PHYSICS

Kinematics

SI system (Kinematics)

Kinematics: The basic units of measurement of values ​​in the SI system are as follows:

  1. length unit – meter (1 m),
  2. time – second (1 s)
  3. weight – kilogram (1 kg),
  4. the amount of substance is mol (1 mol),
  5. temperature – kelvin (1 K),
  6. electric current – ampere (1 A)
  7. Reference: luminous intensity – candela (1 cd, is not actually used when solving school problems).
Measurement Units
Measurement Units

When performing calculations in the SI system, angles are measured in radians. (Kinematics)

If the problem in physics does not indicate in which units it is necessary to give an answer, it must be given in units of the SI system or in the quantities derived from them corresponding to the physical quantity that is asked in the problem. For example, if the task needs to find the speed, and does not say what it should be expressed, then the answer should be given in m / s.

For convenience, in problems in physics it is often necessary to use longitudinal (reducing) and multiple (increasing) prefixes. they can be applied to any physical quantity. For example, mm – millimeter, kt – kiloton, ns – nanosecond, Mg – megagrams, mmol – mmol, μA – microampere. Remember that in physics there are no double prefixes. For example, micrograms is micrograms, not millikilograms. Note that when adding and subtracting values, you can operate only with values ​​of the same dimension. For example, kilograms can be added only with kilograms, only millimeters can be subtracted from millimeters, and so on. When converting values, use the following table.

name number prefix symbol
Tera 1,000,000,000,000 Too [pull] T
Billion 1,000,000,000 Kyrgyzstan G
million 1,000,000 trillion M
thousand 1,000 thousand k
hundred 100 hundred h
ten 10 ten Da
unit 1
one tenth 0.1 Minute d
One percent 0.01 PCT c
thousandth 0.001 Milli m
millionth 0.000 001 micro- μ
One billionth 0.000 000 001 Nano n
One trillion 0.000 000 000 001 Leather p

Path and moving (Kinematics)

Kinematics is a section of mechanics in which the movement of bodies is considered without clarifying the reasons for this movement.

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

Every body has a certain size. However, in many problems of mechanics there is no need to indicate the positions of individual body parts. If the body size is small compared with the distance to other bodies, then this body can be considered as a material point . So when driving a car over long distances, you can neglect its length, since the length of the car is small compared to the distances it travels.

It is intuitively clear that the characteristics of motion (speed, trajectory, etc.) depend on where we look at it. Therefore, to describe the movement introduces the concept of a reference frame. The reference system (CO) is an aggregate of the reference body (it is considered absolutely solid), a coordinate system attached to it, a ruler (a device measuring distances), a clock and a time synchronizer.

Moving over time from one point to another, the body (material point) describes in this CO some line, which is called the trajectory of the body .

The movement of the body is called a directed segment of a straight line connecting the initial position of the body with its final position. Displacement is a vector quantity. Moving can in the process of movement increase, decrease and become equal to zero.

The traversed path is equal to the length of the trajectory traversed by the body for some time. The path is a scalar. The path can not decrease. The path only increases or remains constant (if the body is not moving). When a body moves along a curved path, the modulus (length) of the displacement vector is always less than the distance traveled.

With a uniform (constant speed) motion, the path L can be found by the formula:

Formula Path with uniform motion
Formula Path with uniform motion

where: v is the speed of the body, t is the time during which it moved. When solving problems in kinematics, displacement is usually found from geometrical considerations. Often, geometric considerations for finding displacement require knowledge of the Pythagorean theorem.

Average speed (Kinematics)

Velocity is a vector quantity characterizing the speed at which a body moves in space. The speed is average and instant. The instantaneous velocity describes the movement at a given specific point in time at a given point in space, and the average velocity characterizes the entire movement as a whole, in general, without describing the details of movement in each particular segment.

The average speed of a path is the ratio of the entire path to the entire travel time:

Formula Average Travel Speed
Formula Average Travel Speed

where: L is full – all the way that the body has gone, t is full – all the time of movement.

The average speed of movement is the ratio of the entire movement to the whole time of movement:

Formula Average Travel Speed
Formula Average Travel Speed

This value is directed in the same way as the complete movement of the body (that is, from the starting point of the movement to the end point). At the same time, do not forget that the full displacement is not always equal to the algebraic sum of displacements at certain stages of the movement. The full displacement vector is equal to the vector sum of displacements at separate stages of movement.

