in plain language: The speed 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 of the most mobile frame of reference relative to a fixed frame.
Examples
- 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 with which it is carried by the record due to its rotation.
- If a person walks along the corridor of the car at a speed of 5 kilometers per hour relative to the car, and the car 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 he 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 a person relative to the train is 55 - 50 = 5 kilometers per hour.
- If the waves move relative to the coast at a speed of 30 kilometers per hour, and the ship also 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.
Relativistic mechanics
In the 19th century, classical mechanics faced the problem of extending this rule for adding velocities to optical (electromagnetic) processes. In essence, there was a conflict between the two ideas of classical mechanics, transferred to a 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 it to electromagnetic waves, then we get a contradiction with observations (see, for example, Michelson's experiment).
The classical 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 to be simultaneous not only when they are registered in one coordinate system, but also in any other inertial frame, 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 of reference - 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, it is impossible to detect its movement by some internal mechanical effects. Does this principle extend to optical effects? Is it possible to detect the absolute motion of the system by the optical or, what is the same, electrodynamic effects caused by this motion? Intuition (fairly explicitly related 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 frames, then this speed will change when moving from one frame to another. This follows from the classical rule for adding velocities. Speaking mathematically, the magnitude of the speed of light will not be invariant under the 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 addition of velocities and the principle of relativity. Moreover, these two positions as applied to electrodynamics turned out to be incompatible.
The theory of relativity provides an answer to this question. It expands the concept of the principle of relativity, extending it to optical processes as well. In this case, the rule for adding velocities is not canceled at all, but is only refined for high velocities using the Lorentz transformation:
It can be seen that in the case when , Lorentz transformations turn into Galilean transformations . The same happens when . This suggests that special theory relativity coincides with Newtonian mechanics either in a world with an infinite speed of light, or at speeds that are small compared to the speed of light. The latter explains how these two theories are combined - the first is a refinement of the second.
see also
Literature
- B. G. Kuznetsov Einstein. Life, death, immortality. - M .: Nauka, 1972.
- Chetaev N. G. Theoretical mechanics. - M .: Nauka, 1987.
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See what the "Velocity Addition Rule" is in other dictionaries:
When considering a complex movement (that is, when a point or body moves in one frame of reference, and it moves relative to another), the question arises about the relationship of velocities in 2 frames of reference. Contents 1 Classical mechanics 1.1 Examples ... Wikipedia
A geometric construction expressing the law of addition of velocities. Rule P. s. consists in the fact that with a complex motion (see Relative motion), the absolute speed of a point is represented as a diagonal of a parallelogram built on ... ...
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A physical theory that considers spatio-temporal patterns that are valid for any physical. processes. The universality of spatiotemporal svs, considered by O. t., allows us to speak of them simply as svs of space ... ... Physical Encyclopedia
- [from Greek. mechanike (téchne) the science of machines, the art of building machines], the science of the mechanical movement of material bodies and the interactions between bodies that occur during this. Under mechanical movement understand change over time... Great Soviet Encyclopedia Mathematical Encyclopedia
BUT; m. 1. Normative act, a decision of the highest body of state power, adopted in the prescribed manner and having legal force. Labor Code. Z. on social security. Z. o military service. Z. about the securities market. ... ... encyclopedic Dictionary
2. SPEED OF THE BODY. RECTILINEAR UNIFORM MOVEMENT.
Speed is a quantitative characteristic of the movement of the body.
average speed- this is physical quantity, equal to the ratio of the point displacement vector to the time interval Δt, during which this displacement occurred. The direction of the average velocity vector coincides with the direction of the displacement vector . average speed is determined by the formula:
Instant Speed, that is, the speed at a given moment of time is a physical quantity equal to the limit to which the average speed tends with an infinite decrease in the time interval Δt:
In other words, the instantaneous speed at a given moment of time is the ratio of a very small movement to a very small period of time during which this movement occurred.
The instantaneous velocity vector is directed tangentially to the trajectory of the body (Fig. 1.6).
Rice. 1.6. Instantaneous velocity vector.
