The average speed vector and the average speed vector. Speed, Vector of speed and trajectory, Folding of speed. Spivvіdnoshennia between units of swedkostі

Shvidkіst - a vector quantity that characterizes how swidkіst the movement of a part is along the trajectory, and th directly, in which part of the skin is crumbling.

Average speed per hour view t1 before t2 long-term relocation for the whole hour until the interim hour, for such a relocation there is little space:

The fact that the same average speed is significant to us, laying the average value in the apex of the bow:<...>like a broken thing.

A higher formula has been introduced for the mean velocity vector, which is a direct result of the mathematical mean value of the mean value<f(x)> sufficient functions f(x) for the middle [ a,b]:

Deisno

The average shvidkіst may appear even as a rough characteristic of the rush. For example, the average speed for the period of coliving is going to zero, irrespective of the nature of these coliving, for these simple reasons, that for the period - for the appointed period - the body, that it is fluctuating, turn around to zero point і, then, moving for the period of the future. Z tsієї and low іnshih reasons to introduce mitteva shvidkіst - shvidkіst at a given moment. Nadali, looming on the verge of mitteva shvidkіst, we will write simply: "shvidkіst", omitting the words "mitteva" or "at a given moment of the hour" start, if you cannot bring it to the point of incomprehensibility. For otrimannya swidkosti at the moment of the hour t it is necessary to make it obvious: calculate the interval between the hours at the right time t2 – t1 nanivets. Zrobimo reassignment: t1 = tі t 2 \u003d t + and we will rewrite the upper spіvіdnoshennia at the sight:

Quickness at the moment t a better time between transfers one hour before the next hour, for which there is little space for the transfer, with the remaining time to zero

Rice. 2.5. Appointment of mitteva shvidkost.

At the moment, we do not consider food about the reason for the boundary, allowing for what it is. Significantly, yakscho є kіntseve remіshchennya i іnѕtіvеy interval аnd hour, then th - їх boundary values: infinitely small displacement and infinitely small interval one hour. So what is the right part of the designation of swedkost?

є nischo іnshe yak drіb - privat vid podіlu na, to that the rest of the spіvvіdshennya can be rewritten and often victorious at the viewer

Behind the geometric zm_stom pokhіdnoї, the vector of swidkost in the skin point of the trajectory of straightening along the dotichnі to traєktorії in the tsіy point of the її bіk ruhu.

Video 2.1. The vector of straightness of straightening along the trajectory. Experiment with a grinder.

Whether a vector can be laid out according to the basis (for single vectors in the basis, in other words, single vectors, which indicate the positive direction of the axes OX,OY,oz vikoristovuєmo oznachennya ,, but obviously). The coefficients of such a layout are the projections of the vector on the axis. Importantly: in the algebra of vectors, it has been proved that the arrangement of the basis is one. Let's spread out on the basis of the radius-vector of the actual material point that is collapsing.

Varying steel of Cartesian single vectors , ,

From the other side, layout according to the basis of the speed vector can be seen

leaving the two remaining variables, with the improvement of the unity of the layout of any vector behind the basis, gives the offensive result: the projections of the swidity vector on the Cartesian axes are adjusted to follow the hour after the corresponding coordinates, tobto

Speed ​​vector module

We take away one more important variable for the modulus of the speed vector.

It was already planned that the value of || the daedals are less permeable to the winding path (div. fig. 2). Tom

and in between (>0)

In other words, the module of speed is the same as a traveled path in an hour.

Remaining maєmo:

The average modulus of the speed vector, appears like this:

The average value of the modulus of the speed vector is the average of the distance traveled up to an hour, the length of which path is passed:

Here s(t1,t2)- way per hour t1 before t2 and apparently, s(t0,t2)- way per hour t0 before t2і s(t0,t2)- way per hour t0 before t1.

Average vector speed or just average speed, as it is indicated

Significantly, the first vector for all, the th module - the module of the average vector of the density does not stray from the average values ​​of the module of the vector of the density. The stench is not equal: the modulus of the mean vector is not equal to the mean modulus of the vector . Two operations: the calculation of the module and the calculation of the average, at the same time, cannot be rearranged by places.

Let's look at an example. Let the point collapse in one beat. On fig. 2.6. the testimony of the schedule of the way she passed s per hour (per hour per hour 0 before t). Vikoristovuyuchi physical sensibility of speed, to know for the help of which schedule the moment to the hour, at some mitteva speed is good for the average road speed for the first second and the turn of the point.

