If the acceleration of a material point at all times is zero, then the speed of its movement is constant in magnitude and in direction. The trajectory in this case is a straight line. The motion of a material point under the formulated conditions is called uniform rectilinear. With rectilinear motion, the centripetal component of acceleration is absent, and since the motion is uniform, the tangential component of acceleration is zero.
If the acceleration remains constant in time (), then the movement is called equally variable or uneven. Equally variable motion can be uniformly accelerated if a > 0, and equally slow if a< 0. В этом случае мгновенное ускорение оказывается равным среднему ускорению за любой промежуток времени. Тогда из формулы (1.5) следует а = Dv/Dt = (v-v o)/t, откуда
(1.7)
where v o - initial speed at t=0, v - speed at time t.
According to formula (1.4) ds = vdt. Then 
Since for uniform motion a=const, then
(1.8)
Formulas (1.7) and (1.8) are valid not only for uniformly variable (nonuniform) rectilinear motion, but also for the free fall of a body and for the motion of a body thrown upwards. In the last two cases, a \u003d g \u003d 9.81 m / s 2.
For uniform rectilinear motion v = v o = const, a = 0, and formula (1.8) takes the form s = vt.
Circular motion is the simplest case of curvilinear motion. The speed v of movement of a material point along a circle is called linear. With a constant modulo linear velocity, the motion in a circle is uniform. There is no tangential acceleration of a material point during uniform motion along a circle, and t \u003d 0. This means that there is no change in speed modulo. The change in the linear velocity vector in the direction is characterized by normal acceleration, and n ¹ 0. At each point of the circular trajectory, the vector a n is directed along the radius to the center of the circle.
and n \u003d v 2 / R, m / s 2. (1.9)
The resulting acceleration is indeed centripetal (normal), since at Dt->0 Dj also tends to zero (Dj->0) and the vectors and will be directed along the radius of the circle to its center.
Along with the linear velocity v, the uniform motion of a material point along a circle is characterized by an angular velocity. The angular velocity is the ratio of the angle of rotation Dj of the radius vector to the time interval during which this rotation occurred,
Rad/s (1.10)
For uneven motion, the concept of instantaneous angular velocity is used
.
The time interval t, during which the material point makes one complete revolution around the circumference, is called the rotation period, and the reciprocal of the period is the rotation frequency: n \u003d 1 / T, s -1.
For one period, the angle of rotation of the radius vector of a material point is 2π rad, therefore, Dt \u003d T, whence the rotation period, and the angular velocity is a function of the period or frequency of rotation
It is known that with a uniform motion of a material point along a circle, the path traveled by it depends on the time of movement and linear speed: s = vt, m. The path that a material point passes along a circle with radius R, for a period, is equal to 2πR. The time required for this is equal to the period of rotation, that is, t \u003d T. And, therefore,
2πR = vT, m (1.11)
and v = 2nR/T = 2πnR, m/s. Since the angle of rotation of the radius vector of a material point during the rotation period T is equal to 2π, then, based on (1.10), with Dt = T, . Substituting into (1.11), we obtain and from here we find the relationship between the linear and angular velocity
Angular velocity is a vector quantity. The angular velocity vector is directed from the center of the circle along which the material point moves with linear velocity v, perpendicular to the plane of the circle according to the rule of the right screw.
With non-uniform motion of a material point along a circle, the linear and angular velocities change. By analogy with linear acceleration, in this case, the concept of average angular acceleration and instantaneous is introduced: 
. The relation between tangential and angular accelerations has the form .
The action of a force on a body in some cases can lead to a change only in the modulus of the velocity vector of this body, and in others - to a change in the direction of the velocity. Let's show this with examples.
Figure 34, a shows a ball lying on the table at point A. The ball is tied to one of the ends of the rubber cord. The second end of the cord is attached to the table at point O. If the ball is moved to point B, the cord will stretch. In this case, an elastic force F will appear in it, acting on the ball and tending to return it to its original position.
