– Lerenta contraction of lenglh and alowing of a moving clock:
whete $L_{4}$ is the proper length and $\Delta_{4}$ is the proper time of the moving clock.
– Leresta transformation\”:
– Interval $a_{2 x}$ is an invariant:
\[
\mathrm{t}_{\mathrm{a}}=e^{0} \mathrm{t}_{\mathrm{a}}-\mathrm{H}_{\mathrm{a}}-\text { iav, }
\]

Dhere $t_{13}$ is the time interval between events 1 and $2, b_{3}$ is the distance between ithe points at which thes events eccurred.
– Translormation of velocity*:
– Relativistic mass asd melativistic momentum:
– Pielativistic equation of dysamies for a particle:
\[
\frac{d \mathrm{p}}{\mathrm{di}}-\boldsymbol{F} \text {. }
\]
where $p$ is the relativistic momentum of the particle.
– Total asd kiaetic ebergies of a relativistic particle:
\[
E=m r^{2}=m r^{2}+r, \quad r=(m-m) d .
\]
– The relerence frame $K^{*}$ is aumed to move vith s, velocity $V$ in the posi: tive direction of the $x$ axis of the frame $K$, will the $x^{\prime}$ and $x$ axes coinciding and the $y^{\prime}$ and $y$ axes parsiliel.
s.

tiele
– Relationahip between the energy and momentum of a relativistie par-
– Whan considering the collisios of particles it helps to use the followIng invariant quantity:
where $E$ and $p$ are the total energy and aomentum of the system prior to the collisios, and $\mathrm{m}$, is the rest mase of the particle (or the systea) formed.
1.340. A rod moves lengthwise with a constant velocity $v$ relative to the inertial reference frame $K$. At what value of $v$ will the length of the rod in this frame be $\eta=0.5 \%$ less than its proper length?
1.341. In a triangle the proper length of each side equals a. Find the perimeter of this triangle in the reference frame moving relative to it with a constant velocity $V$ along one of its
(a) bisectors; (b) sides.

Investigate the results ebtained at $V<c$ and $V \rightarrow c$, where $c$ is the velocity of light.
1.342. Find the proper length of a rod if is the laboratory frame of reference its velocity is $v=c / 2$, the length $l=1.00 \mathrm{~m}$, and the angle between the rod and its direction of motion is $\theta=45^{\circ}$.
1.343. A stationary upright cone has a taper angle $8^{\circ}=45^{\circ}$, and the area of the lateral surface $S_{0}=4.0 \mathrm{~m}$ ‘. Find: (a) its taper angle; (b) its lateral surface area, in the reference frame moving with a velocity $v=(W / 5)$ along the axis of the cone.
1.34. With what velocity (relative to the reference frame $K$ ) did the clock move, if during the time interval $t=5.0 \mathrm{~s}$, masured by the clock of the frame $K$, It became slow by $\Delta t=0.10$ s?
1.345. A rod fies with constant velocity past a mark which is stationary in the reference frame $K$. In the frame $K$ it takes $\Delta t=$ – 20 ns for the rod te $1 \mathrm{y}$ past the mark. In the reference frame fixed to the rod the mark moves past the rod for $\Delta r^{\prime}=25$ ns. Find the proper length of the rod.
1.346. The proper lifetime of an unstable particle is equal to $\Delta t_{4}=10$ as. Find the distance this particle will traverse till its decay in the laboratery frame of reference, where its lifetime is equal te $\Delta t=20$ as.
1.347. In the relerence frame $K$ a maon moving with a velocity $v=0.990$ travelled a distance $l=3.0 \mathrm{~km}$ from its birthplace to the point where it decayed. Find:
(a) the proper lifetime of this muon;
(b) the distance travelled by the muen in the frame $K$ \”from the muon’s standpoint\”.
1.343. Two particles moving in a laboratory frame of reterence along the same straight line with the same velocity $v=(3 / 4)$ strike against a stationary target with the time interval $\Delta t=50$ ns. Find
the proper distance between the particles prior to their hitting the target.
f.349. A rod moves along a ruler with a constant velocity. When the positions of both ends of the rod are marked simultaneously in the relerence frame fixed to the ruler, the difference of readings on the ruler is equal to $\Delta x_{1}=4.0 \mathrm{~m}$. But when the positions of the rod’s ends se marked slmultaneously in the reference frame fixed to the rod, the diflerence of readings on the same ruler is equal to $\Delta x_{2}=$ $=9.0 \mathrm{~m}$. Find the proper length of the rod and its velocity relative to the ruler.
