For the decay Λ^0 →p+π^-, calculate (a) the Q -value (energy...

For the decay $\Lambda^{0} \rightarrow \mathrm{p}+\pi^{-},$ calculate $(a)$ the $Q$ -value (energy released), and ( $b$ ) the kinetic energy of the $\mathrm{p}$ and $\pi^{-}$, assuming the $\Lambda^{0}$ decays from rest. (Use relativistic formulas.)

Key Concepts

Two-Body Decay Kinematics

Two-body decay kinematics examines the specific scenario where a particle decays into exactly two products. When the original particle is at rest, the decay products are emitted in opposite directions to conserve momentum, and their individual energies are determined uniquely by the masses involved and the Q-value of the decay.

Conservation of Energy and Momentum

In particle decays, the laws of conservation of energy and momentum require that the total energy (including rest mass energy and kinetic energy) and the total momentum before and after the decay remain equal. This principle is fundamental when determining the kinetic energies and momenta of the decay products, especially in relativistic contexts.

Relativistic Kinematics

Relativistic kinematics involves applying the principles of special relativity to systems where speeds are comparable to the speed of light. In the context of particle decays, it ensures that energy and momentum are conserved using formulas that account for the relativistic relationship between energy, mass, and momentum.

Q-value in Particle Decay

The Q-value represents the total energy released during a decay process and is calculated as the difference between the rest mass energy of the initial particle and the combined rest mass energies of the decay products. This energy difference, multiplied by c² if using mass units, manifests as the kinetic energy imparted to the decay products.

Transcript

00:01 In this problem, we're looking at the decay of a lambda particle into a proton and an antipion.

00:08 We're given that the lambda particle begins at rest, so it's purely a conversion of rest mass energy into new particles and connect energy.

00:17 We want to find the connect energy of the product particles.

00:21 So that's just the leftover energy from the formation of the particles becomes the connect energy.

00:28 So the first thing we do is find the q value of the reaction.

00:38 So that is the mass of the lambda minus the product masses, which can be equivalently written as 1 -115 .7 minus 938 .3 plus 139 .6.

01:13 Times c squared so that cancels we have energy so it's mega electron volts that works out to 37 .8 mega electron volts so our kinetic energy is going to sum up to 37 .8 so uh so lambda is at rest so p total equals 0 is p1 plus p2 so that means that proton momentum the magnitude of the momentum are going to be equal so we can just use the letter p in the energy equations since those magnitudes will be the same so the proton kinetic energy becomes p c squared 938 dot 3m.

03:19 Minus 139.

03:23 Dot 6.

03:27 So this just comes from the definition of energy.

03:30 The rest mass plus the connect energy.

03:32 You rearrange and it's the e squared form.

03:37 So there's a square root.

03:40 So let's see...

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