Example. Let's go back to example (1.13):

x = 1 + 12t 3t2

(coordinate is measured in meters, time in seconds). Consistently differentiating twice, we get:

vx = x = 12 6t;

ax = vx = 6:

As we can see, the acceleration is constant in absolute value and equal to 6 m/s2. The acceleration is directed in the direction opposite to the X axis.

The given example is the case of uniformly accelerated motion, in which the magnitude and direction of acceleration are unchanged. Uniformly accelerated motion is one of the most important and frequently occurring types of motion in mechanics.

From this example it is easy to understand that with uniformly accelerated motion the projection of velocity is linear function time, and the coordinate quadratic function. We will talk about this in more detail in the corresponding section on uniformly accelerated motion.

Example. Let's consider a more exotic case:

x = 2 + 3t 4t2 + 5t3 :

Let's differentiate:

vx = x = 3 8t + 15t2 ;

ax = vx = 8 + 30t:

This movement is not uniformly accelerated: acceleration depends on time.

Example. Let the body move along the X axis according to the following law:

We see that the coordinate of the body changes periodically, ranging from 5 to 5. This movement is an example of harmonic oscillations, when the coordinate changes over time according to the sine law.

Let's differentiate twice:

vx = x = 5 cos 2t 2 = 10 cos 2t;

ax = vx = 20 sin 2t:

The velocity projection changes according to the cosine law, and the acceleration projection again according to the sine law. The quantity ax is proportional to the x coordinate and opposite in sign (namely, ax = 4x); in general, a relation of the form ax = !2 x is characteristic of harmonic oscillations.

1.2.8 Law of addition of speeds

Let there be two reference systems. One of them is associated with a stationary reference body O. We will denote this reference system by K and call it stationary.

The second reference system, denoted by K0, is associated with the reference body O0, which moves relative to the body O with a speed of ~u. We call this reference system moving. Additionally

We assume that the coordinate axes of the system K0 move parallel to themselves (there is no rotation of the coordinate system), so that the vector ~u can be considered the speed of the moving system relative to the stationary one.

The fixed reference frame K is usually related to the ground. If a train moves smoothly along the rails with a speed of ~u, then the reference frame associated with the train car will be a moving reference frame K0.

Note that the speed of any point in car3 is ~u. If a fly sits motionless at some point in the carriage, then relative to the ground the fly moves with a speed of ~u. The fly is carried by the carriage, and therefore the speed ~u of the moving system relative to the stationary one is called the portable speed.

Now suppose that a fly crawled along the carriage. Then there are two more speeds that need to be considered.

The speed of the fly relative to the car (that is, in the moving system K0) is denoted by ~v0 and

called relative speed.

The speed of the fly relative to the ground (that is, in a stationary K frame) is denoted by ~v and

called absolute speed.

Let's find out how these three speeds - absolute, relative and portable - are related to each other.

In Fig. 1.11 the fly is indicated by point M. Next:

~r radius vector of point M in a fixed system K; ~r0 radius vector of point M in the moving system K0 ;

~ radius vector of the body of reference 0 in a stationary system.

Rice. 1.11. To the conclusion of the law of addition of velocities

As can be seen from the figure,

~ 0 ~r = R + ~r:

Differentiating this equality, we get:

The derivative d~r=dt is the speed of point M in the K system, that is, the absolute speed:

d~r dt = ~v:

Similarly, the derivative d~r 0 =dt is the speed of point M in the K0 system, that is, the relative

speed:

d~r dt 0 = ~v0 :

3 In addition to rotating wheels, but we do not take them into account.

What is ~? This is the speed of point0 in a stationary system, that is, portable dR=dt O

speed ~u of a moving system relative to a stationary one:

dR dt = ~u:

As a result, from (1.28) we obtain:

~v = ~u + ~v 0 :

The law of addition of speeds. The speed of a point relative to a stationary reference frame is equal to the vector sum of the speed of the moving system and the speed of the point relative to the moving system. In other words, absolute speed is the sum of portable and relative speeds.

Thus, if a fly crawls along a moving carriage, then the speed of the fly relative to the ground is equal to the vector sum of the speed of the carriage and the speed of the fly relative to the carriage. Intuitively obvious result!

1.2.9 Types of mechanical movement

The simplest types of mechanical motion of a material point are uniform and rectilinear motion.

The movement is called uniform if the magnitude of the velocity vector remains constant (the direction of the velocity can change).

