Two rods of the same length move towards each other. Relativistic mechanics

M.: Higher school, 2001. - 669 p.
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Solution. The length of the rod in the reference systems "Earth" and "rocket" is equal to
1o \u003d ^ (x2-x]) 2 + (y2-y1) 2 "
/ = V(^-x,")2 + (v"-y;)2 respectively (Fig. 16.3).
Since the effect of the reduction of the daina appears only in the direction
movement, then _________
X," = X, ^ 1 - U2/c2 *2 = X2 V 1 - U2/c2
U1=U(", Ug~Ug Therefore,
K K
Ug ¦-Ug
U\-U
Oh Oh"
XX"
Rice. 16.3
I=< (Xj-х,)2(1 -и2/с2) + (y2-yf.
From rns. 16.3 shows that
x1-xi = l0cos a0, y2 ~y1 = /0 sin a0.
Then
/ \u003d / 0 V (1 - u2 / s2) cos2 a + sin2 a \u003d / 0 V 1 - u2 / s2 cos2 a * 0.88 m, a
desired angle
. . >2-^1 . *8" about
a = arctg ------- = arctg ------- ¦**- y, = arctg
*2~x (*2 - Xj) \ 1 - u /c
2/s2
53.3°
Answer: / \u003d / 0 V 1 - u2 / c2 cos2 a * 0.88 m; a \u003d arctg -y- ^ I y * 53.3 °.
^1-uW
16.3. The spacecraft, flying past the observer, has a speed and =
2.4 108 m/s. According to the measurements of the observer, the length of the ship is / = 90 m.
What is the length of the ship at rest?
16.4. The length of the leg AB of a right triangle is a = 5 m, and the angle
between this leg and the hypotenuse - a \u003d 30 °. Find the value of this angle,
the length of the hypotenuse and its relation to its own length in the frame of reference,
moving along leg AB with speed u = 2.6-108 m/s.
16.5. How long will it take for an earthly observer and for astronauts
space travel to a star and back on a rocket flying from
speed u = 2.9108 m/s? Distance to the star (for an earthly observer)
equals 40 light years.
Solution. The distance S that the spacecraft will fly in the system
reference associated with the Earth is 5= 40 3-108 m / s-365-24-3600 s * 3.8
1017 m. Therefore, according to the clock of an earthly observer, the flight of the ship
last
D / \u003d - * 2.62-10 (r) s * 83 years old, and
The time measured by the clock on board the spacecraft
according to the time dilation effect
A/0 = A/V 1-u2/s\
16.6. At what speed must a pion fly to fly before decay
distance / = 20 m? The mean lifetime of a pion at rest is
AtQ = 26 no.
16.7. The spacecraft has clocks synchronized before the flight
with earthly ones. How far will the clock on the ship, according to the measurements of the earth
observer, for the time D/0 = 0.5 years, if the speed of the ship is o = 7.9
km/s?
16.8. In the reference frame K, two parallel rods have the same
own length /0 = 1 m and move in the longitudinal direction towards
to each other with equal speeds o = 2-10 * m / s, measured in this system
reference. What is the length of each rod in the reference frame associated with
another rod?
Solution. For a stationary observer in the motion of extended bodies
with high speeds, their dimensions in the direction of motion are significantly
are shrinking. Let us connect the reference frame K" with one of the rods by directing one
from the axes along the rod (Fig. 16.4). Then in this system rod 1 will be
be at rest and its length will be equal to its own length /0. Length /
rod 2 relative-Fig. 16.4 Reference systems K"
/=/oVi-i4/^
where rel is the speed of the rod 2 relative to the system K".
Velocity rel can be found by the formula for adding velocities
IV + and"
rel ~ , . 2
1 + l)0 them,/s
Since the reference frame K" is connected with one of the rods, the speed u0
motion of the frame of reference K relative to the frame K" in magnitude will be
equal to speed and rod
1 and is directed in the opposite direction. Speed ​​and*. rod 2,
moving relative to the frame K", in the frame of reference K is also equal to u.
If the O "X" axis is directed along the movement of the rod 1, then the projections of the velocities
u0 and \> x, on this axis will be negative (Fig. 16.4). Therefore, the speed
rod moving relative to the system K\ will be equal to
- and - and iotn -,. 2¦
1 + i i / s
Consequently,
,= / V7Tl^I-i
1 - "P
I c4 + 2 u2 c2 + u4 - 4 u2 c2
/2 2H2 - "o 2 2 -" 0 2 2 * ^ SM"
(c + u) C+U C+U
c2-u2
Answer". I = 10 --- x * 38 cm.
16.9. The accelerator imparted to the radioactive nucleus a speed u = 0.4 s (where c =
3-108 m/s). At the moment of departure from the accelerator, the nucleus was ejected in the direction
of its motion a particle with a speed o2 = 0.75 s relative to the accelerator.
What is the speed of the particle relative to the nucleus?
16.10. Two particles are moving at right angles to each other
velocities u, = 0.5 s and o2 = 0.75 s (where c = 3108 m/s) measured
relative
596
relative to the same reference frame K. What is the relative
particle speed?
16.11. When the body moved, its longitudinal dimensions decreased by n = 2
times. How many times has the body weight changed?
Solution. Prn of particle motion with velocity and its relativistic
the mass m increases in comparison with the rest mass m0 in V 1 - u2 / c2 times:
t0
"Vi-u2/c2"
It is known that in the transition from one reference system to another, the dimensions of the body
change, in this case the Lorentz contraction occurs only in the direction
movement. If in the frame of reference associated with the body, its longitudinal
dimensions have some value /0, then in the reference system, relative to
which the body moves with speed and, they are reduced in V 1 - u2 / c2 times:

