Question

1. A muon and an antimuon are moving in opposite directions with velocities of -0.6 c and 0.6 c, respectively. When they collide, they can annihilate each other and turn into two photons. Both muon and antimuon have a mass of 105.66 MeV/c2. What is the total energy before the collision?

2. What is the energy of each photon?

3. What is the wavelength of each photon?

Answer 1

Given

Magnitude of velocities of muon v = 0.6c

Their rest mass m_o = 105.66 MeV/ c²

Solution

The energy associated with the muons while they are travelling

E = γE_o

Where,

γ = 1/√{1-(v²/c²)}

and

E_o = m_oc²

γ = 1/√{1-(v^2‑/c²)}

γ = 1/√{1-[(0.6c)²/c²]}

γ = 1/√{1-[(0.6c)²/c²]}

γ = 1/√{1-0.36}

γ = 1/√{0.64}

γ = 1/0.8

γ = 1.25

E_o = m_oc²

E_o = 105.66 MeVc^-2 x c²

E_o = 105.66 MeV

E = γE_o

E = 1.25 x 105.66

E = 132.075 MeV

This is the energy associated with each muon.

Energy of the muon is 132.075 MeV

As per the conservation of energy the total energy of the each created photon will be 132.075 MeV

The energy of a photon

E = hν

Where h is the Planck’s constant and v is the frequency of the photon

Since frequency ν = c/λ

E= hc/λ

132.075 MeV = 6.626 x10^-34 x 2.99 x10⁸ / λ

132.075 x 10⁶ x 1.6 x 10^-19Joules = 6.626 x10^-34 x 2.99 x10⁸ / λ

λ = 6.626 x10^-34 x 2.99 x10⁸ / 132.075 x 10⁶ x 1.6 x 10^-19

λ = 9.3752 10^-15 m

Wavelength of each photon is 9.3752 x 10^-15 m