Movement is called rectilinear if it occurs along a certain straight line (the magnitude of the speed may change). In other words, the trajectory of rectilinear motion is a straight line.

For example, a car that is traveling with constant speed along a winding road, makes a uniform (but not rectilinear) movement. A car accelerating on a straight section of highway moves in a straight line (but not uniformly).

But if, during the movement of a body, both the magnitude of the velocity and its direction remain constant, then the motion is called uniform rectilinear. So:

uniform motion, j~vj = const;

uniform rectilinear movement, ~v = const.

The most important special case uneven movement is uniformly accelerated motion, in which the magnitude and direction of the acceleration vector remain constant:

uniformly accelerated motion, ~a = const.

Along with the material point, another idealization is considered in mechanics - a rigid body.

A rigid body is a system of material points, the distances between which do not change over time. Model solid is used in cases where we cannot neglect the size of the body, but can not take into account the change in the size and shape of the body during movement.

The simplest types of mechanical motion of a solid body are translational and rotational motion.

The movement of a body is called translational if any straight line connecting any two points of the body moves parallel to its original direction. During translational motion, the trajectories of all points of the body are identical: they are obtained from each other by a parallel shift.

So, in Fig. Figure 1.12 shows the forward motion of a gray square. An arbitrarily chosen green segment of this square moves parallel to itself. The trajectories of the ends of the segment are depicted with blue dotted lines.

Rice. 1.12. Forward movement

The motion of a body is called rotational if all its points describe circles lying in parallel planes. In this case, the centers of these circles lie on one straight line, which is perpendicular to all these planes and is called the axis of rotation.

In Fig. Figure 1.13 shows a ball rotating around a vertical axis. This is how the globe is usually drawn in the corresponding dynamics problems.

Rice. 1.13. Rotational movement

Let the body in the reference frame K" have a speed v", directed along the x" (and x) axis: . In the reference frame K, the speed of this body will be
. Let's find out what the relationship is between the velocities v" and v. Consider the derivative as the ratio of the differentials dx and dt, which we find using Lorentz transformations:

Divide the numerator and denominator of the right side by dt" and get

those. unlike Galileo's transformations, the total speed is not equal to the sum of the speeds, but in
times lower. Let the body move in the rocket at the speed of light v" x = c, and the rocket moves at the speed of light relative to the fixed coordinate system v 0 = c. At what speed v x does the body move relative to the fixed coordinate system?

According to the Galileo transformation, this speed is v = v" x + v 0 = 2c. According to the Lorentz transformation

The concept of relativistic dynamics. Laws of relationship between mass and energy. Total and kinetic energy. The relationship between the total energy and momentum of a particle.

The movement of not too small bodies with not very high speeds obeys the laws of classical mechanics. IN late XIX century, it was experimentally established that the mass of a body m is not a constant quantity, but depends on the speed v of its movement. This dependence has the form

where m 0 is the rest mass.

If v = 300 km/s, then v 2 /c 2 = 1∙ 10 -6 and m > m 0 by an amount of 5 ∙ 10 -7 m 0 .

The rejection of one of the basic provisions (m = const) of classical mechanics led to the need for a critical analysis of a number of its other foundations. The expression of momentum in relativistic dynamics has the form

The laws of mechanics retain their form in relativistic dynamics. Momentum change d(mv ) equal to the force impulse Fdt

dp = d(mv) = F dt.

Hence dp/dt = F- is the expression of the basic law of relativistic dynamics for a material point.

In both cases, the mass included in these expressions is a variable quantity (m ≠ const) and it also needs to be differentiated with respect to time.

Let us establish the connection between mass and energy. The increase in energy, as in classical mechanics, is caused by the work of force F. Therefore, dE = Fds. Dividing the left and right sides by dt, we get

Substitute here

Multiplying the left and right sides of the resulting equality by dt, we get

From the expression for mass
let's define

.

Let's differentiate the expression v 2 .

Let's substitute v 2 and d(v 2) into the expression for dE

Integrating this expression, we get E = mc 2.

The total energy of the system E is equal to the mass multiplied by the square of the speed of light in vacuum. The relationship between energy and momentum for particles without rest mass in relativistic dynamics is given by the relation

which is easy to obtain mathematically: E=mc 2 ,p=mv . Let's square both equalities and multiply both sides of the second by c 2

E 2 = m 2 c 4, p 2 c 2 = m 2 v 2 c 2.