Rice. 2.6. The designation of the mitt and the middle swidkost of the body

Security module at a given moment

being a good way by the hour, reaching the top coefficient of the coliva to the graph of the fallow of the point, which is indicative of the moment of the hour. t*. The average module of air traffic per hour 0 before t*є kutovy sіchnoy coefficient, scho to pass through the points of the same graph, scho t = 0 i kіntsyu t = t* time interval. We need to know such a moment t*, if insults are kutovі koefіtsієnti zbіgayutsya. For this, a straight line is drawn through the cob of coordinates, so that the trajectory is drawn. As you can see from the little one, the torsional point of the straight line graphics s(t) and yes t*. Our butt comes out

To characterize the speed, the concept of speed is required.

Appointment: The average speed of the point for the interval of one hour from to
the vector value is called the increase in the increase in the radius vector of the point for the entire interval of an hour before the th trivality
.

- Medium speed.

Appointment: Shvidkіst (or mitteva shvidkіst) of a point is called a vector quantity, which is equal to the first hour of the hour in the radius vector.

The speed vector characterizes both by magnitude and by direct. vector

Appointment: The speed module is the first good hour from the passed route.

Let's decompose the vector of flatness behind the basis of a rectangular Cartesian coordinate system:

, de V x , V y , V z

de
- central projection of the radius vector of the material point.

At the coordinate input, the speed vector can look like:

Modulus of the velocity vector in the coordinate file:

Zvorotne spіvvіdshennya.

Let us represent the radius vector of the density with the help of the simple and undefined integral:

de t, t 0 - First and last moment of the hour.

Submission of the passed path through the module of speed for the help of a simple and undefined integral.

§four. Vector sigh.

To characterize the speed of changing the vector of the speed of the point at the mechanics, it is necessary to introduce a quicker understanding.

Appointment: Average rate per hour interval before
is called a vector quantity, which is the best way to increase the vector of the speed of a point for a given interval of an hour to the th value.

Appointment: Accelerating (abo mittve priskorennya) points are called a vector quantity, numerically equal to the first hourly relative speed of the point, which can be seen either, which are the same, the other is similar after the hour to the radius vector of the point:

The boost can be entered through the middle of the middle boost:

Two introductions of accelerated entries are equivalent.

We decompose the acceleration vector behind the basis of a rectangular Cartesian coordinate system:

where a x, a y, a z are the projections of the acceleration vector on the whole.

Coordinates of the accelerating vector modulus:

Zvorotni spіvvіdnoshennia:

;

Let's take a look at the material points of the flat curve. Quickly start straightening the middle of the curve or trajectory. We introduce two single vectors: , which directs along dotichny to trajectory that - straightening perpendicular to the trajectory to the center of the curve.

;

Let's spread the vector of accelerating for the tasks of the directions.

- Dotichne priskorennya.

Appointment: Dotichne accelerated - a vector value that characterizes the speed of change of the speed vector for the module.

- Vector submission.

- scalar appearance.

- Normally quickened.

Appointment: Normally accelerated characterizes the speed of change of the speed vector directly and is calculated according to the formula:

-de R-radius of curvature of the trajectory at point M

If the trajectory is a stake, then R is the radius of the stake.

For a scalar file:

From the powers of warehouses, it is more quickly vyplyvaє, scho is more quickly straightened out, killing the trajectory.

The module of total acceleration is more advanced:

Similarly for the vector full hearted:

The term "sweetness" is vicorous in science and in sensi, rozumіyuchi according to it swidkity change whether or not the value (not ob'yazykovo radius-vector) fallow in the other ). So, for example, we can talk about the coolness, the change in temperature, the change in temperature, the change in chemical reaction, the group’s change, the change in temperature, etc. Mathematically, “change in change” is characterized by a similar analyzed value.

Expansion of the understanding of swidkosti є chotirivimirna swidkіst, or swidkіst in relativistic mechanics of narrowed coordinates.

Swiftness of the point at the classical mechanics

de τ → ≡ d r → /d s (\displaystyle (\vec (\tau ))\equiv \mathrm (d) (\vec (r))/\mathrm (d) s)- a single vector of dotichno ї, scho to pass through the flow point of the trajectory s (\displaystyle s) ruhomoy point), and v τ ≡ s ˙ (\displaystyle v_(\tau )\equiv (\dot (s)))- projection of the speed vector on a straight line of a given single vector, which is the most favorable arc coordinate for the hour and is called algebraic skill points. Vіdpovіdno before guiding formulas, the vector of the sharpness of the point of the head of the straightening of the dot is dot, and the variability of the algebra of the dots can be adjusted in the form of the module v (\displaystyle v) which vector is no longer familiar. With whom:

Do not change the arc coordinate and passing by the point way. The way, passing by a point for the interval of an hour from before t (\displaystyle t), can be found like this:

if the algebraic speed of the point is not visible all the time, it is simple to reach the path and the arc coordinate: t 0 (\displaystyle t_(0)) before t (\displaystyle t)(however, if the cob follows the arc coordinates, it fluctuates from the cob positions of the point that collapses, then s ~ (\displaystyle (\tilde (s))) will run away s (\displaystyle s)).