If we now release the ball, then under the action of force F it will accelerate towards point A. In this case, the speed of the ball at any point of the trajectory (for example, at point C) is co-directed with the elastic force and the acceleration resulting from the action of this force. In this case, only the modulus of the ball's velocity vector changes, while the direction of the velocity vector remains unchanged, and the ball moves in a straight line.
Rice. 34. If the speed of the body and the force acting on it are directed along one straight line, then the body moves rectilinearly, and if they are directed along intersecting lines, the body moves curvilinearly
Now consider an example in which, under the action of an elastic force, the ball moves curvilinearly (i.e., the trajectory of its movement is a curved line). Figure 34, b shows the same ball on a rubber cord, lying at point A. Let's push the ball to point B, i.e., give it an initial speed directed perpendicular to the segment O A. If no forces acted on the ball, then it would retain the magnitude and direction of the resulting velocity (remember the phenomenon of inertia). But, moving to point B, the ball moves away from point O and slightly stretches the cord. Therefore, an elastic force F arises in the cord, seeking to shorten it to its original length and at the same time bring the ball closer to point O. As a result of this force, the direction of the ball’s velocity at each moment of its movement changes slightly, so it moves along a curvilinear trajectory AC. At any point of the trajectory (for example, at point C), the speed of the ball v and the force F are directed along intersecting lines: the speed is tangential to the trajectory, and the force is directed towards point O.
The considered examples show that the action of a force on a body can lead to different results depending on the direction of the velocity and force vectors.
If the speed of the body and the force acting on it are directed along one straight line, then the body moves rectilinearly, and if they are directed along intersecting lines, then the body moves curvilinearly.
The converse statement is also true: if the body moves curvilinearly, then this means that some kind of force acts on it, changing the direction of the velocity, and at each point the force and velocity are directed along intersecting straight lines.
There are countless different curvilinear trajectories. But often curved lines, such as the line ABCDEF (Fig. 35), can be represented as a set of arcs of circles of different radii.
Rice. 35. Trajectory ABCDEF can be represented as a set of arcs of circles of different radii
Therefore, in many cases, the study of the curvilinear motion of a body is reduced to the study of its motion in a circle.
Questions
- Consider Figure 34, and answer the questions: under the influence of what force does the ball acquire speed and move from point B to point A? What caused this power? What is the direction of the acceleration, the speed of the ball and the force acting on it? What is the trajectory of the ball?
- Consider Figure 34, C answer the questions: why did the elastic force arise in the cord and how is it directed in relation to the cord itself? What can be said about the direction of the ball's velocity and the elastic force of the cord acting on it? How does the ball move - straight or curved?
- Under what condition does a body move in a straight line under the action of a force, and under what condition does it move in a curvilinear direction?
Exercise 17
With the help of this lesson, you will be able to independently study the topic “Rectilinear and curvilinear motion. The motion of a body in a circle with a constant modulo velocity. First, we characterize rectilinear and curvilinear motion by considering how the velocity vector and the force applied to the body are related in these types of motion. Next, we consider a special case when the body moves along a circle with a constant modulo speed.
In the previous lesson, we considered issues related to the law of universal gravitation. The topic of today's lesson is closely related to this law, we will turn to the uniform motion of a body in a circle.
Earlier we said that motion - this is a change in the position of a body in space relative to other bodies over time. Movement and direction of movement are characterized, among other things, by speed. The change in speed and the type of movement itself are associated with the action of a force. If a force acts on a body, then the body changes its speed.
If the force is directed parallel to the movement of the body, then such a movement will be straightforward(Fig. 1).
Rice. 1. Rectilinear motion
curvilinear there will be such a movement when the speed of the body and the force applied to this body are directed relative to each other at a certain angle (Fig. 2). In this case, the speed will change its direction.
