Question about relativistic energy

In summary, a pion at rest with a mass of 273 times the mass of an electron decays into a muon with a mass of 207 times the mass of an electron and an antineutrino with no rest mass. The energy of the antineutrino is linked to its relativistic momentum through the equation E_{\bar{\nu}_{\mu}}=pc. This is due to the conservation of energy and relativistic momentum. According to Einstein's formula, E^{2}=m^{2}c^{4}+\left|\vec{p}\right|^{2}c^{2}.
  • #1
andrew410
59
0
A pion at rest ([tex]m_pi = 273m_e[/tex]) decays to a muon (mass = 207[tex]m_e[/tex] 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!
 
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  • #2
If the muonic antineutrino has no rest mass,then his energy & rel.momentum vector in modulus are linked through

[tex] E_{\bar{\nu}_{\mu}}=pc [/tex]

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 [tex] E^{2}=m^{2}c^{4}+\left|\vec{p}\right|^{2}c^{2} [/tex].
 
  • #3


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
 

Related to Question about relativistic energy

What is relativistic energy?

Relativistic energy is the energy possessed by an object that is moving at a significant fraction of the speed of light. It takes into account both the object's mass and its velocity, and is described by Einstein's famous equation, E=mc^2.

How is relativistic energy different from classical energy?

Classical energy is based on Newtonian physics and only accounts for an object's mass and velocity relative to the observer. Relativistic energy, on the other hand, takes into account the effects of time dilation and length contraction at high speeds. This means that the energy of a fast-moving object will be significantly greater when calculated using the relativistic equations compared to classical equations.

Can an object have infinite relativistic energy?

No, an object cannot have infinite relativistic energy. According to Einstein's theory of special relativity, an object with a finite mass cannot reach the speed of light, and therefore cannot have infinite energy. As an object approaches the speed of light, its energy will continue to increase, but it will never reach infinity.

What practical applications does relativistic energy have?

Relativistic energy has many practical applications, especially in modern physics and engineering. It is used in particle accelerators to produce high-energy collisions, in nuclear reactors and weapons to harness the energy from atomic nuclei, and in space travel to describe the energy needed for a spacecraft to reach high speeds.

How does relativistic energy relate to the concept of mass-energy equivalence?

Relativistic energy and mass-energy equivalence are closely related. Einstein's famous equation, E=mc^2, links the two concepts, showing that mass and energy are two forms of the same thing. This means that mass can be converted into energy and vice versa, and that the total energy of a system is equivalent to its total mass.

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