  • When solving problems in kinematics, do not make a very common mistake. The average speed, as a rule, is not equal to the arithmetic average of the body velocities at each stage of movement. The arithmetic average is obtained only in some special cases.
  • And even more so the average speed is not equal to one of the speeds with which the body moved in the process of movement, even if this speed had an intermediate value relative to other speeds with which the body moved.

Equal Accelerated Rectilinear Motion (Kinematics)

Acceleration is a vector physical quantity that determines the speed of a change in the speed of a body. The acceleration of the body is the ratio of the change in speed to the period of time during which the change in speed occurred:

Determination of acceleration at uniformly accelerated motion
Determination of acceleration at uniformly accelerated motion

where: 0 is the initial velocity of the body, v  is the final velocity of the body (that is, after a period of time t ).

Further, unless otherwise specified in the problem statement, we believe that if the body moves with acceleration, then this acceleration remains constant. Such a body movement is called uniformly accelerated (or uniformly variable). With a uniformly accelerated motion, the velocity of the body changes by the same amount at any equal time intervals.

Uniformly accelerated motion is actually accelerated when the body increases the speed of movement, and slowed down when the speed decreases. For simplicity, it is convenient to solve problems for slow motion to take acceleration with a “-” sign.

From the previous formula, there follows another more common formula describing the change in speed with time with uniformly accelerated motion:

Formula Dependence of speed on time with uniformly accelerated motion.
Formula Dependence of speed on time with uniformly accelerated motion.

The displacement (but not the path) with the uniformly accelerated motion is calculated by the formulas:

Formula: Moving with a uniformly accelerated rectilinear motion.
Formula: Moving with a uniformly accelerated rectilinear motion.
Formula: Moving with a uniformly accelerated rectilinear motion.
Formula: Moving with a uniformly accelerated rectilinear motion.
Formula: Moving with a uniformly accelerated rectilinear motion.
Formula: Moving with a uniformly accelerated rectilinear motion.

The last formula uses one feature of uniformly accelerated motion. With a uniformly accelerated motion, the average velocity can be calculated as the arithmetic mean of the initial and final velocities (this property is very convenient to use when solving some problems):

Formula Average speed with uniformly accelerated motion
Formula Average speed with uniformly accelerated motion

With the calculation of the path all the more difficult. If the body does not change the direction of motion, then with uniformly accelerated straight-line motion, the path is numerically equal to the displacement. And if it has changed, then it is necessary to separately consider the path to the stop (turning moment) and the path after stopping (turning moment). A simple substitution of time in the formula for moving in this case will lead to a typical error.

Coordinate with uniformly accelerated motion changes according to the law:

Formula Coordinate with uniformly accelerated motion
Formula Coordinate with uniformly accelerated motion

The projection of speed with a uniformly accelerated motion changes according to the following law:

Formula Projection of speed with uniformly accelerated motion.
Formula Projection of speed with uniformly accelerated motion.

Similar formulas are obtained for the remaining coordinate axes. Formula for the stopping distance of the body:

Formula for the stopping distance of the body
Formula for the stopping distance of the body

Vertical free fall (Kinematics)

All bodies in the field of the Earth, the force of gravity. In the absence of support or suspension, this force causes the body to fall to the surface of the Earth. If we neglect the air resistance, then the movement of bodies only under the action of gravity is called free fall. Gravity communicates to any bodies, regardless of their shape, mass and size, the same acceleration, called free fall acceleration. Near the Earth’s surface, the acceleration of free fall is:

Acceleration of gravity
Acceleration of gravity

This means that the free fall of all bodies near the surface of the Earth is a uniformly accelerated (but not necessarily rectilinear) motion. First, we consider the simplest case of free fall, when the body moves strictly vertically. Such a motion is a uniformly accelerated straight-line motion, therefore all the patterns and foci of such a motion studied earlier are also suitable for free fall. Only acceleration is always equal to the acceleration of free fall.

Traditionally, with a free fall, the vertical axis OY is used. There is nothing wrong here. It is just necessary in all formulas instead of the index ” x ” to write ” y “. The meaning of this index and the rule for determining characters is preserved. Where to direct the axis OY – your choice, depending on the convenience of solving the problem. Options 2: up or down.