Rice. 2. Curvilinear motion
So, at rectilinear motion the velocity vector is directed in the same direction as the force applied to the body. BUT curvilinear movement is such a movement when the velocity vector and the force applied to the body are located at some angle to each other.
Consider a special case of curvilinear motion, when the body moves in a circle with a constant speed in absolute value. When a body moves in a circle at a constant speed, only the direction of the speed changes. Modulo it remains constant, but the direction of the velocity changes. Such a change in speed leads to the presence of an acceleration in the body, which is called centripetal.
Rice. 6. Movement along a curved path
If the trajectory of the body's motion is a curve, then it can be represented as a set of motions along arcs of circles, as shown in Fig. 6.
On fig. 7 shows how the direction of the velocity vector changes. The speed during such a movement is directed tangentially to the circle along the arc of which the body moves. Thus, its direction is constantly changing. Even if the modulo speed remains constant, a change in speed leads to an acceleration:
In this case acceleration will be directed towards the center of the circle. That is why it is called centripetal.
Why is centripetal acceleration directed towards the center?
Recall that if a body moves along a curved path, then its velocity is tangential. Velocity is a vector quantity. A vector has a numerical value and a direction. The speed as the body moves continuously changes its direction. That is, the difference in speeds at different points in time will not be equal to zero (), in contrast to a rectilinear uniform motion.
So, we have a change in speed over a certain period of time. Relation to is acceleration. We come to the conclusion that, even if the speed does not change in absolute value, a body that performs uniform motion in a circle has an acceleration.
Where is this acceleration directed? Consider Fig. 3. Some body moves curvilinearly (in an arc). The speed of the body at points 1 and 2 is tangential. The body moves uniformly, that is, the modules of the velocities are equal: , but the directions of the velocities do not coincide.
Rice. 3. Movement of the body in a circle
Subtract the speed from and get the vector . To do this, you need to connect the beginnings of both vectors. In parallel, we move the vector to the beginning of the vector . We build up to a triangle. The third side of the triangle will be the velocity difference vector (Fig. 4).
Rice. 4. Velocity difference vector
The vector is directed towards the circle.
Consider a triangle formed by the velocity vectors and the difference vector (Fig. 5).
Rice. 5. Triangle formed by velocity vectors
This triangle is isosceles (velocity modules are equal). So the angles at the base are equal. Let's write the equation for the sum of the angles of a triangle:
Find out where the acceleration is directed at a given point of the trajectory. To do this, we begin to bring point 2 closer to point 1. With such an unlimited diligence, the angle will tend to 0, and the angle - to. The angle between the velocity change vector and the velocity vector itself is . The speed is directed tangentially, and the velocity change vector is directed towards the center of the circle. This means that the acceleration is also directed towards the center of the circle. That is why this acceleration is called centripetal.
How to find centripetal acceleration?
Consider the trajectory along which the body moves. In this case, this is an arc of a circle (Fig. 8).
Rice. 8. Movement of the body in a circle
The figure shows two triangles: a triangle formed by the velocities, and a triangle formed by the radii and the displacement vector. If points 1 and 2 are very close, then the displacement vector will be the same as the path vector. Both triangles are isosceles with the same vertex angles. So the triangles are similar. This means that the corresponding sides of the triangles are in the same ratio:
The displacement is equal to the product of speed and time: . Substituting this formula, you can get the following expression for centripetal acceleration:
Angular velocity denoted by the Greek letter omega (ω), it indicates at what angle the body rotates per unit time (Fig. 9). This is the magnitude of the arc, in degrees, traversed by the body in some time.
Rice. 9. Angular speed
Note that if a rigid body rotates, then the angular velocity for any points on this body will be a constant value. The point is closer to the center of rotation or farther - it does not matter, that is, it does not depend on the radius.