1.350. Two rods of the same proper length $t_{4}$ move toward each other parallel to a common horizontal axis. In the reference frame fixed to one of the rods the time interval between the moments, when the right and left ends of the rods coincide, is equal to $\Delta t$. What is the velocity of one rod relative to the other?
1.351. Two unstable particles move in the reference frame $K$ slong a straight line in the same direction with a velocity $v=0.990$. The distance between them in this reference frame is eqnal to $t=$ – $120 \mathrm{~m}$. At a certain moment both particles decay simultaneously in the reference frame fixed to them. What time interval between the moments of decay of the two particles will be observed in the frame $\boldsymbol{K}$ ? Which particle decays later in the frame $\boldsymbol{K}$ ?
1.352. $A$ rod $A B$ oriented along the $x$ axis of the reference frame $K$ moves in the positive direction of the $x$ axis with a constant velocity $v$. The point $A$ is the forward end of the rod, and the point $B$ its rear end. Find:
(a) the proper length of the rod, if at the moment $t_{A}$ the coordinate of the point $A$ is equal to $x_{A}$, and at the moment $t_{g}$ the coordinate of the point $B$ is equal to $x_{n}$ :
(b) what time interval should separate the markings of coordinates of the rod’s ends in the frame $K$ for the difference of coordinates to become equal to the proper length of the rod.
1.353. The rod $A^{\prime} B^{\prime}$ moves with a constant velocity $v$ relative to the rod $A B$ (Fig. 1.91). Both rods have the same proper length $l_{\text {, and }}$
Fig. 1.91.
at the ends of each of them clocks are mounted, which are synchronized pairwise: $\boldsymbol{A}$ with $\boldsymbol{B}$ and $\boldsymbol{A}^{\prime}$ with $\boldsymbol{B}^{\prime}$. Suppose the moment when the clock $B^{\prime}$ zets opposite the clock $A$ is taken for the begianing of the time count in the reference frames fixed to each of the rods. Determine:
(a) the readings of the clocks $B$ and $B^{\prime}$ at the moment when they are opposite each other;
(b) the same for the clecks $A$ and $A^{\prime}$.
1.354. There are two groups of mutually synchronired elocks $K$ and $K^{\prime}$ moving relative to each other with a velocity $v$ as shown in Fig. 1.92. The moment when the clock $A^{\prime}$ gets opposite the clock $A$
กie. 1.92.
is taken for the beginning of the time count. Draw the approximate position of hands of all the elocks at this moment \”in terms of the $K$ elocks\”; \”in terms of the $K^{\prime}$ elecks\”.
1.355. The reference frame $K^{\prime}$ moves in the positive direction of the $x$ axis of the frame $K$ with a relative velocity $\boldsymbol{V}$. Suppose that at the moment when the origins of coordinates $O$ and $O^{\prime}$ colneide, the clock readings at these points are equal to zere in beth frames. Find the displacement velocity $\dot{x}$ of the point (in the frame $\boldsymbol{K}$ ) at which the readings of the clocks of both reference frames will be permanently identical. Demonstrate that $\dot{x}<\boldsymbol{V}$.
1.356. At two points of the reference frame $K$ two events occurred separated by a time interval $\Delta t$. Demonstrate that if these events obey the cause-and-effect relationship in the frame $K(6 . g$. a shot fired and a bullet hitting a target), they obey that relationship in any other inertial reference frame $\boldsymbol{K}^{\prime}$.
1.357. The space-time diagram of Fig. 1.93 shews three events $A$, $B$, and $C$ which eccurred on the $x$ axis of some inertial reference frame. Find:
(a) the time interval between the events $A$ and $B$ in the reference frame where the two events occurred at the same point;
(b) the distance between the points at which the events $A$ and $C$ eccurred in the reference frame where these two events are simultaneous.
1.358. The velocity components of a particle meving in the $x y$ plane of the relerence Irame $K$ are equal to $v_{x}$ and $v_{v}$. Find the velocity $v^{\prime}$ of this particle in the frame $\boldsymbol{K}^{\prime}$ which moves with the velocity $\boldsymbol{V}$ relative to the frame $\boldsymbol{K}$ in the positive direction of its $x$ axis. 1.359. Two particles move toward each other with velocities $y_{1}=0.50 \mathrm{e}$ and $v_{1}=0.75 \mathrm{e}$ relative to a laboratery frame of reference. Find:
70
(a) the approach velocity of the particles in the laboratory frame of reference:
(B) their relative velocity.