Subtract term by term from the first equality the second

E 2 – p 2 c 2 = m 2 c 4 -m 2 v 2 c 2 = m 2 c 4 (1-v 2 / c 2).

Considering that
we get

Since the rest mass m 0 and the speed of light c are quantities invariant to Lorentz transformations, the relation (E 2 - p 2 c 2) is also invariant to Lorentz transformations. From this relationship we obtain an expression for the total energy

Thus, from this equation we can conclude:

Material particles that do not have rest mass (photons, neutrinos) also have energy. For these particles, the formula for the relationship between energy and momentum is E = pc.

From the above transformations we obtained dE=c 2 dm. Integrating the left side from E 0 to E, and the right side from m 0 to m, gives

E – E 0 = c 2 (m – m 0) = mc 2 – m 0 c 2 ,

where E = mc 2 is the total energy of the material point,

E 0 =m 0 c 2 - rest energy of a material point.

The difference E – E 0 is the kinetic energy T of the material point.

At speeds v « c , we expand
in a row:

=
.

Considering that v « c, we restrict ourselves to the first two terms in the series.

Then

those. at speeds v much lower than the speed of light in vacuum, the relativistic formula for kinetic energy turns into the classical formula for kinetic energy
.

And this reference system, in turn, moves relative to another system), the question arises about the connection between the velocities in the two reference systems.

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Addition of velocities (kinematics) ➽ Physics grade 10 ➽ Video lesson

Lesson 19. Relativity of motion. Formula for adding speeds.

Physics. Lesson No. 1. Kinematics. Law of addition of speeds

Subtitles

Classical mechanics

V → a = v → r + v → e. (\displaystyle (\vec (v))_(a)=(\vec (v))_(r)+(\vec (v))_(e).)

This equality represents the content of the statement of the theorem on the addition of velocities.

In simple terms: The speed of movement of a body relative to a fixed frame of reference is equal to the vector sum of the speed of this body relative to a moving frame of reference and the speed (relative to a fixed frame) of that point of the moving frame of reference at which this moment time the body is located.

Examples

1. The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed that the point of the record under the fly has relative to the ground (that is, with which the record carries it due to its rotation).
2. If a person walks along the corridor of a carriage at a speed of 5 kilometers per hour relative to the carriage, and the carriage moves at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of the train, and at a speed of 50 - 5 = 45 kilometers per hour when it goes in the opposite direction. If a person in the carriage corridor moves relative to the Earth at a speed of 55 kilometers per hour, and a train at a speed of 50 kilometers per hour, then the speed of the person relative to the train is 55 - 50 = 5 kilometers per hour.
3. If the waves move relative to the shore at a speed of 30 kilometers per hour, and the ship also moves at a speed of 30 kilometers per hour, then the waves move relative to the ship at a speed of 30 - 30 = 0 kilometers per hour, that is, they become motionless relative to the ship.

Relativistic mechanics

In the 19th century, classical mechanics was faced with the problem of extending this rule for adding velocities to optical (electromagnetic) processes. Essentially, there was a conflict between two ideas of classical mechanics, transferred to the new field of electromagnetic processes.

For example, if we consider the example with waves on the surface of water from the previous section and try to generalize to electromagnetic waves, then a contradiction with observations will result (see, for example, Michelson’s experiment).

The classic rule for adding velocities corresponds to the transformation of coordinates from one system of axes to another system moving relative to the first without acceleration. If with such a transformation we retain the concept of simultaneity, that is, we can consider two events simultaneous not only when they are registered in one coordinate system, but also in any other inertial system, then the transformations are called Galilean. In addition, with Galilean transformations, the spatial distance between two points - the difference between their coordinates in one inertial frame - is always equal to their distance in another inertial frame.

The second idea is the principle of relativity. Being on a ship moving uniformly and rectilinearly, its movement cannot be detected by any internal mechanical effects. Does this principle apply to optical effects? Is it not possible to detect the absolute motion of a system by the optical or, what is the same thing, electrodynamic effects caused by this motion? Intuition (related quite clearly to the classical principle of relativity) says that absolute motion cannot be detected by any kind of observation. But if light propagates at a certain speed relative to each of the moving inertial systems, then this speed will change when moving from one system to another. This follows from the classical rule of adding velocities. Speaking mathematical language, the speed of light will not be invariant under Galilean transformations. This violates the principle of relativity, or rather, does not allow the principle of relativity to be extended to optical processes. Thus, electrodynamics destroyed the connection between two seemingly obvious provisions of classical physics - the rule of adding velocities and the principle of relativity. Moreover, these two provisions in relation to electrodynamics turned out to be incompatible.