7. RELATIVISTIC MECHANICS

Velocity addition rule:

1 V c2

speeds in two

inertial coordinate systems moving relative to each other with a speed V .

Lorentz length contraction and deceleration of a moving clock:

								Where		own length,

own time of the moving clock.

Relativistic mass and relativistic momentum:

The rest mass of the particle.

Total and kinetic energy of a relativistic particle:

				T E E0
					Where E 0			The rest energy of a particle.

7.1. The volume of water in the ocean is V=1.37·109 km3. How much will the mass of water in the ocean change if its temperature is increased by 1o C.

7.2. The ratio of the charge of a moving electron to its mass, determined from experience q/m=0.88 10 11 C/kg. Determine the relativistic mass of the electron and its speed. Answer: m=2m0 ; v=0.87c.

7.3. In the laboratory reference system, one of two identical

particle with mass m0 is at rest, the other one moves with speed v=0.8c towards the resting particle. Determine Relativistic Mass

moving particle in the laboratory frame of reference and its kinetic energy. Answer: m=1.67 m0 ; E \u003d 0.67 m0 s2.

7.4. The electron moves with speed v=0.6s. Define it

relativistic momentum and kinetic energy E. Answer:

p=2.05 10-22 kg m/s; E=0.128 MeV.

7.5. The momentum p of a relativistic particle is equal to m 0 c (m0 - rest mass). Determine the speed of the particle v in fractions of the speed of light and

the ratio of the mass of a moving particle to its rest mass m/m0 . Answer: v=0.71s; m/m0 =1.41.

7.6. The total energy of the α-particle increased during the accelerated

particle motion?

E=56.4 MeV. How fast is it moving

will the mass of the particle change? rest mass α-

particles m0 =4 a.m.u. Answer: m=1.5m0 ; v=0.917s.

7.7. Let's assume that we can measure the length of the rod with an accuracy of l = 0.1 µm. At what relative speed u of two inertial reference frames could one detect a relativistic shortening of the length of a rod whose own length l 0 =1 m? How many times will the mass of the rod change when it moves with the calculated speed u relative to a fixed frame of reference?

Answer: u=134 km/s; m/m0 = 1.114.

7.8. The proper lifetime of some unstable particle

20ns Answer: v = 0.87c; S = 5.2m.

7.9. The μ meson, born in the upper layers of the earth's atmosphere, moves at a speed of V = 0.99 s relative to the earth and flies from the place of its birth to the point of decay, the distance l = 3 km. Determine the proper lifetime of this meson and the distance it will fly in this frame of reference "from its point of view". Answer: τ0 =1.4 µs; l 0 \u003d 420 m.