The unit of measurement in this case will be either degrees per second (), or radians per second (). Often the word "radian" is not written, but simply written. For example, let's find what the angular velocity of the Earth is. The earth makes a full rotation in one hour, and in this case we can say that the angular velocity is equal to:
Also pay attention to the relationship between angular and linear velocities:
The linear speed is directly proportional to the radius. The larger the radius, the greater the linear speed. Thus, moving away from the center of rotation, we increase our linear speed.
It should be noted that motion in a circle at a constant speed is a special case of motion. However, circular motion can also be uneven. The speed can change not only in direction and remain the same in absolute value, but also change in its value, i.e., in addition to changing direction, there is also a change in the speed module. In this case, we are talking about the so-called accelerated circular motion.
What is a radian?
There are two units for measuring angles: degrees and radians. In physics, as a rule, the radian measure of an angle is the main one.
Let's construct a central angle , which relies on an arc of length .
mechanical movement. Relativity of mechanical motion. Reference system
Mechanical movement is understood as a change over time in the relative position of bodies or their parts in space: for example, the movement of celestial bodies, fluctuations in the earth's crust, air and sea currents, the movement of aircraft and vehicles, machines and mechanisms, deformation of structural elements and structures, movement liquids and gases, etc.
Relativity of mechanical motion
We have been familiar with the relativity of mechanical motion since childhood. So, sitting in a train and watching a train moving away, which had previously stood on a parallel track, we often cannot determine which of the trains actually started moving. And here it should immediately be clarified: to move relative to what? Regarding the Earth, of course. Because we started moving relative to the neighboring train, regardless of which of the trains started its movement relative to the Earth.
The relativity of mechanical motion lies in the relativity of the speeds of movement of bodies: the speeds of bodies relative to different reference systems will be different (the speed of a person moving in a train, steamer, airplane will differ both in magnitude and in direction, depending on which reference system these speeds are determined: in the frame of reference associated with a moving vehicle, or with a stationary Earth).
The trajectories of the motion of the body in different frames of reference will also be different. So, for example, raindrops falling vertically on the ground will leave a trail in the form of oblique jets on the window of a rushing train. In the same way, any point on the rotating propeller of a flying aircraft or a helicopter descending to the ground describes a circle relative to the aircraft and a much more complex curve - a helix relative to the Earth. Thus, in mechanical motion, the trajectory of motion is also relative.
The path traveled by the body also depends on the frame of reference. Returning to the same passenger sitting on the train, we understand that the distance traveled by him relative to the train during the trip is equal to zero (if he did not move around the car) or, in any case, much less than the distance that he covered together with the train relative to the Earth. Thus, in mechanical motion, the path is also relative.
Awareness of the relativity of mechanical motion (that is, the fact that the motion of a body can be considered in different frames of reference) led to the transition from the geocentric system of the world of Ptolemy to the heliocentric system of Copernicus. Ptolemy, following the movement of the Sun and stars in the sky observed since ancient times, placed the motionless Earth in the center of the Universe with the rest of the celestial bodies rotating around it. Copernicus also believed that the Earth and other planets revolve around the Sun and simultaneously around their axes.
Thus, the change in the reference system (the Earth - in the geocentric system of the world and the Sun - in the heliocentric one) led to a much more progressive heliocentric system, which makes it possible to solve many scientific and applied problems of astronomy and change the views of mankind on the Universe.
The coordinate system $X, Y, Z$, the body of reference with which it is connected, and the device for measuring time (clock) form a reference frame, relative to which the movement of the body is considered.
reference body a body is called, with respect to which a change in the position of other bodies in space is considered.
The reference system can be chosen arbitrarily. In kinematic studies, all frames of reference are equal. In problems of dynamics, any arbitrarily moving frames of reference can also be used, but inertial frames of reference are most convenient, since the motion characteristics in them have a simpler form.
Material point
A material point is an object of negligible size, having a mass.
The concept of "material point" is introduced to describe (with the help of mathematical formulas) the mechanical motion of bodies. This is done because it is easier to describe the motion of a point than of a real body, the particles of which, moreover, can move at different speeds (for example, during rotation of the body or deformations).