Since the speed of the algebra of a point does not change from time to time (otherwise, the modulus of speed is constant), then the speed of the point is called equal(algebraically dotically faster s ¨ (\displaystyle (\ddot (s))) under which it is equal to zero).

Let's assume that s ¨ ≥ 0 (\displaystyle (\ddot (s))\geqslant (0)). However, in equal Russian, the speed of the point (algebraic) is more expensive than the passing path s ~ (\displaystyle (\tilde (s))) before the hour t − t 0 (\displaystyle t-t_(0)), for some way tsey buv passages:

In the wild type there are similar blues

signify vidpovidno medium speed specks that її middle school algebra; as the term "middle speed" is used, then about the values ​​\u200b\u200band s ˙ (\displaystyle (\dot (s))) to speak mittevikh shvidkostakh.

Illustration of medium and mitt

Not a trace of zmishuvati two introductions are more understandable of the average speed. First, v → c p (\displaystyle (\vec (v))^(\,\,\mathrm (cp) ))- Vector as well s ˙ c p (\displaystyle (\dot (s))^(\,\mathrm (cp) ))- Scalar. In another way, q values ​​can be increased modulo. So, let the point collapse; then the module of the average smoothness of the point is more expensive kroku the screw line (tobto vіdstanі mіzh її turns) for an hour of ruin, and the module of the average algebraic swiftness - for an hour of ruin.

For the body of lingering expansions, the understanding of the sharpness (the body is like this, and not just one point) cannot be assigned; winyatok to become a vipadok of a mittevo-progressive rush. It seems that the body of the mittevo-progressive movement is absolutely firm, as if at a given moment the speed of all storage points is equal; then you can, wisely, put the tightness of the body evenly tightness, be it some point. So, for example, equal speeds of all the points of the cabin of the sight wheel (like, obviously, swaying the cabin with the knocks).

At the savage mood of swidness there is a point that makes the body firm, not equal to itself. So, for example, for a wheel that rolls without slipping, the modulus of speed at the point on the front of the road takes the value from zero (at the point of the dot from the road) to the double value of the speed of the center of the wheel (at the point, diametrically opposite the point of the dot). The point of an absolutely rigid body has been subdivided into sharpness and is described by Euler's kinematic formula.

In Cartesian coordinates

At the same hour r = x i + y j + z k , (\displaystyle \mathbf(r) =x\mathbf(i) to that

In this order, the coordinates of the speed vector are the speed of changing the coordinates of the material point:

For cylindrical coordinates

Speed ​​in polar coordinates

V φ (\displaystyle v_(\varphi)) bear the name of the transverse swidkost, v R (\displaystyle v_(R))- radial.

For spherical coordinates

Uzagalnennya

The sharpness of the understanding of swidkosti є chotirivimirna shvidkіst, or the swidkіst of relativist mechanics, and the sharpness of swidkіst, or the swidkіst in narrowed coordinates.

Chotirivimirna shvidkist

Chotirivimirny vector of lightness is a time-like vector, so that it lies in the middle of the light cone.

In fixed coordinates

Sliding apart the coordinate and physical dimensions. With the introduction of curvilinear numbers of specified coordinates, the position of bodies is described by the current fallow time. Changes in body coordinates by the hour are called coordinate shifts.

Change of speed

In the classical mechanics of Newton, the shifts are transformed every hour of the transition from one inertial system to a different direction with the transformations of Galileo. How firm the body is in the system S (\displaystyle S) dorivnyuvala v → (\displaystyle (\vec (v))), and the speed of the system S′ (\displaystyle S") S (\displaystyle S) check the security of the body when switching to the system S′ (\displaystyle S") be equal

For those who are close to being light, the transformations of Galileo become unfair. When moving out of the system S (\displaystyle S) at the system S′ (\displaystyle S") it is necessary to twist Lorenz's transformation for shvidkost:

at popuschennі, sho swidkіst u → (\displaystyle (\vec (u))) straightened vzdovzh axis x (\displaystyle x) systems S (\displaystyle S). It is easy to cross over between the non-relativistic shifts of Lorentz's transformation to Galileo's transformation.

Po'yazanі understanding

A number of understanding of classical mechanics are expressed through the speed.

de U μ (\displaystyle U^(\mu ))- Uzagalnena chotirivimirna shvidkіst.

de m(\displaystyle\m)- body weight, v (\displaystyle\v)- Shvidkist to the center of the mass body, I (\displaystyle (\mathcal (I)))- moment of inertia of the body, ω → (\displaystyle (\vec (\omega )))- Kutova shvidkіst tіla.

Changes in speed by the hour are characterized by quickening. Speeding up the change of speed as for the magnitude (tangentially speeding up), so for direct (speeding up the pre-center):

de r(\displaystyle\r)- Radius of curvature of the point trajectory.