1.360. Two rods having the same proper length $t_{0}$ move lengthwise toward each other parallel to a cemmon axis with the same velocity
Fig. 1.93.
v relative to the laboratery frame of reference. What is the length of each rod in the reference frame fixed to the other rod?
1.361. Two relativistic particles move at right angles to each other in a laboratory frame of reference, one with the velocity $v_{1}$ and the other with the velocity $v_{v}$. Find their relative velocity.
1.362. An unstable particle moves in the reference frame $\boldsymbol{K}^{\prime}$ along its $y^{\prime}$ axis with a velocity $v^{\prime}$. In its turn, the frame $K^{\prime}$ moves relative to the frame $K$ in the positive direction of its $x$ axis with a velocity $\boldsymbol{V}$. The $\boldsymbol{x}^{\prime}$ and $x$ axes of the two reference frames coineide, the $y$ and $y$ axes are parallel. Find the distance which the particle traverses in the frame $\boldsymbol{K}$, if its proper lifetime is equal to $\Delta t_{0}$.
1.363. A particle moves in the frame $K$ with a velocity $v$ at an angle $\theta$ to the $x$ axis. Find the corresponding angle in the frame $K^{*}$ moving with a velocity $V$ relative to the frame $K$ in the positive $d i$ rection of its $x$ axis, if the $x$ and $x^{\prime}$ axes of the two frames coincide.
1.364. The rod $A B$ oriented parallel to the $x^{\prime}$ axis of the reference frame $\tilde{K}^{\prime}$ weves in this frame with a velocity $v^{\prime}$ along its $y^{\prime}$ axis. In its turn, the frame $\boldsymbol{K}^{\prime}$ moves with a velocity $\boldsymbol{V}$ relative to the frame $K$ as shown in Fig. 1.94. Find the angle $\theta$ between the rod and the $x$ axis in the frame $K$.
1.365. The frame $K^{\prime}$ meves with a constant velocity $\mathbf{V}$ relative to the frame $\boldsymbol{K}$. Find the acceleration $w^{\prime}$ of a particle in the frame $\boldsymbol{K}^{\prime}$, If in the frame $\boldsymbol{K}$ this particle moves with a velocity $v$ and accelerstion $w$ along $a$ straight line
(a) in the direction of the vector $\mathbf{y}$;
(b) perpendicular to the vector $\mathbf{v}$.
1.366. An imaginary space rocket launched from the Earth moves with an acceleration $w^{\prime}=10 \mathrm{~g}$ which is the same in every instantsneous co-moving inertial reference frame. The boost stage lasted
Fie. 1.94.
₹ $=\mathbf{1 . 0}$ year of terrestrial time. Find how much (in per cent) does the rocket velocity differ from the velocity of light at the end of the boost stage. What distance does the rocket cever by that moment?
1.367. From the conditions of the foregoing problem determine the boost time $\tau_{0}$ in the reference frame fixed to the rocket. Remember that this time is defined by the formula
\[
\mathrm{x}_{0}=\int_{0}^{1} \sqrt{1-(v / \epsilon)^{2}} d t \text {. }
\]
where $d t$ is the time in the geocentric reference frame.
1.368. How many times does the relativistie mass of a particle whose velocity differs from the velocity of light by $0.010 \%$ exceed its rest mass?
1.369. The density of a stationary body is equal to $p_{0}$. Find the velocity (relative to the bedy) of the reference frame in which the density of the body is $\eta=25 \%$ greater than $p_{0}$.
1.370. A proton moves with a momentum $p=10.0 \mathrm{GeV} / \mathrm{c}$, where $c$ is the velocity of light. How much (in per cent) does the proton velocity differ from the velocity of light?
1.371. Find the velocity at which the relativistic momentum of a particle exceeds its Newtonian momentum $\eta=2$ times.
1.372. What work has to be performed in order to increase the velocity of a particle of rest mass $m_{0}$ from 0.60 e to $0.80 \mathrm{e}$ ? Compare the result obtained with the value calculated from the classical formula.
1.373. Find the velocity at which the kinetic energy of a partiele equals its rest energy.
1.374. At what values of the ratio of the kinetic energy to rest energy can the velocity of a particle be calculated from the classical formula with the relative error less than $\varepsilon=0.010$ ?