If a real body is replaced by a material point, then the mass of this body is attributed to this point, but its dimensions are neglected, and at the same time, the difference in the characteristics of the movement of its points (velocities, accelerations, etc.), if any, is neglected. In what cases can this be done?
Almost any body can be considered as a material point if the distances traveled by the points of the body are very large compared to its dimensions.
For example, the Earth and other planets are considered material points when studying their movement around the Sun. In this case, the differences in the movement of various points of any planet, caused by its daily rotation, do not affect the quantities describing the annual movement.
Therefore, if in the studied motion of a body its rotation around an axis can be neglected, such a body can be represented as a material point.
However, when solving problems related to the daily rotation of the planets (for example, when determining the sunrise in different places on the surface of the globe), it makes no sense to consider a planet as a material point, since the result of the problem depends on the size of this planet and the speed of movement of points on its surface.
It is legitimate to consider an aircraft as a material point if, for example, it is required to determine the average speed of its movement on the way from Moscow to Novosibirsk. But when calculating the air resistance force acting on a flying aircraft, it cannot be considered a material point, since the drag force depends on the size and shape of the aircraft.
If a body moves forward, even if its dimensions are comparable to the distances it travels, this body can be considered as a mass point (since all points of the body move in the same way).
In conclusion, we can say: a body whose dimensions can be neglected under the conditions of the problem under consideration can be considered a material point.
Trajectory
A trajectory is a line (or, as they say, a curve) that a body describes when moving relative to a selected reference body.
It makes sense to talk about a trajectory only when the body can be represented as a material point.
Trajectories can have different shapes. It is sometimes possible to judge the shape of the trajectory by the apparent trace left by a moving body, for example, a flying plane or a meteor rushing through the night sky.
The shape of the trajectory depends on the choice of the reference body. For example, relative to the Earth, the trajectory of the Moon is a circle, relative to the Sun - a line of a more complex shape.
When studying mechanical motion, as a rule, the Earth is considered as a reference body.
Methods for specifying the position of a point and describing its movement
The position of a point in space is specified in two ways: 1) using coordinates; 2) using the radius vector.
The position of a point with the help of coordinates is given by three projections of the point $x, y, z$ on the axes of the Cartesian coordinate system $ОХ, ОУ, OZ$, connected with the body of reference. To do this, from point A it is necessary to lower the perpendiculars on the plane $YZ$ (coordinate $x$), $XZ$ (coordinate $y$), $XY$ (coordinate $z$), respectively. It is written like this: $A(x, y, z)$. For the specific case, $(x=6, y=10.2, z= 4.5$), the point $A$ is denoted by $A(6; 10; 4.5)$.
On the contrary, if specific values of the coordinates of a point in a given coordinate system are given, then to image the point itself, it is necessary to plot the coordinate values on the corresponding axes ($x$ on the $OX$ axis, etc.) and construct a parallelepiped on these three mutually perpendicular segments. Its vertex, opposite to the origin $O$ and lying on the diagonal of the parallelepiped, will be the desired point $A$.
If a point moves within a certain plane, then it suffices to draw two coordinate axes through the points chosen on the reference body: $ОХ$ and $ОУ$. Then the position of the point on the plane is determined by two coordinates $x$ and $y$.
If the point moves along a straight line, it is enough to set one coordinate axis OX and direct it along the line of motion.
Setting the position of the point $A$ using the radius vector is carried out by connecting the point $A$ with the origin $O$. The directed segment $OA = r↖(→)$ is called the radius vector.
Radius vector is a vector connecting the origin to the position of a point at an arbitrary point in time.
A point is given by a radius vector if its length (modulus) and direction in space are known, i.e. the values of its projections $r_x, r_y, r_z$ on the coordinate axes $OX, OY, OZ$, or the angles between the radius vector and coordinate axes. For the case of motion on a plane, we have:
Here $r=|r↖(→)|$ is the modulus of the radius vector $r↖(→), r_x$ and $r_y$ are its projections on the coordinate axes, all three quantities are scalars; xxy - coordinates of point A.