In the relativistic mechanics, between dottic and luminous lines, the parts and at all times in the basic system should be called swidkost (indicated by θ (\displaystyle \theta)). The speed is expressed by the formula:

de A r t h x (\displaystyle \mathrm (Arth) \,x)- Areatangent, or hyperbolic arctangent. Shvidkіst pragne neskіnchennosti if shvidkіst pragne shvidkostі light. On the vіdmіnu vіd shvidkostі, on yak it is necessary to koristuvatisya Lorenz's transformations, the swidіkіstіn is additive, tobto

de θ 0 (\displaystyle \theta _(0))- speed of the system S′ (\displaystyle S") schodo system vіdlіku S (\displaystyle S).

Deyakі shvidkostі

Space swissness

Light swedishness

Speed ​​of gravity

• Radiani per second, adopted in CІ and CGS systems. Physical expansion 1/s.
• Wrap in a second (technically)
• degrees per second degrees per second

Spivvіdnoshennia between units of swedkostі

• 1 m/s = 3.6 km/year
• 1 vuzol \u003d 1.852 km / year \u003d 0.514 m / s
• Mach 1 ~ 330 m/s ~ 1200 km/year
• c= 299 792 458 m/s

historical drawing

Two stages of the body thrown to the floor according to the theory of Avetsenni: AB – the period of “violent exercise”, BC – the period of “natural exercise” (falling vertically down)

In 1328, after reading the book “A Treatise on Proportions or on the Proportions of Sweeps in Russia” by Thomas Bradwardin, he knew the inconsistency of Aristotle’s physics and the connection of sweds with ferocious forces. Bradvardin, who was, for the verbal formula of Aristotle Yakshcho Rushiina, the force of Dorivnyuh, then Shvidkіst Dorivnyu 1, Todi Yak Vona Dorivnuvati 0. Vin, introducing his formula for the snakes, was not overwhelmed staleness of dryness due to the reasons for the collapse. Bradwardin called swidkist "kіlkіstyu ruhu". William Haytsbury, in the treatise "About Mistsevy Rukh", instilling understanding of Mity's shvidkost. In 1330-1340, the years of the wines and other scholars of Bradwardin brought it to the title of "Merton's rule", as it means the equanimity of the way in the case of an evenly accelerated Rus and an equal Rus from the middle swidkist.

Whether it’s the breadth of the fluff, which is uniformly bathed or worn out, it’s consistent with its average degree, so that the styles will definitely pass the winds of this latitude, which is to be bathed, the skils and the winds of the middle degree, yakby the middle degree collapsed the whole hour.

In 1609, Kepler formulated the law of area in the work of "Nova Astronomy", for which the sectoral speed of the planet (the area described by the planet - the Sun, for one hour) became. In "The Cobs of Philosophy", Descartes formulated the law of saving the amount of ruha, as in the case of some sensible one, he added the amount of matter to the swedishness, while Descartes did not take to respect the fact that the amount of ruhu could not be more than magnitude, but directly. Nadali understood "a lot of movement" having developed Guk, a kind of mind in yoga like "steps of speed, powerful singing of speech." Huygens, Wallis and Rehn got straight to the point of being appointed. In such a look, in the other half of the 17th century, a lot of movement has become an important understanding of dynamics, zokrema of robots

The position of a material point in space at a given moment is determined according to the time to be some other body, as it is called tіlom vіdlіku.

Contact him feedback system- The complexity of the coordinate system and years, connected with the body, according to the ratio to which the ruh is twisted, whether there are any other material points. The choice of the system should be deposited on the date of the due date. In case of kinematic achievements, the systems are considered equal (cartesian, polar). In the tasks of dynamics, the role of play is more important inertial systems, In terms of some kind of differential rіvnyannya Rukh may have a more simple look.

In the Cartesian coordinate system, the position of the point BUT at a given moment, according to the date to the point of the system, three coordinates are indicated X, atі z, which is the radius vector (Fig. 1.1). With Russian material points її coordinates change from time to time. Zagalom її ruh is recognized as equal

or vector equals

=(t). (1.2)

Qi equal are called kinematic equals to the movement material point.

including hour t in the system equals (1.1), equals are taken away trajectories of movement material point. For example, as a kinematic equal to the movement of the task points in the form:

those including t, We take:

tobto. the point collapses near the plane z= 0 along an elliptical trajectory with equal trajectories aі b.

Trajectory of movement material point is called a line, which is described by a point in space. Fallow in the form of a trajectory ruh can buti rectilinearі curvilinear.

