A particle decays in flight into two pions, each having a rest energy of 140.0 MeV. The pions tr

step 1 So given in the question a particle decays in flight into two pions. The rest energy of each pio

A particle decays in flight into two pions, each having a...

Key Concepts

Invariant Mass

The invariant mass, sometimes referred to as the rest mass, is a fundamental property of a particle or system that remains the same in all inertial reference frames. In particle physics, when a particle decays, the invariant mass of the system of decay products is used to infer properties of the original particle, irrespective of the observers' frame of reference.

Lorentz Factor

The Lorentz factor, denoted by ? (gamma), quantifies the relativistic effects experienced by a moving particle, such as time dilation and length contraction. It is defined by the equation ? = 1/sqrt(1 - v^2/c^2) and is used to adjust quantities like mass, momentum, and energy to their relativistic expressions.

Relativistic Kinetic Energy

Relativistic kinetic energy is the extra energy a particle possesses due to its motion, taking into account relativistic effects at high speeds. It is calculated as the difference between the total energy and the rest energy, ensuring that the calculations remain accurate even when the particle's speed approaches the speed of light.

Energy-Momentum Conservation

This principle states that the total energy and momentum of a closed system remain constant in time, even if the system undergoes transformations like particle decay. In the context of particle decays, it requires that the sum of the energy and momentum of all decay products equals the initial energy and momentum of the particle before decay.

Relativistic Energy-Momentum Relationship

In relativistic mechanics, energy and momentum are linked through the relation E^2 = (pc)^2 + (m_0c^2)^2, which accounts for both the rest mass energy and the kinetic energy of a particle. This equation is essential for analyzing particles moving at speeds close to the speed of light, where classical formulas no longer hold.

Transcript

00:01 So given in the question a particle decays in flight into two pions.

00:10 The rest energy of each pions let it be represented by e naught is given to be 140 .0 million electron volts.

00:23 The pions travel at right angles to each other and the speed of each pion is given to be v that is equal to 0 .900 c where c is the speed of light.

00:39 It has been asked to find the momentum magnitude of the original particle.

00:47 The second part is to find its kinetic energy and third part is to find its mass in units of mev per c square.

01:02 Now this is the original particle and this has been split into two pions.

01:08 Let one pion goes in this direction the other one in this direction making an angle theta and phi relative to the original direction of motion.

01:20 This is the particle and these are the pions.

01:26 The rest mass energy or the rest energy is 140 mev of both the pions and the velocity is 0 .900 c.

01:38 Now let a let the magnitude the momentum magnitude of the original particle be p naught.

01:53 The momentum of each pion let it be represented by p will be given as gamma mv where gamma is the lorentz factor and is given as 1 by under root 1 minus v square by c square.

02:10 V is 0 .900 c putting the value of v we get the lorentz factor to be equal to 2 .294.

02:24 So the momentum of each pion would be 2 .294 times m that is 140 mev per c square times v that is 0 .900 c.

02:36 So it comes out to be 289 .04 mev per c.

02:43 Now if you look into the diagram the along the horizontal direction the momentum add up along the vertical direction they cancel each other which means sin theta is equals to sin phi implies that theta is equal to phi given in the question that theta plus phi is 90 degree therefore theta must be equal to phi must be equal to 45 degree.

03:05 So the momentum or magnitude of the original particle that is p naught would be 2 p will be given using the conservation of linear momentum that is p cos theta plus p cos phi which is 2 p cos 45 degrees and 2 times 289 .04 mev per c times 1 by root 2 and it comes out to be 408 .8 mev per c that is 409 mev per c.

03:41 So this is the answer to the first part of the question.

03:44 The second part is we have to ask we have been asked to find the kinetic energy of the original particle.

03:51 So let the kinetic energy of the original particle be k...

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