The last equations demonstrate the connection between the coordinate and vector methods of specifying the position of a point.
The vector $r↖(→)$ can also be decomposed into components along the $X$ and $Y$ axes, i.e. represented as the sum of two vectors:
$r↖(→)=r↖(→)_x+r↖(→)_y$
Thus, the position of a point in space is given either by its coordinates or by the radius vector.
Methods for describing the movement of a point
In accordance with the methods of specifying coordinates, the movement of a point can be described: 1) in a coordinate way; 2) in a vector way.
With the coordinate method of describing (or setting) the movement, the change in the coordinates of a point over time is written as functions of all three of its coordinates from time:
The equations are called kinematic equations of motion of a point, written in coordinate form. Knowing the kinematic equations of motion and the initial conditions (i.e., the position of the point at the initial moment of time), it is possible to determine the position of the point at any moment in time.
With the vector method of describing the motion of a point, the change in its position with time is given by the dependence of the radius vector on time:
$r↖(→)=r↖(→)(t)$
The equation is an equation of point motion written in vector form. If it is known, then for any moment of time it is possible to calculate the radius vector of a point, i.e., to determine its position (as in the case of the coordinate method). Thus, setting three scalar equations is equivalent to setting one vector equation.
For each case of motion, the form of the equations will be quite definite. If the trajectory of the point is a straight line, the movement is called rectilinear, and if the curve is curvilinear.
Movement and path
Movement in mechanics is a vector connecting the positions of a moving point at the beginning and at the end of a certain period of time.
The concept of a displacement vector is introduced to solve the kinematics problem - to determine the position of a body (point) in space at a given time, if its initial position is known.
On fig. the vector $(M_1M_2)↖(-)$ connects two positions of the moving point - $M_1$ and $M_2$ at times $t_1$ and $t_2$, respectively, and, according to the definition, is a displacement vector. If the point $M_1$ is given by the radius vector $r↖(→)_1$, and the point $M_2$ is given by the radius vector $r↖(→)_2$, then, as can be seen from the figure, the displacement vector is equal to the difference of these two vectors , i.e., the change in the radius vector over the time $∆t=t_2-t_1$:
$∆r↖(→)=r↖(→)_2-r↖(→)_1$.
The addition of displacements (for example, on two neighboring sections of the trajectory) $∆r↖(→)_1$ and $∆r↖(→)_2$ is carried out according to the vector addition rule:
$∆r=∆r↖(→)_2+∆r↖(→)_1$
The path is the length of the trajectory section traveled by a material point in a given period of time. The module of the displacement vector is generally not equal to the length of the path traveled by the point in the time $∆t$ (the trajectory can be curvilinear, and, in addition, the point can change the direction of movement).
The module of the displacement vector is equal to the path only for rectilinear movement in one direction. If the direction of rectilinear motion changes, the magnitude of the displacement vector is less than the path.
With curvilinear motion, the modulus of the displacement vector is also less than the path, since the chord is always less than the length of the arc that it subtends.
Material point speed
Speed characterizes the speed with which any changes occur in the world around us (the movement of matter in space and time). The movement of a pedestrian on the sidewalk, the flight of a bird, the propagation of sound, radio waves or light in the air, the flow of water from a pipe, the movement of clouds, the evaporation of water, the heating of an iron - all these phenomena are characterized by a certain speed.
In the mechanical motion of bodies, the speed characterizes not only the speed, but also the direction of motion, i.e. is vector quantity.
