A particle at rest, with a rest energy of m c^2, decays into...

Key Concepts

Conservation of Energy

The principle of conservation of energy states that the total energy in an isolated system remains constant over time. In relativistic particle decays, this includes not only the rest energies of particles but also their kinetic energies. When a particle decays, the sum of the energy of the resulting fragments equals the energy of the original particle, allowing one to solve for unknown quantities such as kinetic energy.

Conservation of Momentum

Conservation of momentum holds that the total momentum of an isolated system is conserved. In a decay process where the initial particle is at rest, the momentum of the resulting particles must cancel out, meaning they have equal and opposite momenta. This condition is crucial in setting up equations that, in conjunction with energy conservation, help derive relations for the kinetic energies of decay products.

Relativistic Energy-Momentum Relation

This key relation, given by E² = (pc)² + (m?c²)², links a particle’s total energy, its momentum, and its rest mass. It is fundamental in relativistic mechanics and is used to connect the kinetic energy of decay fragments to their momenta and rest energies, enabling the derivation of formulas that account for relativistic effects in high-energy particle decays.

Invariant Mass

Invariant mass, also known as rest mass, is a measure of the total mass of a system that remains the same in all reference frames. In particle decays, the invariant mass of the decaying system is related to the masses and energies of the decay products. This concept allows one to analyze the decay process consistently, using conserved quantities to derive expressions for observable parameters such as kinetic energy.

Transcript

00:01 Okay, so problem number 66.

00:03 It wants us to, we have two particles.

00:06 We have basically that decays that shows this particular decay path.

00:12 So you have some original particle and it decays into two new particles given by these two mass energies, rest energies.

00:23 We want to show that the kinetic energy of particle 1 is equal to, so k1 is equal to, 1 over 2 m c squared times m c squared minus m1 c squared squared minus m2 c squared squared squared all right so to do this we need to start with first the conservation of momentum and that makes it so from conservation momentum this side is at rest and it's a decay.

01:04 So that means that the total momentum on this side is 0, which means the total magnitude of momentum on this side, i'm sorry, the total momentum on this side has to be equal to zero, which means that the momentum of particle 1 has to be equal in magnitude, but opposite in direction of the momentum in particle 2.

01:24 So we can say p1 is equal to p2.

01:28 Now, from the, where? the relativistic energy equation, e squared, is equal to p .c.

01:35 Squared.

01:37 I'm going to write this on the other side, so give myself some more room.

01:41 So the relativistic energy equation is given by e squared is equal to pc squared plus m squared c to the fourth.

01:52 And then rearranging that for pc squared.

01:54 You get p c squared is equal to e squared minus m squared c to the fourth okay so now from that we can put this in the terms of p1c squared is equal to p2c squared now the total energy we can use conservation of energy now that we have the conservation of momentum laws we can use the conservation of energy laws to state that the total energy on this side, the left -hand side, is going to be equal to the total energy on the right -hand side.

02:32 So we have e -0, which is the total energy on the left -hand side, which is just equal to the rest -mass, the rest energy of the original particle, is going to be equal to e1 plus e.

02:48 Okay.

02:50 Now, we can rearrange this a little bit.

02:53 And show that e2, giving myself some more room over here, e2 is equal to m0 squared, c squared minus e1.

03:12 Okay, so now we have, now we have, excuse me, so we have now the equation of p1c squared equals p2c squared.

03:33 So where p1c squared is equal to is going to be equal to e1 squared squared minus m1 squared c to the fourth, and p2c squared is equal to p2 squared minus m2 squared c to the fourth.

04:10 Okay, so now we can plug in what we know.

04:16 E2 is equal to this.

04:19 So we have e2 can be given by this.

04:35 So we have e2 squared is going to be equal to m0 c squared minus e1 squared which is going to be equal to p2 squared plus m2 squared c to the fourth.

05:09 So that's e2 squared, which is also equal to this.

05:14 So i will go ahead and put this in red.

05:21 Okay.

05:24 So now we can state that p2 from conservation and momentum, i put this in green, p2 squared, is just going to be equal to, let's do this, from conservation and momentum is just going to be equal to p1 squared.

05:53 So we can then plug that in.

05:56 So this is going to be equal to p1 squared c squared plus m2, c2 squared, m2 squared, c to the fourth...

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