectory, then the movement is called uniform.
The question of running speed is important in sports biomechanics. It is known that the speed of running for a certain distance depends on the value of this distance. A runner can only maintain top 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. The average speed (V) versus distance length (S) is shown below.
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 h 21 min 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 the convenience of calculations, the average speed can also be written in terms of a change in the coordinates of the body. In rectilinear motion, 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 ∆X = X1 - X0, and the time of movement ∆t = t1 - t0 (the symbol ∆ denotes difference of the same type of values or to designate very small intervals). In this case:
The unit of speed in SI is m/s. When overcoming long distances, the 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.
The average speeds on different sections of the path differ significantly even with a relatively uniform distance: starting acceleration, overcoming the distance with intra-cycle speed fluctuations (during repulsion, the speed increases, during free gliding in skating or the flight phase in l / a run, it decreases) , finishing. As the interval over which the speed is calculated decreases, it is possible to determine the speed at a given point in the trajectory, which is called the instantaneous speed.
Or the speed at a given point of the trajectory is the limit to which the movement of the body in the vicinity of this point tends to time with an unlimited decrease in the interval:
Instantaneous speed is a vector quantity.
If the speed value (or the module of the velocity vector) does not change, the movement is uniform, if the module of speed changes, it is uneven.
Uniform called a movement in which a body travels the same distance in equal intervals of time. In this case, the magnitude of the speed remains unchanged (the direction of the speed may change if the movement is curvilinear).
Straightforward called movement in which the path is a straight line. In this case, the direction of the speed remains unchanged (the magnitude of the speed may change if the movement is not uniform).
Uniform rectilinear is called a 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 fast these changes occur, a special quantity is used - acceleration.
Acceleration – this is a value equal to the ratio of the change in the speed of the body to the duration of the time interval during which this change in speed occurred. The average acceleration based on this definition is, m/s²:
Instant acceleration called a physical quantity equal to the limit to which the average acceleration over the interval tends∆t → 0, m/s²:
Since the speed can change both in magnitude and in 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 of the trajectory, is called normal acceleration. It characterizes the change in the velocity vector in the direction 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 equally variable if the speed of the body changes by the same amount over any interval of time. 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 equally slow. In nature, there is a natural uniformly accelerated movement - this is free fall.
free fall- is called the fall of a body, if only one force acts on it - the force of gravity. Experiments conducted by Galileo showed that in free fall all bodies move with the same free fall acceleration and are denoted by the letter ĝ. Near the Earth's surface ĝ = 9.8 m/s². Free fall acceleration is due to gravity from the Earth and is directed vertically downwards. Strictly speaking, such motion is possible only in a vacuum. Falling 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 the body is thrown vertically downwards, then the trajectory is a vertical segment, and the motion is called equally variable. If a body is thrown vertically upwards, then the trajectory consists of two vertical segments. First, the body rises, moving uniformly slow. At the point of the highest rise, the speed becomes equal to zero, after which the body descends, moving with uniform acceleration.
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 jumping long, 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- this is one of the forms of determining the position of the body in space, which can be expressed:
Analytically, that is, using formulas. This kind of law of motion is given by 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;
Tabularly, that is, in the form of a data vector, when numerical time readings are entered in one column of the table, and the coordinates of a point or points of the body are entered in the other in comparison with the first.
Curvilinear motion of the body
Curvilinear motion of a body definition:
Curvilinear motion is a type of mechanical motion in which the direction of velocity changes. The speed modulus can change.
Uniform body movement
Uniform body motion definition:
If a body travels equal distances in equal intervals of time, then such a movement is called. With uniform motion, the modulus of velocity is a constant value. And it can change.
Uneven body movement
Uneven body movement definition:
If a body travels different distances in equal intervals of time, then such a movement is called uneven. With uneven movement, the speed modulus is a variable. The direction of speed can change.
Uniform body movement
Equal-variable motion of a body definition:
There is a constant value in uniformly variable motion. If at the same time the direction of the velocity does not change, then we get a rectilinear uniformly variable motion.
Uniformly accelerated motion of the body
Uniformly accelerated motion of a body definition:
Equally slow motion of the body
Uniformly slow motion of a body definition:
When we talk about the mechanical motion of a body, we can consider the concept of translational motion of a body.