In the SI system, speed is measured in meters per second, that is, the unit of speed is considered to be the speed of such uniform rectilinear motion, in which in one second the body travels a distance of one meter. The unit of speed is denoted m/s. Often speed is measured in other units. For example, when measuring the speed of a car, train, etc. The commonly used unit of measure is kilometers per hour:
1 km/h = 1000 m / 3600 s = 1 m / 3.6 s
1 m/s = 3600 km / 1000 h = 3.6 km/h
Addition of speeds (perhaps not necessarily the same question will be in 5).
The velocities of the body in different reference systems are connected by the classical law of addition of speeds.
body speed relative to fixed frame of reference is equal to the sum of the velocities of the body in moving frame of reference and the most mobile frame of reference relative to the fixed one.
For example, a passenger train is moving along a railroad at a speed of 60 km/h. A person is walking along the carriage of this train at a speed of 5 km/h. If we consider the railway to be motionless and take it as a frame of reference, then the speed of a person relative to the frame of reference (that is, relative to railway), will be equal to the addition of the speeds of the train and the person, that is
60 + 5 = 65 if the person is walking in the same direction as the train
60 - 5 = 55 if the person and the train are moving in different directions
However, this is only true if the person and the train are moving along the same line. If a person moves at an angle, then this angle will have to be taken into account, remembering that speed is vector quantity.
An example is highlighted in red + The law of displacement addition (I think this does not need to be taught, but for general development you can read it)
Now let's look at the example described above in more detail - with details and pictures.
So, in our case, the railway is fixed frame of reference. The train that is moving along this road is moving frame of reference. The car on which the person is walking is part of the train.
The speed of a person relative to the car (relative to the moving frame of reference) is 5 km/h. Let's call it C.
The speed of the train (and hence the wagon) relative to a fixed frame of reference (that is, relative to the railway) is 60 km/h. Let's denote it with the letter B. In other words, the speed of the train is the speed of the moving reference frame relative to the fixed frame of reference.
The speed of a person relative to the railway (relative to a fixed frame of reference) is still unknown to us. Let's denote it with a letter.
We will associate the XOY coordinate system with the fixed reference system (Fig. 1.7), and the X P O P Y P coordinate system with the moving reference system. Now let's try to find the speed of a person relative to the fixed reference system, that is, relative to the railway.
For a short period of time Δt, the following events occur:
Then for this period of time the movement of a person relative to the railway:
it displacement addition law. In our example, the movement of a person relative to the railway is equal to the sum of the movements of a person relative to the wagon and the wagon relative to the railway.
Rice. 1.7. The law of addition of displacements.
The law of addition of displacements can be written as follows:
= ∆ H ∆t + ∆ B ∆t
The speed of a person relative to the railroad is:
The speed of a person relative to the car:
Δ H \u003d H / Δt
The speed of the car relative to the railway:
Therefore, the speed of a person relative to the railway will be equal to:
This is the lawspeed addition:
Uniform movement- this is movement at a constant speed, that is, when the speed does not change (v \u003d const) and there is no acceleration or deceleration (a \u003d 0).
Rectilinear motion- this is movement in a straight line, that is, the trajectory of rectilinear movement is a straight line.
Uniform rectilinear motion is a movement in which the body makes the same movements for any equal intervals of time. For example, if we divide some time interval into segments of one second, then with uniform motion the body will move the same distance for each of these segments of time.
The speed of uniform rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the displacement vector coincides in direction with the velocity vector. In this case, the average speed for any period of time is equal to the instantaneous speed:
Speed of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body for any period of time to the value of this interval t:
Thus, the speed of uniform rectilinear motion shows what movement a material point makes per unit of time.
moving with uniform rectilinear motion is determined by the formula:
Distance traveled in rectilinear motion is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity on the OX axis is equal to the velocity and is positive:
v x = v, i.e. v > 0
The projection of displacement onto the OX axis is equal to:
s \u003d vt \u003d x - x 0
where x 0 is the initial coordinate of the body, x is the final coordinate of the body (or the coordinate of the body at any time)
Motion equation, that is, the dependence of the body coordinate on time x = x(t), takes the form:
If the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body velocity on the OX axis is negative, the velocity is less than zero (v< 0), и тогда уравнение движения принимает вид.