Let's look at the material points of the fair trajectory AB(Fig. 1.2). We’ll wait for an hour, if the point changed in position BUT (t= 0). Dovzhina dilyanka trajectory AB passed by the material point at the moment t= 0, called dozhina way and є scalar function of the hour. The vector , drawn from the cob position of the point, which collapses into position її at a given moment, is called moving vector. In the case of a rectilinear Russian, the vector of displacement changes from a straight line trajectory and the same module to the passed path.

Shvidkist- tse vector physical quantity, the designation of speed is introduced and її directly for a given hour.

Let the material point collapse in a curvilinear trajectory and at the moment of the hour tїй відпоідє radius-vector. (Fig. 1.3). With a small interval of an hour, the point passed the way and took away a little less movement. Razrіznyayut the middle and mitteva shvidkostі.

Average speed vector called the increase in the radius-vector of the point before the interval of the hour:

The straightening vector is so self, yak. With an unrestricted change in the average speed, the value of the load is up to the limit value, as it is called mitteva shvidkistya or just shvidkistyu:

In this rank, speed is a vector quantity, which is the best of the first pohіdnіy radius-vector of a point, which collapses after an hour. So, as a rule, in between, it moves from the dot, the vector of speed of directing along the dot to the trajectory in the bіk rush.

In the world of change, the dozhina of the arc is getting closer and closer to the douzhina of the chord, which draws it, tobto. the numerical value of the security of the material point of the road to the first good day of the її way by the hour:

in such a manner,

From the point of view (1.5) we can integrate over an hour from before, we know a long way, passed by a material point in an hour:

Even though the vector of the mitt's speed does not change directly, the material point does not change, which means that the point is collapsing along the trajectory, dotichni up to that at all points, one and the same straight line. Such power can be less than straight-line trajectories. Otzhe, there will be analyzes rectilinear.

Even though the vector of the speed of the material point changes from time to time, the point is described curvilinear trajectory.

If the numerical value of the mitte of the point is lost for an hour, the ruh is constant, such a ruh is called equal. In what direction

Tse means that for a fairly equal time, the material point is to pass the path of a equal time.

As for quite a few equal intervals, the point to pass the paths of different times, the numerical value of the speed changes with the hour. Such a ruh is called uneven. At this point, they are coryzed with a scalar value, as they are called average swedishness of uneven movement on this distance trajectory. There is more to the numerical value of the speed of such an even rіvnomіrny ruh, with the same hour on the passage of the road, as for a given rіvnomirny rukh:

As a material point, one hour taking a fate from a few revolutions, then independence lawїї resulting displacement of one vector sumi relocation, which zdіysnyuyutsya her for that very hour at the skin s ruhіv okremo. For this reason, the speed of the resultant movement is like a vector sum of the strengths of all such flows, for which a material point takes its fate.

Nature tends to be more wary of changes, in some, the speed changes like in magnitude (modulus), and in a straight line, tobto. to be brought to the right of the mother from the nerves of the hands. To characterize the change in the speed of such changes, an understanding is required prikorennya.

Let in an hour the point that is collapsing has crossed from the camp BUT at the camp At(Fig. 1.4). The vector sets the speed of the point at the position BUT. At the position At the point priddbala shvidkіst, vіdminnu vіd yak for size, so i for direct i became equal. Transferable vector u mottle BUT i know.

Middle aged of uneven movement in the interval of hours from to the vector value is called, which is good for changing the speed up to the interval of the hour:

It is obvious that the vector zbіgaєtsya at the straight line of the vector zmіni swidkosti .

Mittevim priskorennyam or I'm sorry material points at the time of the hour will be between the average acceleration:

In this rank, sooner is a vector quantity, which is more expensive than the first good windfall by the hour.

We will store the vector in two warehouses. For which point BUT for direct swidkost_ vіdklademo the vector, for the modulus equal. Same vector, equal, signifying change of speed behind the module(values) pid hour, tobto. . Another warehouse vector characterizes the change in speed per hour off straight - .

Warehouse accretion, which means the change of speed for the size, is called tangential warehouse. Numerically won't cost the first hour after the hour from the speed module:

We know a friend of a warehouse, as soon as it is called normal warehouse. Let's say, what's the point At close to the point BUT that way can be vvazhat an arc of a stake of a certain radius. r, little vіdіznyаєєєі vіd khordi AB. Z similar trikutnikov AOBі ЄAD screaming what

zvіdki That friend's warehouse has an early morning:

Won behind the straight line and straightened to the center of curvature of the trajectory behind the normal. Її call it to the center of the roots.

Out of my mind body is the geometric sum of tangential and normal warehouses:

3 fig. 1.5 next, that the booster module is good:

Directly povnogo priskrennya vyznaetsya kutom mizh vectors ta. Obviously what

The fallow value of the tangential and normal warehouse accelerations of the body is classified differently. Yakshcho (the size of the speed does not change with the size), ruh є equal. If > 0, ruh is called hurry up, like< 0 - uplifting. Yaxcho = const0, then ruh is called equal. Zreshtoy, be it a straightforward Russian (you can’t change your swidkost straight away).

