# Kinematics

## SI system (Kinematics)

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

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

**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:

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:

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:

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:

where: *v *_{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:

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

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):

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:

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

Similar formulas are obtained for the remaining coordinate axes. **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:

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:

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

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

Horizontal throw (Kinematics)

With a horizontal throw with an initial speed *v *_{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 *v *_{x} = *v *_{0} . A vertical increases according to the laws of accelerated motion *v *_{y} = *gt* . In this case, the **total speed of the body** can be found by the formulas:

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:

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:

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

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:

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:

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

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:

### 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):

Rise 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):

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

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:**

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.

**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:

**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:

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:

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

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

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:

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

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).

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

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.