We said that the speed of light is the maximum possible speed signal propagation. But what happens if light is emitted by a moving source in the direction of its speed V? According to the law of addition of velocities, which follows from Galileo's transformations, the speed of light must be equal to c+V. But in the theory of relativity this is impossible. Let's see what law of velocity addition follows from the Lorentz transformations. To do this, we write them for infinitesimal quantities:
By definition of the speed of its components in the frame of reference K are found as ratios of the corresponding displacements to time intervals:
Similarly, the speed of an object in a moving frame of reference is determined K", only spatial distances and time intervals must be taken relative to this system:
Therefore, dividing the expression dx to the expression dt, we get:
Dividing the numerator and denominator by dt", we find a connection x- component of velocities in different frames of reference, which differs from the Galilean rule for adding velocities:
In addition, in contrast to classical physics, the velocity components that are orthogonal to the direction of motion also change. Similar calculations for other velocity components give:
Thus, formulas for the transformation of velocities in relativistic mechanics have been obtained. The formulas for the inverse transformation are obtained by replacing primed quantities with unprimed ones and vice versa, and by replacing V on the –V.
Now we can answer the question posed at the beginning of this section. Let at the point 0" moving reference frame K" a laser is installed that sends a pulse of light in the positive direction of the axis 0"x". What will be the momentum velocity for a stationary observer in the frame of reference To? In this case, the speed of the light pulse in the frame of reference TO" has components
Applying the law of relativistic addition of velocities, we find for the components of the momentum velocity relative to the stationary system To :
We get that the speed of the light pulse and in a fixed frame of reference, relative to which the light source moves, is equal to
The same result will be obtained for any direction of propagation of the pulse. This is natural, since the independence of the speed of light from the motion of the source and the observer is inherent in one of the postulates of the theory of relativity. The relativistic law of velocity addition is a consequence of this postulate.
Indeed, when the speed of the moving reference frame V<<c, the Lorentz transformations turn into Galilean transformations, we get the usual law of addition of velocities
In this case, the course of the flow of time and the length of the ruler will be the same in both reference systems. Thus, the laws of classical mechanics are applicable if the speed of objects is much less than the speed of light. The theory of relativity did not cross out the achievements of classical physics; it established the framework for their validity.
Example. body with speed v 0 hits a wall perpendicular to it, moving towards it with a speed v. Using the formulas for the relativistic addition of velocities, we find the speed v 1 body after bounce. The impact is absolutely elastic, the mass of the wall is much greater than the mass of the body.
Let us use the formulas expressing the relativistic law of addition of velocities.
Let's direct the axis X along the initial velocity of the body v 0 and associate the frame of reference K" with a wall. Then v x= v 0 and V= –v. In the reference frame associated with the wall, the initial velocity v" 0 body equals
Now let's go back to the laboratory frame of reference To. Substituting into the relativistic law of addition of velocities v" 1 instead v" x and considering again V = –v, we find after transformations:
Let us derive a law relating the projections of the particle velocity in the IFR K and K".
Based on the Lorentz transformations (1.3.12), for infinitely small increments of particle coordinates and time, one can write
Dividing in (1.6.1) the first three equalities by the fourth, and then the numerators and denominators of the right-hand sides of the resulting relations by dt" and taking into account that
are the projections of the particle velocities on the CO axes K and K", we arrive at the desired law:
If the particle makes a one-dimensional motion along the axes OX and O"X", then, in accordance with (1.6.2),
Example 1. ISO K" moving at a speed V relatively ISO K. at an angle 0" to the direction of travel ISO K" bullet fired at a speed v". What is this angle 0 in ISO K?