The theory of relativity provides the answer to this question. It expands the concept of the principle of relativity, extending it to optical processes. In this case, the rule for adding velocities is not canceled completely, but is only refined for high velocities using the Lorentz transformation:

It can be noted that in the case when v / c → 0 (\displaystyle v/c\rightarrow 0), Lorentz transformations turn into Galilean transformations. This suggests that special relativity reduces to Newtonian mechanics at speeds small compared to the speed of light. This explains how these two theories relate - the first is a generalization of the second.

Which were formulated by Newton at the end of the 17th century, for about two hundred years it was considered everything explaining and infallible. Until the 19th century, its principles seemed omnipotent and formed the basis of physics. However, by this period, new facts began to appear that could not be squeezed into the usual framework of known laws. Over time, they received a different explanation. This happened with the advent of the theory of relativity and the mysterious science of quantum mechanics. In these disciplines, all previously accepted ideas about the properties of time and space have undergone a radical revision. In particular, the relativistic law of addition of velocities eloquently proved the limitations of classical dogmas.

Simple addition of speeds: when is this possible?

Newton's classics in physics are still considered correct, and its laws are used to solve many problems. You just have to take into account that they operate in the world that is familiar to us, where the speeds of various objects, as a rule, are not significant.

Let's imagine a situation where a train is traveling from Moscow. Its speed is 70 km/h. And at this time, in the direction of travel, a passenger travels from one carriage to another, running 2 meters in one second. To find out the speed of its movement relative to the houses and trees flashing outside the train window, the indicated speeds should simply be added up. Since 2 m/s corresponds to 7.2 km/h, the desired speed will be 77.2 km/h.

World of high speeds

Photons and neutrinos are another matter; they obey completely different rules. It is for them that the relativistic law of addition of velocities operates, and the principle shown above is considered completely inapplicable for them. Why?

According to the special theory of relativity (STR), any object cannot move faster than light. In extreme cases, it can only be approximately comparable to this parameter. But if we imagine for a second (although in practice this is impossible) that in the previous example the train and the passenger are moving approximately in this way, then their speed relative to the objects resting on the ground, past which the train is passing, would be equal to almost two times the speed of light. And this shouldn't happen. How are calculations made in this case?

The relativistic law of addition of velocities, known from the 11th grade physics course, is represented by the formula given below.

What does it mean?

If there are two reference systems, the speed of a certain object relative to which is V 1 and V 2, then for calculations you can use the specified relationship, regardless of the value of certain quantities. In the case when both of them are significantly less than the speed of light, the denominator on the right side of the equality is practically equal to 1. This means that the formula for the relativistic law of addition of velocities turns into the most common one, that is, V 2 = V 1 + V.

It should also be noted that when V 1 = C (that is, the speed of light), for any value of V, V 2 will not exceed this value, that is, it will also be equal to C.

From the realm of fantasy

C is a fundamental constant, its value is 299,792,458 m/s. Since the time of Einstein, it has been believed that no object in the Universe can surpass the movement of light in a vacuum. This is how we can briefly define the relativistic law of addition of velocities.

However, science fiction writers did not want to come to terms with this. They have invented and continue to invent many amazing stories, the heroes of which refute such organic ones. In the blink of an eye spaceships moving to distant galaxies located many thousands of light years from the old Earth, thereby nullifying all the established laws of the universe.

But why are Einstein and his followers sure that this cannot happen in practice? We should talk about why the light limit is so unshakable and the relativistic law of adding velocities is inviolable.

Relationship of cause and effect

Light is a carrier of information. It is a reflection of the reality of the Universe. And the light signals reaching the observer recreate pictures of reality in his mind. This happens in the world that is familiar to us, where everything goes on as usual and obeys the usual rules. And from birth we are accustomed to the fact that it cannot be otherwise. But what if we imagine that everything around has changed, and someone has gone into space, traveling at superluminal speed? Since he is ahead of the photons of light, the world begins to appear to him as if it were a film replayed in reverse. Instead of tomorrow, yesterday comes for him, then the day before yesterday, and so on. And he will never see tomorrow until he stops, of course.