We present several formulas that are the solution of some specific problems on the kinematics of free fall vertically. For example, the speed with which a body falls from a height h without an initial velocity:

Formula The speed at which a body falls from a height without initial velocity
Formula The speed at which a body falls from a height without initial velocity

Time of fall of a body from height h without initial speed:

Formula Time the fall of the body from a height without the initial speed
Formula Time the fall of the body from a height without the initial speed

The maximum height at which the body will rise, thrown vertically upwards with the initial speed 0 , the time of raising this body to the maximum height, and the total flight time (before returning to the starting point):

Formula Maximum height at which the body will rise, thrown vertically upwards.
Formula Maximum height at which the body will rise, thrown vertically upwards.
Formula Time of lifting the body thrown vertically up to the maximum height
Formula Time of lifting the body thrown vertically up to the maximum height
Formula Full time of flight of the body thrown vertically up (before returning to the starting point)
Formula Full time of flight of the body thrown vertically up (before returning to the starting point)

Horizontal throw (Kinematics)

With a horizontal throw with an initial speed 0, it is convenient to consider the body movement as two movements: uniform along the OX axis (along the OX axis there are no forces preventing or helping the movement) and uniformly accelerated motion along the OY axis.

The speed at any time is directed tangentially to the trajectory. It can be decomposed into two components: horizontal and vertical. The horizontal component always remains unchanged and is equal to x =  0 . A vertical increases according to the laws of accelerated motion y = gt . In this case, the total speed of the body can be found by the formulas:

Formula Full speed of the body thrown vertically
Formula Full speed of the body thrown vertically
Formula Full speed horizontal throw
Formula Full speed horizontal throw

It is important to understand that the time of the body falling to the earth in no way depends on the horizontal speed with which it was thrown, but is determined only by the height from which the body was thrown. The time of the fall of the body to the earth is according to the formula:

Formula Time the fall of the body in a horizontal throw
Formula Time the fall of the body in a horizontal throw

As the body falls, it simultaneously moves along the horizontal axis. Consequently, the range of the body or the distance that the body can fly along the axis OX will be equal to:

Formula Range of body flight at horizontal throw
Formula Range of body flight at horizontal throw

The angle between the horizon and the speed of the body is easy to find from the relation:

Angle between horizon and speed when throwing horizontally
Angle between horizon and speed when throwing horizontally

Also, sometimes in tasks, one may be asked about the point in time at which the full speed of the body will be tilted at a certain angle to the vertical . Then this angle will be from the relation:

Angle between vertical and horizontal throw speed
Angle between vertical and horizontal throw speed
The flight path of the body in a horizontal throw
The flight path of the body in a horizontal throw

It is important to understand exactly which angle appears in the problem (with a vertical or with a horizontal). This will help you choose the right formula. If, however, this problem is solved by the coordinate method, then the general formula for the law of coordinate change with uniformly accelerated motion is:

The law of change of the coordinates of uniformly accelerated motion
The law of change of the coordinates of uniformly accelerated motion

It is transformed into the following law of motion along the OY axis for a body thrown horizontally:

The law of change of OY coordinates for a freely falling body
The law of change of OY coordinates for a freely falling body

With her help, we can find the height at which the body will be at any time. At the same time at the time of the fall of the body to the ground, the coordinate of the body along the axis OY will be zero. Obviously, the body moves along the OX axis uniformly, therefore, within the framework of the coordinate method, the horizontal coordinate will change according to the law:

The law of change of OX coordinates for a freely falling body
The law of change of OX coordinates for a freely falling body

Throw at an angle to the horizon (from earth to earth)

Maximum lift height when throwing at an angle to the horizon (relative to the initial level):

Formula Maximum lifting height when throwing at an angle to the horizon
Formula Maximum lifting height when throwing at an angle to the horizon

Rise time to the maximum height when throwing at an angle to the horizon:

Formula lifting time to the maximum height when throwing at an angle to the horizon
Formula lifting time to the maximum height when throwing at an angle to the horizon

Flight distance and full time of flight of the body abandoned at an angle to the horizon (provided that the flight ends at the same height from which it began, ie, the body was thrown, for example, from the ground to the ground):