Solution. When moving, there is not only a reduction in spatial, but also a stretching of time intervals. To find tg0 = v y / v x it is necessary in (1.6.2) to divide the second formula by the first, and then the numerator and denominator of the resulting fraction - by v "x = v" cos0 " Considering that v " y / v" x = tg0 ", we find
For speeds that are small compared to the speed of light, formulas (1.6.2) turn into the well-known law of classical mechanics (1.1.4):
From the formulas for the transformation of particle velocity projections (1.6.2), it is not difficult to determine the velocity modulus and its direction in the IFR K through the particle velocity in the IFR K. , and in the X"0"Y" plane), and denote by 0 (0") the angle between
V (V") and the axis OX (O "X"). Then
v x = vcos0, v = vsin0, v" x = v"cos©", v* = v"sin©", v z = v" z = 0 (1.6.4) or
As for the direction of the particle velocity in CO K (angle 0), it is determined by term-by-term division in (1.6.5) of the second formula by the first one:
and substitution (1.6.4) into (1.6.2) gives
After squaring both equalities (1.6.5) and adding them, we obtain
The inverse transformation formulas are obtained by replacing primed values with unprimed ones and vice versa and replacing V with -V.
Task 2. Determine relative speed v 0TH rendezvous of two spacecraft 1 and 2 moving towards each other with speedsX And V2-
Solution. Let's connect the mobile CO K" with the spacecraft 1. Then V = Vi, and the required relative speed v 0TH will be the speed of the craft 2 in this CO. Applying the relativistic law of velocity addition (1.6.3) to the second craft, taking into account the direction of its velocity (v "2 = -v 0TH) we have
Numerical estimates for v, = v 2 = 0.9 s give
Task 3. body with speed v0 hits a wall perpendicular to it, moving towards it with speed. Using the relativistic law of addition of velocities, find the speed v 0Tp body after rebound. The impact is absolutely elastic, the mass of the wall is much greater than the mass of the body. Find v 0Tp , if v 0 \u003d v \u003d c / 3. Analyze extreme cases.
where V is the speed of CO K "relative to CO K. Let's connect CO K" with the wall. Then V \u003d -v and in this CO the initial velocity of the body, according to the expression for v",
Let us now return back to the laboratory CO K. Substituting into
(1.6.3) v" 0Tp instead of v" and taking into account again that V = -v, after simple transformations we obtain the desired result:
Let us now analyze the limiting cases.
If the velocities of the body and the wall are small (v 0 « s, v « s), then we can neglect all the terms where these velocities and their product are divided by the speed of light. Then from the general formula obtained above we arrive at the well-known result of classical mechanics: v 0Tp = -(v 0 + 2v) -
the speed of the body after the rebound increases by twice the speed of the wall; it is directed, of course, opposite to the initial one. It is clear that in the relativistic case this result is incorrect. In particular, when v 0 =v = c/3, it follows from it that the speed of the body after the rebound will be equal to - c, which cannot be.
Let now a body moving at the speed of light hit the wall (for example, a laser beam is reflected from a moving mirror). Substituting v 0 \u003d c into the general expression for v, we get v \u003d -c.
This means that the speed of the laser beam has changed direction, but not its absolute value - in full accordance with the principle of invariance of the speed of light in vacuum.
Let us now consider the case when the wall moves with a relativistic velocity v -> With. In this case
The body after the bounce will also move at a speed close to the speed of light.
- Finally, we substitute into the general formula for v 0Tp the values
v n \u003d v \u003d c / 3. Then = -s * -0.78 s. Unlike the classical
mechanics, the theory of relativity gives a value for the speed after the bounce, less than the speed of light.
In conclusion, let's see what happens if the wall moves away from the body with the same speed v = -v 0 . In this case general formula for v 0Tp leads to the result: v = v 0 . As in classical mechanics, the body will not catch up with the wall and, therefore, its speed will not change.
The results of the experiment were described by the formulas
where n is the refractive index of water, and V is the speed of its flow.
Prior to the creation of SRT, the results of the Fizeau experiment were considered on the basis of the hypothesis put forward by O. Fresnel, within which it was necessary to assume that moving water partially entrains the "world ether". Value 
was called the drag coefficient of the ether, and formulas (1.7.1) and (1.7.2) with this approach directly follow from the classical law of addition of velocities: c/n is the speed of light in water relative to the ether, kV is the speed of the ether relative to the experimental setup.
Lorentz transformations give us the opportunity to calculate the change in the coordinates of an event when moving from one frame of reference to another. Let us now pose the question of how, when the reference system changes, the speed of the same body will change?
