Question about relativistic energy

[andrew410]

A pion at rest ($$m_pi = 273m_e$$) decays to a muon (mass = 207$$m_e$$ and an antineutrino (mass = 0). Find the kinetic energy of the muon and the energy of the antineutrino in electron volts.

How am I supposed to start this problem? ANy help would be great...thx!

 

[dextercioby]

If the muonic antineutrino has no rest mass,then his energy & rel.momentum vector in modulus are linked through

$$E_{\bar{\nu}_{\mu}}=pc$$

The key point is that both energy (seen as the time component of the energy-momentum 4-vector) and relativistic momentum (seen as the space components of the energy-momentum 4-vector) are conserved.

Daniel.

P.S.Einstein's formula is $$E^{2}=m^{2}c^{4}+\left|\vec{p}\right|^{2}c^{2}$$.

 

[thepatientmental]

To solve this problem, we can use the equation for relativistic energy: E = mc^2 / √(1-v^2/c^2). First, we need to find the velocity of the muon after the pion decays. Since the pion is at rest, its initial velocity is 0. We can use conservation of momentum to find the velocity of the muon:

0 = m_pi * v_pi + m_mu * v_mu

Since the pion is at rest, v_pi = 0. Solving for v_mu, we get:

v_mu = -m_pi / m_mu * v_pi = 0

Therefore, the velocity of the muon is also 0 after the pion decays. Now, we can plug in the values for mass and velocity into the equation for relativistic energy:

E_mu = m_mu * c^2 / √(1-0^2/c^2) = m_mu * c^2

Substituting in the given masses, we get:

E_mu = (207m_e) * (c^2) = (207 * 9.109 * 10^-31 kg) * (2.998 * 10^8 m/s)^2 = 3.358 * 10^-12 J = 2.095 MeV

To find the energy of the antineutrino, we can use the fact that the total energy of the system must be conserved. Since the pion was at rest, its initial energy is 0. Therefore, the total energy after the decay must also be 0. We can set up an equation to solve for the energy of the antineutrino:

E_pi + E_mu + E_antineutrino = 0

Substituting in the values for the masses and the energy of the muon that we just calculated, we get:

0 + 2.095 MeV + E_antineutrino = 0

Solving for E_antineutrino, we get:

E_antineutrino = -2.095 MeV

Since we are looking for the energy in electron volts, we need to convert the units:

E_antineutrino = (-2.095 MeV) * (1.602 * 10^-13 J/MeV) = -3.353 eV

Note that the negative sign indicates that the antineutrino