Irodov Problem 1.376

The number of particles impinging upon the target each second is given by



The momentum of each particle is given by the solution in Problem 1.375 as,




The total change in momentum of all the n particles each second is the force imparted in the target and is given by,




The total energy of the n particles getting absorbed into the target is

Irodov Problem 1.375

From the solution to Problem 1.374 we know that given that the particle has a kinetic energy T and a rest mass of m0,











Also we know that the momentum of the particle is given by,





From (1), (2) and (3) we can write,

Irodov Problem 1.374

Let T be the kinetic energy of the particle. Let its rest mass be mo. Let the ratio of its kinetic energy to rest mass energy be equal to,




If we used Newtonian mechanics to determine the velocity we would have,





If we used Relativistic mechanics on the other hand we would have,
















The relative error in the estimation of velocity would then be,














For small values of we can consider only the linear terms in equation (4) and we have,

Irodov Problem 1.373

When the particle moves at velocity v, its kinetic energy is given by




Thus, if its kinetic energy is equal to its rest mass energy, then we have,

Irodov Problem 1.372

All the work done on the particle translates into kinetic energy of the particle.
The Newtonian kinetic energy is given by,



The relativistic kinetic energy is given by the change in total energy in the particle given by,

Irodov Problem 1.371

The ratio of relativistic momentum to Newtonian momentum is given by,

Irodov Problem 1.370

As the particle moves its mass increases. When the velocity of the particle is v, its mass is given by, .

Consequently its momentum is given by



Now we can solve for v from equation from (1) and obtain,