7.10. Two rods of the same intrinsic length l 0 move in the longitudinal direction towards each other parallel to the common axis with the same speed v=0.8s relative to the laboratory reference system. How many times does the length of each rod l in the reference system associated with another rod differ from its own length? Answer: l 0 / l \u003d 4.6.

7.11. On the spacecraft-satellite there are clocks synchronized with the earth clock before the flight. Satellite speed v=7.9 km/s. How far will the clock on the satellite according to the measurements of an earth observer over a time interval of 0.5 years. How do the values ​​of the kinetic energy of the satellite differ if the calculation is carried out according to the classical and

relativistic formulas? The rest mass of the satellite is 10 tons. Answer: τ=5.4 10-3 s; they do not differ.

7.12. What relative error will be allowed if the calculation of the momentum of a particle moving at a speed of: 1) 10 km/s, 2) 103 km/s, 3) 105 km/s, 4) 0.9 s. Produce within the framework of classical mechanics?

Answer: 1) prel /pclass =1; 2) prel /pclass = 1; 3) prel / pclass = 1.06; 4) rrel /rclass =

7.13. What work needs to be done so that the speed of particles with a rest mass m0 changes from 0.6s to 0.8s? Compare the result obtained with the value of the work calculated according to the classical formula. Answer: Arel \u003d 0.417m0 c2; A class =0.14 m0 c2 .

7.14. The photon rocket moves relative to the earth at such a speed that, according to the clock of an observer on Earth, the passage of time in it slows down by 1.25 times. What fraction of the speed of light is the speed of the rocket? By how much will its linear dimensions change in the direction of motion, if the original length of the rocket was

35m? Answer: v=0.6c; l = 7m.

7.15. Particle with rest mass m 0 at time t = 0 begins to move under the action of a constant force F. Find the dependence of the velocity V of the particle on time t. Construct a qualitative graph V(t).

7.18. The kinetic energy of the accelerated proton increased to 3 10-10 J. How many times has the mass of the proton changed in this case? What is the speed of a proton? Answer: m/m0 =3; v=2.8∙108 m/s.

7.19. Two relativistic particles are moving in a laboratory

reference system with velocities v1 =0.6s and v2 =0.9s along one straight line. Determine their relative speed in two cases: 1) the particles move in opposite directions, 2) the particles move in the same direction. What is the kinetic energy of the first particle in the reference frame associated with the second, if the first particle is a proton?

Answer: 1) v=0.974s, E1.2=510 pJ; 2) v=0.195s, E1.2=300 pJ.

7.20. With what speed (in fractions of the speed of light) should an electron move in order for its mass to increase by 6·10-31 kg. What is the kinetic energy of an electron at this speed? Answer: v=0.8c; E = 0.34 MeV.

7.21. The kinetic energy of a moving body is twice the rest energy. By how many times does the apparent size decrease?

body in the direction of travel? What is the speed of the body? Answer: l 0 /l =3; v=0.94c.

7.22. The mass of the moving particle increased by 1.5 times. What is the speed of the particle? What relative error will be made if the kinetic energy of a particle under these conditions is calculated in a classical way? Answer: v=0.75 c; E/Erel =0.44.

7.23. An electron is accelerated in an electric field with a potential difference of U=106 V. Calculate the electron speed and its kinetic energy using the following methods: 1) classical mechanics, 2)

relativistic mechanics. Evaluate the received data. Answer: 1) v=6 108 m/s; E=106 eV. 2) v=0.94c; E=10 6 eV.

7.24. The electron in the accelerator has passed the accelerating difference

potentials U=102 kV. How many times has the mass of the particle increased? Calculate its kinetic energy. Answer: m/m0 =1.2; 1.6∙10-14 J.

7.25. Initially, the kinetic energy of a relativistic particle was equal to its rest energy, and then increased by 4 times during accelerated motion. How much will the momentum of the particle increase? With what speed (in fractions of the speed of light) did the particle move initially? Answer: r 2 /p1 = 2.84; v=0.87c.

