Q. 356. At two points of the reference frame K two events occurred separated by a time interval Δt. Demonstrate that if these events obey the causeandeffect relationship in the frame K (e.g. a shot fired and a bullet hitting a target), they obey that relationship in any other inertial reference frame K'.
Solution. 356. We can take the coordinates of the two events to be
For B to be the effect and A to be cause we must have
In the moving frame the coordinates of A and B become
Since
Q. 357. The spacetime diagram of Fig. 1.93 shows three events A, B, and C which occurred 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 occurred in the reference frame where these two events are simultaneous.
Solution. 357. (a) The fourdimensional interval between A and B (assuming Δy = Δz = 0) is :
5^{2}  3^{2} bs 16 units
Therefore the time interval between these two events in the reference frame in which the events occurred at the same place is
(b) The four dimensional interval between A and C is (assuming Δy = Δz = 0)
3^{2}  5^{2} =  16
So the distance between the two events in the fram in which they are simultaneous is 4 units = 4m.
Q. 358. The velocity components of a particle moving in the xy plane of the reference frame K are equal to v_{x} and v_{y}. Find the velocity v' of this particle in the frame K' which moves with the velocity V relative to the frame K in the positive direction of its x axis.
Solution. 358. By the velocity addition formula
and
Q. 359. Two particles move toward each other with velocities v_{1} = 0.50c and v_{2} = 0.75c relative to a laboratory frame of reference.
Find:
(a) the approach velocity of the particles in the laboratory frame of reference;
(b) their relative velocity.
Solution. 359. (a) By definition the velocity of apporach is
in the reference frame K .
(b) The relative velocity is obtained by the transformation law
Q. 360. Two rods having the same proper length l_{0} move lengthwise toward each other parallel to a common axis with the same velocity v relative to the laboratory frame of reference. What is the length of each rod in the reference frame fixed to the other rod?
Solution. 360. The velocity of one of the rods in the reference frame fixed to the other rod is
The length of the moving rod in this frame is
Q. 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_{2}. Find their relative velocity.
Solution. 361. The approach velocity is defined by
in the laboratory frame.
On the other hand, the relative velocity can be obtained by using the velocity addition formula and has the components
Q. 362. An unstable particle moves in the reference frame K' along its y' axis with a velocity v'. In its turn, the frame K' moves relative to the frame K in the positive direction of its x axis with a velocity V. The x' and x axes of the two reference frames coincide, the y' and y axes are parallel. Find the distance which the particle traverses in the frame K, if its proper lifetime is equal to Δt_{0}.
Solution. 362. The components of the velocity of the unstable particle in the frame K are
so the velocity relative to K is
The life time in this frame dilates to
and the distance traversed is
Q. 363. A particle moves in the frame K with a velocity v at an angle θ 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 direction of its x axis, if the x and x' axes of the two frames coincide.
Solution. 363. In the frame K' the components of the velocity of the particle are
Q. 364. The rod AB oriented parallel to the x' axis of the reference frame K' moves in this frame with a velocity v' along its y' axis. In its turn, the frame K' moves with a velocity V relative to the frame K as shown in Fig. 1.94. Find the angle θ between the rod and the x axis in the frame K.
Solution. 364. In K' the coordinates of A and B are
After performing Lorentz transformation to the frame K we get
By translating we can write
the coordinates of B as B : t = γ t'
Thus
Hence
Q. 365. The frame K' moves with a constant velocity V relative to the frame K. Find the acceleration w' of a particle in the frame K', if in the frame K this particle moves with a velocity v and acceleration w along a straight line
(a) in the direction of the vector V;
(b) perpendicular to the vector V.
Solution. 365.
In K the velocities at time t and t + dt are respectively v and v + wdt along x  axis which is parallel to the vector In the frame K' moving with velocity with respect to K, the velocities are respectively,
The latter velocity is written as
Also by Lorentz transformation
Thus the acceleration in the K' frame is
(b) In the K frame the velocities of the particle at the time t and t + di are repectively
where is along jtaxis. In the K frame the velocities are
and
Thus the acceleration
Q. 366. An imaginary space rocket launched from the Earth moves with an acceleration w' = 10g which is the same in every instantaneous comoving inertial reference frame. The boost stage lasted τ =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 cover by that moment?
Solution. 366. In the instantaneous rest frame v = V and
So,
w' is constant by assumption. Thus integration gives
Integrating once again
Q. 367. From the conditions of the foregoing problem determine the boost time τ_{0} in the reference frame fixed to the rocket. Remember that this time is defined by the formula where dt is the time in the geocentric reference frame.
Solution. 367. The boost time τ_{0} in the reference frame fixed to the rocket is related to the time τ elapsed on the earth by
Q. 368. How many times does the relativistic mass of a particle whose velocity differs from the velocity of light by 0.010% exceed its rest mass?
Solution. 368.
For
Q. 369. The density of a stationary body is equal to p_{0}. Find the velocity (relative to the body) of the reference frame in which the density of the body is η = 25% greater than p_{0}.