In classical mechanics, as is known, the speed of a body is simply added to the speed of the frame of reference. Now we will make sure that in the theory of relativity the speed is transformed according to a more complex law.
We will again restrict ourselves to the one-dimensional case. Let two frames of reference S and S` "observe" the motion of some body that moves uniformly and rectilinearly parallel to the axes X and x` both reference systems. Let the speed of the body measured by the reference system S, there is and; the speed of the same body, measured by the system S`, will be denoted by and` . letter v we will continue to denote the speed of the system S` relatively S.
Suppose that two events occur with our body, the coordinates of which in the system S
essence x 1 ,t 1 , andX 2
,
t 2
.
Coordinates of the same events in the system S`
let them be x` 1,
t` 1 ;
x` 2
,
t` 2
.
But the speed of a body is the ratio of the path traveled by the body to the corresponding period of time; therefore, to find the speed of the body in both frames of reference, it is necessary to divide the difference in the spatial coordinates of both events by the difference in time coordinates
which can, as always, be obtained from the relativistic one if the speed of light is assumed to be infinite. The same formula can be written as
For small, "ordinary" velocities, both formulas—relativistic and classical—give practically identical results, which the reader can easily verify if he wishes. But at speeds close to the speed of light, the difference becomes quite noticeable. So, if v=150,000 km/s, u`=200 000 km/Withek, km/s the relativistic formula gives u = 262 500 km/Withec.
S
with speed v = 150,000 km/s S`
gives the result u
=200 000 km/s km/Withec.
km/sec, and the second - 200,000 km/sec, km.
With. It is not difficult to prove this assertion quite rigorously. Indeed, it is easy to check.
For small, "ordinary" velocities, both formulas—relativistic and classical—give practically identical results, which the reader can easily verify if he wishes. But at speeds close to the speed of light, the difference becomes quite noticeable. So, if v=150,000 km/s, u`=200 000 km/Withek, then instead of the classical result u = 350 000 km/s the relativistic formula gives u = 262 500 km/Withec. According to the meaning of the velocity addition formula, this result means the following.
Let the reference frame S` move relative to the reference frame S
with speed v = 150,000 km/s Let a body move in the same direction, and the measurement of its speed by the reference system S`
gives the result u`
=200 000 km/s If we now measure the speed of the same body using the reference frame S, we get u=262 500 km/Withec.
It should be emphasized that the formula obtained by us is intended specifically for recalculating the velocity of the same body from one frame of reference to another, and by no means for calculating the “speed of approach” or “removal” of two bodies. If we observe two bodies moving towards each other from the same reference frame, and the speed of one body is 150,000 km/sec, and the second - 200,000 km/sec, then the distance between these bodies will decrease by 350,000 every second km.
The theory of relativity does not abolish the laws of arithmetic.
The reader has already understood, of course, that by applying this formula to velocities not exceeding the speed of light, we again obtain a speed not exceeding With. It is not difficult to prove this assertion quite rigorously. Indeed, it is easy to check that the equality
Because i` ≤ c
and v <
c,
then on the right side of the equality the numerator and denominator, and with them the whole fraction, are non-negative. Therefore, the square bracket is less than one, and therefore and ≤ c
.
If a and`
=
With,
then and and=With. This is nothing but the law of the constancy of the speed of light. One should not, of course, consider this conclusion as a "proof" or at least a "confirmation" of the postulate of the constancy of the speed of light. After all, from the very beginning we proceeded from this postulate and it is not surprising that we came to a result that does not contradict it, otherwise this postulate would be refuted by proof from the contrary. At the same time, we see that the law of velocity addition is equivalent to the postulate of the constancy of the speed of light, each of these two statements follows logically from the other (and other postulates of the theory of relativity).
When deriving the law of addition of velocities, we assumed that the speed of the body is parallel to the relative speed of reference systems. This assumption could not be made, but then our formula would refer only to the velocity component that is directed along the x axis, and the formula should be written in the form
With the help of these formulas, we will analyze the phenomenon aberrations(see § 3). We restrict ourselves to the simplest case. Let some luminary in the reference system S
motionless, let, further, the frame of reference S`
moves relative to the system S
with speed v
and let the observer, moving along with S`, receive the rays of light from the luminary just at the moment when it is exactly above his head (Fig. 21). The velocity components of this beam in the system S
will
u x = 0, u y = 0, u x = -c.