$n$
1.375. Find how the momentum of a particle of rest mass $m$, depends on its kinetic energy. Calculate the momentum of a proton whose kinetic energy equals $5 \mathrm{C} 0 \mathrm{MeV}$.
1.376. A beam of relativistic particles with kinetic energy $T$ strikes against an sbsorbing target. The beam current equals $I$, the charge and rest mass of each particle are equal to $e$ and $m$, respectively. Find the pressure developed by the beam on the target surface, and the power liberated there.
1.377. A sphere moves with a relativistic velocity $v$ through a gas whose unit volume contains $n$ slowly moving particles, each of mass $m$. Find the pressyre $p$ exerted by the gas on a spherical surface element perpendicular to the velocity of the sphere, provided that the particles scatter elastically. Show that the pressure is the same both in the reference frame fixed to the sphere and in the reference frame fixed to the gas.
1.378. A particle of rest mass $m$, starts moving at a moment $t=0$ due to a constant force F. Find the time dependence of the particle’s velecity and of the distance covered.
1.379. A particle of rest mass $m_{\text {, }}$ moves along the $x$ axis of the frame $K$ in accordance with the law $x=\sqrt{a^{2}+C C^{I}}$, where $a$ is a constant, $c$ is the velocity of light, and $t$ is time. Find the force acting on the particle in this reference frame.
1.350. Proceeding from the fundamental equation of relativistic dynamics, find:
(a) under what circumatances the acceleration of a particle coincides in direction with the force $\mathbf{F}$ acting on it:
(b) the proportionality factors relating the force $\boldsymbol{F}$ and the acceleration $\mathbf{w}$ in the cases when $\boldsymbol{F} \perp$, and $\boldsymbol{F} \| \mathbf{v}$, where $\mathbf{v}$ is the velocity of the particle.
1.381. A relativistie particle with momentum $p$ and total energy $E$ moves along the $x$ axis of the frame $K$. Demonstrate that in the frame $\boldsymbol{K}^{\prime \prime}$ moving with a constant velocity $\boldsymbol{V}$ relative to the frame $\boldsymbol{K}$ in the positive direction of its axis $z$ the mementum and the tetal energy of the given particle are defined by the formulas:
where $B=V / e$.
1.382. The photon energy in the frame $K$ is equal to $\varepsilon$. Making use of the transformation formulas cited in the foregoing problem, find the energy $\varepsilon^{\prime}$ of this photon in the frame $\boldsymbol{K}^{\prime}$ moving with a velocity $V$ relative to the frame $K$ in the photon’s motion direction. At what value of $V$ is the energy of the photon equal to $\varepsilon^{\prime}=\varepsilon / 2$ ?
1.383. Demonstrate that the qunntity $E^{2}-p^{4} c^{2}$ for a partiele is an invariant, i.e. it has the same magnitude in all inertial relerence fromes. What is the magnitude of this invariant?
1.384. A neutron with kinetic energy $T=2 m_{4}$, where $m_{0}$ is its rest mass, strikes another, stationary, neutron. Determine:
(a) the combined kinetic energy $\widetilde{T}$ of both neutrons in the frame of their centre of inertia and the momentum $\tilde{p}$ of each neutron in that frame;
(b) the velocity of the centre of inertia of this system of particles.
Instruction. Make use of the invariant $E^{2}-p^{2} c^{2}$ remaining constant on transition from one inertial reference frame to another ( $E$ is the total energy of the system, $p$ is its composite momentum).
1.385. A particle of rest mass $m_{0}$ with kinetic energy $T$ strikes a stationary particle of the same rest mass. Find the rest mass and the velocity of the compound particle formed as a result of the collision.
1.386. How high must be the kinetic energy of a proton striking another, stationary, proton for their combined kinetic energy in the frame of the centre of inertia to be equal to the total kinetic energy of two protons moving toward each other with individual kinetic energies $T=25.0 \mathrm{GeV}$ ?
1.387. A stationary particle of rest mass $m_{0}$ disintegrates into three particles with rest masses $m_{1}, m_{2}$, and $m_{3}$. Find the maximum total energy that, for example, the particle $m_{1}$ may possess.
1.388. A relativistic rocket emits a gas jet with non-relativistic velocity $\mathbf{u}$ constant relative to the rocket. Find how the velocity $v$ of the rocket depends on its rest mass $m$ if the initial rest mass of the rocket equals $m_{0}$.