In this order, the momentum of a material point can be attacked:

1) - rectilinear equal movement ();

2) - rectilinear equal movement. With such a sight

Yakshcho pochatkovy hour, and pochatkovy swidkіst, then, knowing i, we take:

stars. (1.16)

Having integrated the virase at the borders from zero to a sufficient hour, we take away the formula for the recognition of the long way passed by the point in equal time Russia:

3) - rectilinear movement from the change of speed;

4) - the modulus of curvature does not change, the stars show that the radius of curvature can be constant. Otzhe, tsey ruh according to the stake є equal;

5) - rіvnomirny curvilinear movement;

6) - curvilinear rіvnozminniy ruh;

7) - curvilinear movement from the change of speed.

Kinematics of the wraparound motion of a solid body

As it was meant to be, the wrapping motion of an absolutely solid body is called such a motion, in which all points of the body collapse at the planes perpendicular to the non-violent straight line, called the wrapping, and describe the stake, the centers of which lie on the axis.

It is hard to look at the body, which wraps around a long, unbreakable axis (Fig. 1.6). Also, around the points of this body, describe the number of different radii, the centers of which lie on the axis of the wrap. Let deyak point A collapse along the stake radius R. Її position through the interval of an hour at a time.

Kutovoy swidkistyu The wrapping is called a vector, numerically equal to the first step of the turn of the body by the hour and directing the axis of the wrapping following the rule of the right screw:

One unit of the top speed is radians per second (rad/s).

In this rank, the vector signifies directly that swidkiness wrapping. Yakshcho, then the wrapper is called equal.

Kutova swidkіst mozhe buti pov'azana z linear swidkistyu prevіlnoї point A. Let the point for an hour to pass through the arc of the stake dozhina way. The same line speed of the point is more expensive:

With equal wrapping of yoga, one can characterize wrapping period T- an hour, stretching a point of the body to make one turn, tobto. turn on cut 2π:

The number of new turns, which are called by the body in equal Russia according to the stake, is called in one hour wrapping frequency:

To characterize the uneven wrapping of the body, the understanding apex root. The top speed is called the vector quantity, which is the cost of the first windy top speed per hour:

When wrapping the body on a slightly unbreakable axis, the vector of the apex acceleration straightening of the uzdovzh axis is wrapped around the vec- tor of the apex of the smoothness (Fig. 1.7); with accelerated rus, the directing vector for the same bіk, yak і, and y for the protolezhnu bіk with improved wrapping.

Vislovimo tangential and normal warehouse speeding specks BUT the body that wraps around the top of the swedishness and the top of the quickening:

In a different equal movement, points on a stake ():

de pochatkova kutova shvidkіst.

Progressive and wraparound movement of a solid body is just the simplest types of yoga movement. At the tip of the hand of a solid body, we can also fold it. However, theoretical mechanics it can be argued that if a smoother movement of a solid body is possible, it is possible as a succession of progressive and wrapping movement.

Kinematic alignment of translational and overt rotational links in Table. 1.1.

Table 1.1

Short loops:

A part of physics that develops laws mechanical movement and the reasons that call or change this ruh, are called mechanics. Classical mechanics (mechanics of Newton-Galiley) play the laws of macroscopic bodies, the swiddenness of which is small, equal to the swedish light of the vacuum.

- Kinematic- Razdіl mechanics, the subject of vyvchennya є ruh tіl without looking at the reasons, yak tsey ruh umovleno.

In the mechanics for describing the flow of tіl in the fallow, in the minds of specific heads of the vikoristovuyutsya different physical models : material point, absolutely hard body, absolutely spring body, absolutely spring body

Ruh tіl vіdbuvaєtsya at the expanse of that at the hour. Therefore, to describe the flow of the material point, it is necessary to know, in some places the expanse of the point changed and in the same hour it passed those other positions. The sequence of the body according to the coordinate system connected with it and the synchronization between itself is called the year system.

The vector , drawn from the cob position of the point, which collapses into position її at a given moment, is called moving vector. A line that can be described by a rough material point (body) of a chosen system is called trajectory. Fallow in the form of trajectories are divided straightforwardі curvilinear ruh. The length of the trajectory, passed by the material point for the same interval of time, is called dozhina way.