For the reference frame S`, our formulas give
u` x = -v, u` y = 0,
u` z = -c√
(1-v 2
/c 2
)
We get the tangent of the angle of inclination of the beam to the z-axis, if we divide and`X
on the and`z:
tgα = and`X / and`z
\u003d (v / c) / √ (1 - v 2 / c 2)
If the speed v
is not very large, then we can apply the approximate formula known to us, with the help of which we obtain
tan α \u003d v / c + 1/2 * v 2 / c 2
.
The first term is a well-known classical result; the second term is the relativistic correction.
The Earth's orbital speed is about 30 km/sec, so (v/
c)
=
1
0
-4
.
For small angles, the tangent is equal to the angle itself, measured in radians; since the radian contains 200,000 arc seconds in a round count, we get for the angle of aberration:
α = 20°
The relativistic correction is 20,000,000 times smaller and lies far beyond the accuracy of astronomical measurements. Due to aberration, the stars describe annually ellipses in the sky with a semi-major axis of 20".
When we look at a moving body, we see it not where it is at the moment, but where it was a little earlier, because the light needs some time to get from the body to our eyes. From the point of view of the theory of relativity, this phenomenon is equivalent to an aberration and reduces to it in the transition to the frame of reference in which the body under consideration is motionless. Based on this simple consideration, we can obtain the aberration formula in a completely elementary way, without resorting to the relativistic law of addition of velocities.
Let our luminary move in parallel earth's surface from right to left (Fig. 22). When it arrives at the point BUT, an observer exactly below him at point C sees him still at point AT. If the speed of the star is v,
and the time interval during which it passes the segment BUTAT,
equals Δt,
then
AB=Δt
,
BC
=
cΔt
,
sinα
= AB/BC = v/c.
But then, according to trigonometry formula,
Q.E.D. Note that in classical kinematics these two points of view are not equivalent.
The next question is also interesting. As you know, in classical kinematics, the velocities are added according to the parallelogram rule. We have replaced this law with another, more complex one. Does this mean that in the theory of relativity velocity is no longer a vector?
First, the fact that u≠u`+
v
(we designate vectors in bold letters), in itself still does not give grounds to deny the vector nature of the velocity. From two given vectors, the third vector can be obtained not only by their addition, but, for example, by vector multiplication, and in general in an infinite number of ways. It does not follow from anywhere that when the reference system is changed, the vectors and` and v
have to be put together. Indeed, there is a formula expressing and
through and`
and v
using vector calculus operations:
In this regard, it should be recognized that the name "law of addition of velocities" is not entirely apt; it is more correct to speak, as some authors do, not about addition, but about the transformation of velocity when the frame of reference changes.
Secondly, in the theory of relativity one can point out the cases when the velocities add up as before in a vector way. Let, for example, the body move for a certain period of time Δt
with speed u 1 ,
and then - the same period of time with a speed u 2. This complex movement can be replaced by movement at a constant speed u = u 1+ u 2 . Here the speed u 1
and u 2
add up like vectors, according to the parallelogram rule; The theory of relativity does not make any changes here.
In general, it should be noted that most of the "paradoxes" of the theory of relativity are connected in one way or another with a change in the reference frame. If we consider phenomena in the same frame of reference, then the changes introduced by the theory of relativity in their regularities are far from being as drastic as is often thought.
We also note that four-dimensional vectors are a natural generalization of ordinary three-dimensional vectors in the theory of relativity; when the reference system is changed, they are transformed according to the Lorentz formulas. In addition to three spatial components, they have a temporal component. In particular, one can consider a four-dimensional velocity vector. The spatial "part" of this vector, however, does not coincide with the usual three-dimensional velocity, and in general the four-dimensional velocity differs noticeably from the three-dimensional velocity in its properties. In particular, the sum of two four-dimensional velocities will not, generally speaking, be a speed.