(a) The muon can decay into an electron and two neutrinos. The neutrinos\' mass is almost zero so we neglect it here. Let the muon to be at rest before it decays. Calculate the maximal and minimal total kinetic energy of those two neutrinos.

(b) About 10,000 muons reach the earth\'s surface per square meter per minute. Those muons are created by cosmic rays which hit the atmosphere. For Simplicity, let\'s say all the muons are created at a height 10km, and all the muons are then moving towards the earth at v = 0.98c. Calculate how many muons are created per square meter per minute at height 10km, using a frame at rest with the earth.

(c) Redo the calculation in (b), using a frame at rest with the muon (v = 0.98c towards the earth). Use the result to explain the relation between time dilation and length contraction.

The interaction between the muon and the atmosphere can be neglected.

The earth has a much greater radius than the thickness of the atmosphere. Thus when calculating the unit area, we don\'t have to consider the change of area because of height.

Answer these questions as shown above: 

The “moun” is an unstable particle. The mass of the moun is m _μ = 106 MeV/c ² , which is much greater than its cousin electron, whose mass is m _e = 0.511 MeV/c ² .

In it,s own rest frame, the average lifetime of moun is t0 = 2 × 10 ^-6 s. This means that if initially we have N0 mouns, afaer time t, the remaining number of mouns is

N = N ₀ e ^-t/t0 .

 

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