- Shvidkist- is a vector physical quantity that characterizes the speed of movement and yoga directly at a given moment. Mitteva shvidkist is determined by the first moving radius-vector of the moving point by the hour:

vector The module of the mittvoї svidkostі materialії point dоrіvnyuє lіvіy hіdnіy її її її її shlyakhu by o'clock:

- priskorennya- Vector physical quantity for the characteristic uneven rush. It will determine the speed of changing the speed for the module directly. Mitteve is sorry- Vector value, which is the cost of the first cold weather per hour:

Tangential warehouse acceleration characterize the speed of change of speed for the magnitude(Straightened along dotichny to trajectory move):

Normal storage speed characterize the speed of change of speed off straight(Straightened to the center of curvature of the trajectory):

Out of my mind in curvilinear Russia - the geometric sum of tangential and normal warehouses:

3. What is the review system? What is called a displacement vector?

4. Which movement is called progressive? Overwhelming?

5. What characterizes speed and speed? Give the meaning of the middle speed and the middle speed, the mitt's speed and the mitt's speed.

6. Lay down the equal trajectory of the movement of the body, thrown horizontally with a swidkistyu v 0 with a deak height. Opir again do not lie.

7. What characterizes the tangential and normal warehouse acceleration? What are your modules?

8. How can you classify ruh fallow in terms of tangential and normal storage rates?

9. What is called kutovoy shvidkistyu and kutovy prikorennyam? How are these direct lines determined?

10. What formulas are used to describe the linear and cult characteristics of the movement?

Apply the solution of tasks

Head 1. Nehtuyuyuyu support povіtrya, vyznachiti kut, under which the body is thrown to the horizon, as if the maximum height of the body is 1/4 of the distance of the flight (Fig. 1.8).

Shvidkist

The average swedishness often characterizes the swedishness of the rush for the end of the hour. Without changing the gap, we will come to a physical value, which characterizes the speed of movement at a given moment. Such a value is called mitteva shvidkistyu or simply shvidkistyu:

denotes a mathematical operation on the cob between. Under this symbol, the mind is written, for which the boundary crossing is given; in this period of time, exercise to zero in an hour. With the calculation of the average speed, according to this rule, we are changing the change in the hour to the point where, at some stage, the average value of the average speed is reduced less and less, one type of one. Therefore, in practice, with the knowledge of the swidkost, you can surf on final meaning, Sufficiently small to obtain the necessary accuracy of the value of speed.

Speed ​​vector and trajectory.

Looking at the boundary crossing may be clear geometric zmist. If the vector of displacement of straight lines along the chord, which connects two points of the trajectory, then when these points are approached, which is taken at, the position is taken, which leads to the trajectory in this point. Tse means that the vector of straightness of straightening along the trajectory. So it will be at any point of the trajectory (Fig. 14). With a rectilinear trajectory, the vector of the straightness of the straightening of the bridle of the straight line.

Shvidkist passage way.

A similar transition marks the mitteva of the passage of the path:

For a smooth curve, which is the trajectory of any kind of uninterrupted mechanical movement, the length of the arc is less disturbed by the length of the chord, which is shorter, the arc is shorter. At the borders, the dozhini are escaping. So you can vvazhati, scho. Tse means that the speed of the passage of the road is closer to the module of the mitt's speed. Rukh, for which the module of security is immutable, it is called equal. In times of rectilinear trajectory in equal Russia, the vector of rigidity is constant, and in times of curvilinear trajectories it changes less than straight ahead.

Storage of swishes.

Yakshcho body at the same time taking a part in a lot of ruhs, yogo swidkіst dоrіvnyuє vector sumі ї shvidkost dermal z іh ruhіv. Tse without middle vyplivaє z the rules of the folding of the movement: oskіlki, then after podіlu on otrimuєmo

In other words, you can manually show the deuce of folding as a superposition, i.e., the superimposition of two simple ones. For this type of property (3) it is possible to interpret, as a rule, the distribution of the property vector in warehouses.

Manager.

Crossing over the river. The speed of the flow near the rivers with parallel banks is the same and fresh. The width of the river (Fig. 15). The boat can sail zі shvidkіstyu schodo drive. How do you get the boat down the river flow, so when crossing the boat straight across the banks?

The boat takes its fate at the same time at two ruhs: zі swidkіst, straightened across the flow, and at once from the water zі vіdkіstyu, yak straightened parallel to the birch. Vіdpovіdno up to the rule of folded shvidkost povnі kіst of the boat shkodo cherіv dоrіvnіuє vector sumі (Fig. 16). It is obvious that the boat moves in a straight line, directing the bridle of the vector. Vіdstan s, on the way of the boat at the crossing, you can know from the likeness of a tricot, adorned with swidkost vectors:

Tse zavdannya is easy to break and do not go to the folding of vectors of swedes. It is obvious that you can get a better flow of water leaks time by pulling a boat like a river. You can know what time it is by dividing the width of the river on the speed of the boat across the river. Father, we know Mal. 16. Storage of swedes when crossing. However, even with a slight complication of the mind of the task, one can clearly see the advantages of the first method, which is based on the fold of vectors of swidkost.