1.5.1. There is a right triangle with a leg a= 5.00 m and the angle between this leg and the hypotenuse α = 30°. Find in reference system K", moving relative to this triangle with speed = 0.866 c along the leg a:

a) the corresponding value of the angle α";

b) length l" hypotenuse and its relation to its own length.

1.5.2. Find the proper length of the rod if K- reference frame its speed = c/2, length l= 1.00 m and the angle between it and the direction of motion = 45°.

1.5.3. The rod flies at a constant speed past a mark that is stationary in K- reference system. Time of flight = 20 ns in K-system. In the frame of reference associated with the rod, the mark moves along it for t = 25 ns. Find the proper length of the rod.

1.5.4. Two particles moving in the laboratory reference frame along the same straight line with the same speed hit the stationary target with a time interval = 50 ns. Find the proper distance between particles before hitting the target.

1.5.5. Two rods of the same intrinsic length l 0 move towards each other parallel to a common horizontal axis. In the reference frame associated with one of the rods, the time interval between the moments of coincidence of the left and right ends of the rods turned out to be equal to . What is the speed of one rod relative to the other?

1.5.6. Kernel AB, oriented along the axis x K x A, back - dot B. Find the proper length of the rod if at the moment t A point coordinate A is equal to x A , and at the moment t B point coordinate B is equal to X b.

1.5.7. Kernel AB, oriented along the axis x K-frame of reference, moves at a constant speed in the positive direction of the axis x. The front end of the rod is the point A, back - dot B. After what period of time is it necessary to fix the coordinates of the beginning and end of the rod in K‑system so that the difference in coordinates is equal to the proper length of the rod.

1.5.8. K"- the frame of reference moves in the positive direction of the axis x K-systems with speed V regarding the latter. Let at the moment of coincidence of the origins of coordinates O and O" the readings of the clocks of both systems at these points are equal to zero. Find in K-system the speed of movement of a point in which the clock readings of both reference systems will be the same all the time. Make sure that .

1.5.9. At two points K-system events occurred separated by a time interval . Show that if these events are causally related in K-system (for example, a shot and hitting a target), then they are causally related in any other inertial K"- reference system.

1.5.10. In plane xy K-frame of reference a particle is moving, the projections of the velocity of which are equal to and . Find speed " this particle in K'- a system that moves at a speed V relatively K‑systems in the positive direction of its axis x.

1.5.11. Two particles are moving towards each other with velocities = 0.50 c u = 0.75 c with respect to the laboratory reference system. Find:

a) the rate at which the distance between particles decreases in the laboratory frame of reference;

b) relative velocity of particles.

1.5.12. Two relativistic particles are moving at right angles to each other in a laboratory frame of reference, one with a speed and the other with a speed. Find their relative speed.

1.5.13. The particle is moving in K-system with speed at an angle to the axis x. Find the corresponding angle in K"- a system moving at a speed V relatively K-system in the positive direction of its axis x if the axes x and x" both systems are the same.

1.5.14. K"- the system moves at a constant speed V relatively K-systems. Find acceleration a" particles in K"-system, if K-system it moves with speed and acceleration a in a straight line:

a) in the direction of the vector V;

b) perpendicular to the vector V.

1.5.15. What work must be done to increase the speed of a particle with mass m from 0.60 c up to 0.80 c? Compare the result obtained with the value calculated by the non-relativistic formula.

1.5.16. Find the velocity of a particle whose kinetic energy T= 500 MeV and momentum p= 865 MeV/ c, where c is the speed of light.

1.5.17. Mass Particle m moves along the axis x K- reference systems according to the law , where d is some constant c- the speed of light, t- time. Find the force acting on the particle in this frame of reference.

1.5.18. Neutron with kinetic energy T = 2mc 2 , where m- its mass, collides with another resting neutron. Find their center of mass in the system:

a) the total kinetic energy of neutrons;

b) momentum of each neutron.

1.5.19. Mass Particle m in the moment t= 0 starts to move under the action of a constant force F. Find the speed of the particle and the distance traveled by it as a function of time t.

1.5.20. A relativistic rocket ejects a jet of gas at a non-relativistic speed u, constant with respect to the rocket. Find the dependence of the rocket speed on its mass m, if at the initial moment the mass of the rocket is equal to m 0 .