Question

1 a)

Using an example of muons, explain how both time dialation and length contraction are demonstarted by the detection of these cosmic particles on earth.

b) what is meant by the term blueshift and redshift and what do their observation in the spectral lines of a cosmic object imply?

Answer 1

1 a) Muons' half life in the rest frame is 1.54 microseconds. Despite this short half life, many more muons are detected on Earth than were supposed to be. This is a consequence of Relativistic effects at high velocities.

Take, v = 0.98c

So, the Lorentz factor is:

This means, the half-life of Muons in the lab frame will be:

T = 5.025(1.54) = 7.738 microseconds

This longer half-life due to time-dilation means that the muons will take more time to decay and are thus able to travel a longer distance to Earth before decaying.

Take L = 10000 m, to be distance from where the Flux of the muons is measured.

Likewise, in the Muons' rest frame, the distance will be Lorentz contracted to:

L = 10000/5.025 = 1989.97 m

Which must be covered with a speed of v = 0.98c

So, time taken in its own frame is:

T = 1989.97/0.98c = 6.768 microseconds.

Whereas in the lab frame, the time will be:

T' = 10000/0.98c = 34 microseconds.

Since T < T', more Muons will be able to reach the Earth than expected (non-relativistically).