By the way, science fiction writers also actively adopted a similar idea, creating an analogue of a time machine using these principles. Their heroes went back in time and traveled there. However, the cause-and-effect relationships collapsed. And it turned out that in practice this is hardly possible.

Other paradoxes

The reason cannot be ahead is contrary to normal human logic, because there must be order in the Universe. However, SRT also implies other paradoxes. She says that even if the behavior of objects obeys the strict definition of the relativistic law of addition of velocities, it is also impossible for it to exactly match the speed of movement with photons of light. Why? Yes, because truly magical transformations begin to occur. The mass increases endlessly. The dimensions of a material object in the direction of motion indefinitely approach zero. And again, disturbances cannot be completely avoided over time. Although it does not move backward, when it reaches the speed of light it stops completely.

Eclipse of Io

SRT states that photons of light are the fastest objects in the Universe. In this case, how was it possible to measure their speed? It’s just that human thought turned out to be quicker. She was able to solve a similar dilemma, and its consequence was the relativistic law of addition of velocities.

Similar questions were solved back in the time of Newton, in particular, in 1676 by the Danish astronomer O. Roemer. He realized that the speed of ultrafast light can only be determined when it travels enormous distances. This, he thought, was only possible in heaven. And the opportunity to bring this idea to life soon presented itself when Roemer observed through a telescope the eclipse of one of Jupiter’s moons called Io. The time interval between entering the blackout and the appearance of this planet for the first time was about 42.5 hours. And this time everything roughly corresponded to preliminary calculations carried out according to the known orbital period of Io.

A few months later, Roemer again performed his experiment. During this period, the Earth moved significantly away from Jupiter. And it turned out that Io was 22 minutes late to show his face compared to earlier assumptions. What did this mean? The explanation was that the satellite did not delay at all, but the light signals from it took some time to cover a significant distance to the Earth. Having made calculations based on these data, the astronomer calculated that the speed of light is very significant and is about 300,000 km/s.

Fizeau's experience

A harbinger of the relativistic law of addition of velocities, Fizeau's experiment, carried out almost two centuries later, confirmed Roemer's guesses correctly. Only the famous French physicist carried out laboratory experiments in 1849. And to implement them, an entire optical mechanism was invented and designed, an analogue of which can be seen in the figure below.

The light came from the source (this was stage 1). Then it was reflected from the plate (stage 2) and passed between the teeth of the rotating wheel (stage 3). Next, the rays hit a mirror located at a considerable distance, measured at 8.6 kilometers (stage 4). Finally, the light was reflected back and passed through the teeth of the wheel (step 5), entering the eyes of the observer and recorded by him (step 6).

The wheel rotated at different speeds. When moving slowly, the light was visible. As the speed increased, the rays began to disappear without reaching the viewer. The reason is that the beams took some time to move, and during this period, the teeth of the wheel moved slightly. When the rotation speed increased again, the light again reached the observer’s eye, because now the teeth, moving faster, again allowed the rays to penetrate through the gaps.

SRT principles

Relativistic theory was first introduced to the world by Einstein in 1905. Devoted to this work description of events taking place in the most different systems reference, the behavior of magnetic and electromagnetic fields, particles and objects when they move, as close as possible to the speed of light. The great physicist described the properties of time and space, and also examined the behavior of other parameters, the sizes of physical bodies and their masses under the specified conditions. Among the basic principles, Einstein named the equality of any inertial frames of reference, that is, he meant the similarity of the processes occurring in them. Another postulate of relativistic mechanics is the law of addition of velocities in a new, non-classical version.

Space, according to this theory, is represented as emptiness where everything else functions. Time is defined as a certain chronology of ongoing processes and events. It is also for the first time called as the fourth dimension of space itself, now receiving the name “space-time”.

Lorentz transformations

The relativistic law of addition of Lorentz transformation rates is confirmed. That's what they call it mathematical formulas, which are presented in their final version below.

These mathematical relationships are central to the theory of relativity and serve to transform coordinates and time, being written for a quadruple spacetime. The presented formulas received this name at the suggestion of Henri Poincaré, who, while developing the mathematical apparatus for the theory of relativity, borrowed some ideas from Lorentz.

Such formulas prove not only the impossibility of overcoming the supersonic barrier, but also the inviolability of the principle of causality. According to them, it became possible to mathematically substantiate time dilation, shortening the lengths of objects, and other miracles occurring in the world of ultra-high speeds.