The speed $υ↖(→)$ of a point is the limit of the ratio of the displacement $∆r↖(→)$ to the time interval $∆t$ during which this displacement occurred, as $∆t$ tends to zero (i.e., the derivative $∆r↖(→)$ in $t$):
$υ↖(→)=(lim)↙(∆t→0)(∆r↖(→))/(∆t)=r↖(→)_1"$
The components of the velocity vector along the axes $X, Y, Z$ are defined similarly:
$υ↖(→)_x=(lim)↙(∆t→0)(∆x)/(∆t)=x"; υ_y=y"; υ_z=z"$
The concept of speed defined in this way is also called instant speed. This definition of speed is valid for any kind of movement - from curvilinear uneven to rectilinear uniform. When talking about speed during uneven movement, it is understood as instantaneous speed. This definition directly implies the vector nature of the velocity, since moving- vector quantity. The instantaneous velocity vector $υ↖(→)$ is always directed tangentially to the motion trajectory. It indicates the direction in which the body would move if, from the moment of time $t$, the action of any other bodies on it ceased.
average speed
The average speed of a point is introduced to characterize non-uniform movement (ie movement with variable speed) and is defined in two ways.
1. The average speed of the point $υ_(av)$ is equal to the ratio of the entire path $∆s$ traveled by the body to the entire time of motion $∆t$:
$υ↖(→)_(av)=(∆s)/(∆t)$
With this definition, the average speed is a scalar, since the distance traveled (distance) and time are scalar quantities.
This definition gives an idea of average speed on the trajectory section (average ground speed).
2. The average speed of a point is equal to the ratio of the movement of the point to the period of time during which this movement occurred:
$υ↖(→)_(av)=(∆r↖(→))/(∆t)$
The average speed of movement is a vector quantity.
For non-uniform curvilinear motion, such a definition of the average speed does not always allow one to determine even approximately the real speeds along the path of the point. For example, if a point moved along a closed path for some time, then its displacement is zero (but the speed is clearly different from zero). In this case, it is better to use the first definition of the average speed.
In any case, one should distinguish between these two definitions of average speed and know which one is being discussed.
The law of addition of speeds
The law of addition of velocities establishes a connection between the values of the velocity of a material point relative to different frames of reference moving relative to each other. In non-relativistic (classical) physics, when the speeds under consideration are small compared to the speed of light, Galileo's speed addition law is valid, which is expressed by the formula:
$υ↖(→)_2=υ↖(→)_1+υ↖(→)$
where $υ↖(→)_2$ and $υ↖(→)_1$ are the velocities of a body (point) with respect to two inertial reference frames - a stationary reference frame $K_2$ and a reference frame $K_1$ moving with a speed $υ↖(→ )$ with respect to $K_2$.
The formula can be obtained by adding the displacement vectors.
For clarity, consider the movement of a boat with a speed $υ↖(→)_1$ relative to a river (reference system $K_1$), whose waters move at a speed $υ↖(→)$ relative to the shore (reference system $K_2$).
The displacement vectors of the boat relative to the water $∆r↖(→)_1$, the river relative to the coast $∆r↖(→)$ and the total displacement vector of the boat relative to the coast $∆r↖(→)_2$ are shown in Fig..
Mathematically:
$∆r↖(→)_2=∆r↖(→)_1+∆r↖(→)$
Dividing both sides of the equation by the time interval $∆t$, we get:
$(∆r↖(→)_2)/(∆t)=(∆r↖(→)_1)/(∆t)+(∆r↖(→))/(∆t)$
In projections of the velocity vector on the coordinate axes, the equation has the form:
$υ_(2x)=υ_(1x)+υ_x,$
$υ_(2y)=υ_(1y)+υ_y.$
Velocity projections are added algebraically.
Relative speed
It follows from the law of addition of velocities that if two bodies move in the same frame of reference with velocities $υ↖(→)_1$ and $υ↖(→)_2$, then the speed of the first body relative to the second $υ↖(→) _(12)$ is equal to the difference in the velocities of these bodies:
$υ↖(→)_(12)=υ↖(→)_1-υ↖(→)_2$
So, when bodies move in one direction (overtaking), the modulus of relative speed is equal to the difference in speeds, and when moving in the opposite direction, it is the sum of the speeds.
