Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

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: Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

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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,

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

or Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.373. Find the velocity at which the kinetic energy of a particle equals its rest energy.

Solution. 373. 

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

or    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

or    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

So    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Thus    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

But Classically,    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Hence if   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

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 m0 depends on its kinetic energy. Calculate the momentum of a proton whose kinetic energy equals 500 MeV.

Solution. 375. From the formula

we find Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

or    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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 m0 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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

since I = Ne, N being the number of particles striking the target per second. Also,

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

The density of the moving element is, from 1.369,  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

and the momentum transferred per unit time per unit area is

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

In the frame fixed to the gas When the sphere hits a stationary particle, the latter recoils with a velocity

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

The momentum transferred is   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

and the pressure   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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   

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Integrating   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

or using r = 0 at r - 0, we get,  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.379. A particle of rest mass m0 moves along the x axis of the frame K in accordance with the law Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev 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.   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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. 

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Thus   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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: 

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Solution. 381. By definition,

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

where  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev i.s the invariant interval (dy = dz - 0)

Thus,  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev
Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

In the moving frame,  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Note that  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.383. Demonstrate that the quantity E2 — p2c2  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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Similarly  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Then  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.384. A neutron with kinetic energy T = 2m0c2, where m0 is its rest mass, strikes another, stationary, neutron. Determine: 
 (a) the combined kinetic energy 
Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevof both neutrons in the frame of their centre of inertia and the momentu Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev each neutron in that frame;
 (b) the velocity of the centre of inertia of this system of particles. Instruction. Make use of the invariant E2  — p2c2  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 

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

where wc have used the result of problem (Q.375) Then

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

 Total energy in the CM frame is

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

So    Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.385. A particle of rest mass m0 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. Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Also   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


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

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.387. A stationary particle of rest mass m0 disintegrates into three particles with rest masses m1, m2, and m3. Find the maximum total energy that, for example, the particle m1 may possess. 

Solution. 387. We have

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Hence  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

The L.H.S.   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

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   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRevIrodov Solutions: Relativistic Mechanics- 3 Notes | EduRev


Q.388. A relativistic rocket emits a gas jet with non-relativistic 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 m0.

Solution. 388. The velocity of ejected gases is u realtive to the rocket. In an earth centred frame it is

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

in the direction of the rocket The momentum conservation equation then reads

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

or  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Here - dm is the mass of the ejected gases, so 

Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Integrating   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

The constant   Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

Thus  Irodov Solutions: Relativistic Mechanics- 3 Notes | EduRev

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