Solution. 369. We define the density p in the frame K in such a way that p dx dy dz is the rest mass dm_{0} of the element That is p dx dy dz = p_{0} dx_{0} dy_{0} dz_{0}, where p_{0} is the proper density dx_{0}, dy_{0} , dz_{0} are the dimensions of the element in the rest frame K_{0}. Now
if the frame K is moving with velocity, v relative to the frame K_{0}. Thus
Defining
We get
or
Q. 370. A proton moves with a momentum p = 10.0 GeV/c, where c is the velocity of light. How much (in per cent) does the proton velocity differ from the velocity of light?
Solution. 370. We have
or
or
Q.371. Find the velocity at which the relativistic momentum of a particle exceeds its Newtonian momentum η = 2 times.
Solution. 371. By definition of η,
or
Q.372. What work has to be performed in order to increase the velocity of a particle of rest mass mo from 0.60 c to 0.80 c? Compare the result obtained with the value calculated from the classical formula.
Solution. 372. The work done is equal to change in kinetic energy which is different in the two cases Classically i.e. in nonrelativistic mechanics, the change in kinetic energy is
Q.373. Find the velocity at which the kinetic energy of a particle equals its rest energy.
Solution. 373.
or
or
Q.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 ε = 0.010?
Solution. 374. Relativistically
So
Thus
But Classically,
Hence if
the velocity β is given by the classical formula with an error less than ε.
Q.375. Find how the momentum of a particle of rest mass m_{0} depends on its kinetic energy. Calculate the momentum of a proton whose kinetic energy equals 500 MeV.
Solution. 375. From the formula
we find
or
Q.376. A beam of relativistic particles with kinetic energy T strikes against an absorbing target. The beam current equals I, the charge and rest mass of each particle are equal to e and m_{0} respectively. Find the pressure developed by the beam on the target surface, and the power liberated there.
Solution. 376. Let the total force exerted by the beam on the target surface be .F and the power liberated there be P. Then, using the result of the previous problem we see
since I = Ne, N being the number of particles striking the target per second. Also,
These will be, respectively, equal to the pressure and power developed per unit area of the target if I is current density.
Q.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 pressure 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.
Solution. 377. In the tome fixed to the sphere The momentum transferred to the eastically scattered particle is
The density of the moving element is, from 1.369,
and the momentum transferred per unit time per unit area is
In the frame fixed to the gas When the sphere hits a stationary particle, the latter recoils with a velocity
The momentum transferred is
and the pressure
Q.378. A particle of rest mass mo starts moving at a moment t = 0 due to a constant force F. Find the time dependence of the particle's velocity and of the distance covered.
Solution. 378. The equation of motion is
Integrating
or using r = 0 at r  0, we get,
Q.379. A particle of rest mass m_{0} moves along the x axis of the frame K in accordance with the law 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.
Solution. 379.
Q.380. Proceeding from the fundamental equation of relativistic dynamics, find:
(a) under what circumstances the acceleration of a particle coincides in direction with the force F acting on it;
(b) the proportionality factors relating the force F and the acceleration w in the cases when F ⊥ and F II v, where v is the velocity of the particle.
Solution. 380.
Thus
Q.381. A relativistic particle with momentum p and total energy E moves along the x axis of the frame K. Demonstrate that in the frame K' moving with a constant velocity V relative to the frame K in the positive direction of its axis x the momentum and the total energy of the given particle are defined by the formulas:
Solution. 381. By definition,
where i.s the invariant interval (dy = dz  0)
Thus,
Q.382. The photon energy in the frame K is equal to ε. Making use of the transformation formulas cited in the foregoing problem, find the energy ε' of this photon in the frame K' 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 ε' = ε/2?
Solution. 382. For a photon moving in the x direction
In the moving frame,
Note that
Q.383. Demonstrate that the quantity E^{2} — p^{2}c^{2} for a particle is an invariant, i.e. it has the same magnitude in all inertial reference frames. What is the magnitude of this invariant?
Solution. 383. As before
Similarly
Then
Q.384. A neutron with kinetic energy T = 2m_{0}c^{2}, where m_{0} is its rest mass, strikes another, stationary, neutron. Determine:
(a) the combined kinetic energy of both neutrons in the frame of their centre of inertia and the momentu 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).
Solution. 384. (b) & (a) In the CM frame, the total momentum is zero, Thus
where wc have used the result of problem (Q.375) Then
Total energy in the CM frame is
So
Q.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.
Solution. 385.
Also
Q.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 GeV?
Solution. 386. Let T = kinetic energy of a proton striking another stationary particle of the same rest mass. Then, combined kinetic energy in the CM frame
Q.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.
Solution. 387. We have
Hence
The L.H.S.
The R.H.S. is an invariant We can evaluate it in any frame. Choose the CM frame of the particles 2 and 3.
In this frame
Q.388. A relativistic rocket emits a gas jet with nonrelativistic velocity 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}.
Solution. 388. The velocity of ejected gases is u realtive to the rocket. In an earth centred frame it is
in the direction of the rocket The momentum conservation equation then reads
or
Here  dm is the mass of the ejected gases, so
Integrating
The constant
Thus
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