Material point acceleration
Acceleration is a value that characterizes the rate of change of speed. As a rule, the movement is uneven, i.e., it occurs at a variable speed. In some parts of the trajectory, the body can have a greater speed, in others - less. For example, a train leaving a station moves faster and faster over time. Approaching the station, he, on the contrary, slows down his movement.
Acceleration (or instantaneous acceleration) is a vector physical quantity equal to the limit of the ratio of the change in speed to the time interval during which this change occurred, when $∆t$ tends to zero, (i.e., the derivative of $υ↖(→)$ with respect to $ t$):
$a↖(→)=lim↙(∆t→0)(∆υ↖(→))/(∆t)=υ↖(→)_t"$
The components of $a↖(→) (a_x, a_y, a_z)$ are respectively:
$a_x=υ_x";a_y=υ_y";a_z=υ_z"$
Acceleration, like the change in speed, is directed towards the concavity of the trajectory and can be decomposed into two components - tangential- tangential to the trajectory of motion - and normal- perpendicular to the path.
In accordance with this, the projection of the acceleration $а_х$ onto the tangent to the trajectory is called tangent, or tangential acceleration, the projection of $a_n$ onto the normal - normal, or centripetal acceleration.
Tangential acceleration determines the amount of change in the numerical value of the speed:
$a_t=lim↙(∆t→0)(∆υ)/(∆t)$
Normal, or centripetal acceleration characterizes the change in the direction of speed and is determined by the formula:
where R is the radius of curvature of the trajectory at its corresponding point.
The acceleration module is determined by the formula:
$a=√(a_t^2+a_n^2)$
In rectilinear motion, the total acceleration $a$ is equal to the tangential one $a=a_t$, since the centripetal $a_n=0$.
The SI unit of acceleration is the acceleration at which the speed of a body changes by 1 m/s every second. This unit is designated 1 m / s 2 and is called "meter per second squared."
Uniform rectilinear motion
The movement of a point is called uniform if it covers equal distances in any equal intervals of time.
For example, if a car travels 20 km for every quarter hour (15 minutes), 40 km for every half hour (30 minutes), 80 km for every hour (60 minutes), etc., then such movement is considered uniform. With uniform motion, the numerical value (modulus) of the speed of the point $υ$ is a constant value:
$υ=|υ↖(→)|=const$
Uniform motion can occur both along a curvilinear and along a rectilinear trajectory.
The law of uniform motion of a point is described by the equation:
where $s$ is the distance measured along the arc of the trajectory from some point on the trajectory taken as the origin; $t$ - time of a point in a way; $s_0$ - the value of $s$ at the initial time $t=0$.
The path traveled by a point in time $t$ is determined by the summand $υt$.
Uniform rectilinear motion- this is a movement in which the body moves with a constant speed in modulus and direction:
$υ↖(→)=const$
The speed of uniform rectilinear motion is a constant value and can be defined as the ratio of the movement of a point to the period of time during which this movement occurred:
$υ↖(→)=(∆r↖(→))/(∆t)$
Module of this speed
$υ=(|∆r↖(→)|)/(∆t)$
meaning is the distance $s=|∆r↖(→)|$ traveled by the point in the time $∆t$.
The speed of a body in uniform rectilinear motion is a value equal to the ratio of the path $s$ to the time for which this path has been traveled:
Displacement during rectilinear uniform motion (along the X axis) can be calculated by the formula:
where $υ_x$ is the projection of the velocity on the X axis. Hence, the law of uniform rectilinear motion has the form:
If at the initial time $x_0=0$, then
The graph of speed versus time is a straight line parallel to the x-axis, and the distance traveled is the area under this straight line.
The graph of the path versus time is a straight line, the angle of inclination of which to the time axis $Ot$ is the greater, the greater the speed of uniform motion. The tangent of this angle is equal to the speed.