Formula Range of flight of a body abandoned at an angle to the horizon.
Formula Range of flight of a body abandoned at an angle to the horizon.
Formula Full time of flight of the body thrown at an angle to the horizon
Formula Full time of flight of the body thrown at an angle to the horizon

The minimum speed of the body thrown at an angle to the horizon – at the highest point of the rise, and is equal to:

Minimum body speed when throwing at an angle to the horizon
Minimum body speed when throwing at an angle to the horizon

The maximum speed of the body thrown at an angle to the horizon – at the moments of throwing and falling to the ground, and is equal to the initial one. This statement is true only for throwing from the ground to the ground. If the body continues to fly below the level from which it was thrown, then it will gain more and more speed there.

Velocity addition (Kinematics)

The movement of bodies can be described in various reference systems. From the point of view of kinematics, all reference systems are equal. However, the kinematic characteristics of motion, such as trajectory, displacement, speed, in different systems are different. The values ​​depending on the choice of the reference system in which they are measured are called relative. Thus, rest and body movement are relative. The classical law of velocity addition:

The classical law of velocity addition
The classical law of velocity addition

Thus, the absolute velocity of a body is equal to the vector sum of its velocity relative to the moving coordinate system and the velocity of the moving reference system itself. Or, in other words, the velocity of the body in a fixed frame of reference is equal to the vector sum of the velocity of the body in the moving frame and the velocity of the moving frame relative to the fixed frame.

Uniform circular motion (Kinematics)

The movement of the body in a circle is a special case of curvilinear motion. This kind of movement is also considered in kinematics. With curvilinear motion, the velocity vector of the body is always directed tangentially to the trajectory. The same thing happens when moving in a circle (see figure). Uniform movement of the body in a circle is characterized by a number of quantities.

Body movement in a circle
Body movement in a circle

Period – the time for which the body, moving in a circle, makes one complete turn. Unit of measure – 1 s. The period is calculated by the formula:

Definition of rotation period
Definition of rotation period

Frequency – the number of revolutions that the body has made, moving in a circle, per unit of time. Unit of measure – 1 rev / s or 1 Hz. Frequency is calculated by the formula:

Determination of rotational speed
Determination of rotational speed

In both formulas: N is the number of revolutions during time t . As can be seen from the above formulas, the period and frequency values ​​are reciprocal:

Formulas Relationship of the period and frequency
Formulas Relationship of the period and frequency

With a uniform rotation, the speed of the body will be determined as follows:

Formula Linear speed with uniform motion along a circle
Formula Linear speed with uniform motion along a circle

where: l – length of the circumference or path traversed by the body for the time equal to the period T . When a body moves along a circle, it is convenient to consider the angular displacement φ (or the angle of rotation), measured in radians. The angular velocity ω of the body at a given point is the ratio of the small angular displacement Δ φ to the small time interval Δ t . It is obvious that in time equal to the period T the body will pass an angle equal to 2 π , therefore with the uniform motion along the circle the formulas are fulfilled

Formula Angular Rotation Speed
Formula Angular Rotation Speed

Angular velocity is measured in rad / s. Don’t forget to convert angles from degrees to radians. The arc length l is related to the rotation angle by the ratio:

Formula Relationship of the rotation angle and the path with a uniform motion along a circle
Formula Relationship of the rotation angle and the path with a uniform motion along a circle

The relationship between the linear velocity module v  and the angular velocity ω :

Formula Relationship between linear and velocity and angular velocity
Formula Relationship between linear and velocity and angular velocity

When a body moves in a circle with a constant speed modulo, only the direction of the velocity vector changes; therefore, the body motion in a circle with a constant velocity modulus is a motion with acceleration (but not equally accelerated), since the speed direction changes. In this case, the acceleration is directed along the radius to the center of the circle. It is called normal, or centripetal acceleration , since the acceleration vector at any point of the circle is directed to its center (see figure).

Body movement in a circle
Body movement in a circle

The module of centripetal acceleration is connected with linear v  and angular ω speeds by the relations:

Formula Centripetal Acceleration
Formula Centripetal Acceleration

Note that if the bodies (points) are on a rotating disk, ball, rod, and so on, in one word on the same rotating object, then all bodies have the same period of rotation, angular velocity and frequency.

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