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Time measurements in relativity

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At standard Temperature and pressure sound travels at speed u=330m/s relative to the air through which it propagates.
Four students A,B,C and D position themselves as shown if figure 1.20 (see attachment), with A,B,C in a straight line and D vertically above B.
A steady wind is blowing with speed 30m/s along the line ABC.
If B fires a revolver what are the speed with which the sound will travel to A,C, and D. discuss whether the differences in your answers could be detected.

What must be one's speed relative to frame S in order that one's clock will lose 1 second per day as observed from S? What if teh are to lose one minute per day.

A group of Pi-mesons (pions) is observed traveling at speed 0.8c in the laboratory. What is the gamma factor for the pions? If the pions half life is 1.8E-8s what is their half life as observed in the lab frame?
If there are initially 32000 Pions, how many will be left after they have traveled 36m? What would be the answer to the previous question if you ignore time dilation?

Muons have proper half life of 1.5 microsecond. In a certain experiment a muon detector is placed in altitude of 2000m and registers within one hour 650 muons traveling at 0.99c towards earth.
If the detector is placed at sea level how many muons do you expect to register in one hour?

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The solution apply the relativistic concept of time dilation to the problems.

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