2. Crossing across the river. Let's assume that now we need to cross the boat across the same river exactly across, so we can get to point B, which lies opposite the cob point A (Fig. 17). How to direct the boat at the crossing? How long will such a crossing take? Solution. In this way, the speed is clear - the boats are well shored, the vector sum of the speed is equal, but it is straight across the river.

3 fig. 17 it is clearly seen that the vector, which is a pleasure to marvel at from the boat, is guilty of climbing a deaky kut uphill along the river flow straight ahead. The sinus of this kuta is more expensive to use the modules of the flow rate and the boats should be driven well. Crossing across the river without crossing is possible only in the fall, if the speed of the boat can be driven more for the speed of the leak. You can clearly see it from the tricot of shvidkos in fig. 17 (the hypotenuse is greater than the leg), or formulas (the sine of the leg is guilty of less than one). We know the time to cross the river, dividing the width of the river to the full speed of the boat according to the Pythagorean theorem.

3. Znesennya in case of a swedish leak.

It is acceptable now that the speed of the boat should be driven less for the speed of the leak: In such a time, the crossing is impossible without a toll. How to straighten out the boats for an hour to cross, so that the rise was minimal? How do you get a boat? Solution. Povna shvidkіst shkodo cherіv in all fluctuations is given by the formula. However, now it’s better to add the vector in and according to the rule of the tricot (Fig. 18), the first image is the capital for which we know the module directly, and then until the end of the year, we will arrive at the beginning of the vector, only the module, it is also necessary to choose directly. Tsej vybir need to grow so, the vector of the resultant swidkost yaknay less vydhilyavsya in a straight line across the river.

Rice. 19. Dedicated to the course (directly of the vector) ferry to the minimum znesennya 18. Folding the fasteners of the ferry Kіnets be-yakoy directly guilty of lying on the radius of the center of which the end of the vector. The whole is shown So the mind of the task is that the point is clear to the cob to lie in the position of the cob. From the little one it is clear that I am making the smallest direct kut todi, if the wines are straightened dotically. Otzhe, perpendicular to the vector is a right-angle tricot. In such a manner, direct the flow uphill under the cut of the line. it is perpendicular to a straight line, in which to marvel at the boat. Tse means that the boat is collapsing sideways on its trajectory. another birch river to moor points, docks to know similar trikutniks. The module is found in the Pythagorean theorem. results are acceptable

4. Choven cable. A chauvin is lifted by a cable to tie a nose, winding a drum. what kind of swidkistyu chauvin that moment, the cable horizon? The cable is vibrated by the swidkist drum.

Solution.

A speck of rope, de vin tying to a chute, crumbles with tієyu w svidkіst, like a chauvin. Tsya swidk_st straightened horizontally. Sob'yazat її zі svidkіstyu vibirannya cable, it is necessary to bend it, so that the roar of the cable is wound up to the turn around the point, de vin the drum hangs, and forging the air of the wet straight, that is straight. Therefore, it is natural to lay out the swidk_st points on two warehouses, straightening the air and across the cable (Fig. 21). Shvidkist, straightened across, tied with a turn of the cable. The modulus of the straightness of the direct air cable - ce and є given for understanding the specified value of the speed.

The world has more closeness to the shore. Tse means that cos and change and shukana speed increases. The task for the independent vision of Lyudin is to be found in the field on the front of the straight road. Angrily, looking at yourself, you mark the car that is crashing along the highway. In what direction did you go straight to the highway, to get on the road in front of the car and how did you see it? The speed of the car and the speed of the people.

Explain why the vector of the speed of directing along the trajectory.

In some cases, the trajectory of the rush can often be the mother of evils. Bring examples of such ruhіv. What can you say about the straight forwardness at the points, de trajectory maє evils?

In times of uninterrupted mechanical movement, the vector of mobility does not depend on strings, neither behind the module, nor behind the direct one. Appearance of a streamlined swidkost zavzhdy pov'yazani z deakoy іdealіzієyu real protsess. What kind of idealizations were present in the butts you aimed at trajectories from evils?

Find a pardon in the tasks listed below 4. We place the fastness, the points of the cable on the vertical and horizontal warehouses (Fig. 22). Horizontal warehouse and є shukana shvidk_st chovna. To that i (wrong!).

Shvidkіst like pokhіdna.

Let's turn to virazu (1) for mitteva swidkost. When the particles are Russian, the radius-vector r changes, so that it is a function of the hour. Moving Dg for the interval of an hour At є with a difference in radius-vectors at the moment and hour. Therefore, formula (1) can be rewritten in a mathematical way. In mathematics, such a value is called a similar function for an hour. For it, such a value is known. The rest of the designation (a dot over the letter) is more characteristic of itself for the hour. It is significant that in to this particular type similar to a vector, shards appear in the results of differentiation of a vector function behind a scalar argument. For the module of mitt'voї shvidkostі vіdpovіdno up to fair virazu on the cob of statі.